{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-16T00:12:35.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-16T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":49943,"title":"Splitting Hexagon - Problem the third","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 402px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 201px; transform-origin: 407px 201px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 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data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThis hexagon is to be split into two regions (e.g., red and blue). Given the ratio between the two regions and the side of the hexagon, determine the radius of the circle that splits the region. The ratio between the regions (red to blue) is presented through the first two entries in the input. For example, if the ratio is 1 to 2, then these two numbers will be the first two entries in the input. The last entry is the side of the hexagon.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = ratio_polygon(s)\r\n  y = s;\r\nend","test_suite":"%%\r\ns=[1 2 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.5250;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[0 1 1];\r\ny=ratio_polygon(s);\r\ny_correct=0;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[1 7 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.3215;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[3 7 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.4981;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[4 1 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.8134;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[3 5 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.5569;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":5,"created_by":180632,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":33,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-01-22T22:23:48.000Z","updated_at":"2025-12-27T03:32:18.000Z","published_at":"2021-01-22T22:24:53.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider a hexagon sitting in Quadrant I as depicted in an example below:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"273\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"307\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis hexagon is to be split into two regions (e.g., red and blue). Given the ratio between the two regions and the side of the hexagon, determine the radius of the circle that splits the region. The ratio between the regions (red to blue) is presented through the first two entries in the input. For example, if the ratio is 1 to 2, then these two numbers will be the first two entries in the input. The last entry is the side of the 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the pentagon (with the minimum number of triangles)","description":"Problem statement\r\n\r\nAn pentagon is a regular polygon with 5 vertices and 5 edges. Here below is an example of the vertex set V, corresponding to the XYZ coordinates column vectors of a pentagon included in the unit circle.\r\n\r\nV = [1           0            0;\r\n     cos(2*pi/5) sin(2*pi/5)  0;\r\n     cos(4*pi/5) sin(4*pi/5)  0;\r\n     cos(4*pi/5) sin(-4*pi/5) 0;\r\n     cos(2*pi/5) sin(-2*pi/5) 0];\r\n\r\nA triangulated mesh T (stands for triangles here) -or a triangulation- is simply a N x 3 matrix of positive integers where each row contains the vertex indices of a triangle, and where N is the number of triangles. \r\n\r\nYour task here is to mesh this pentagon with the minimum possible number of triangles. To do so, you will list the pentagons/rows in a matrix of faces, F. The row order of the triangles in the list doesn't matter.\r\n\r\nExample\r\nThe first triangle here can be [1, 2, 3] if counterclockwise oriented.\r\n\r\n\r\n\r\n\r\nTip\r\nBeware to avoid self intersecting triangles.\r\n\r\nForbidden functions / expressions\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\nMesh processing\r\nMesh generation toolbox","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1278.4px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 639.2px; transform-origin: 408px 639.2px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 327.525px 8px; transform-origin: 327.525px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn pentagon is a regular polygon with 5 vertices and 5 edges. Here below is an example of the vertex set \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.09167px 8px; transform-origin: 6.09167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eV,\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.2417px 8px; transform-origin: 48.2417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e corresponding to the XYZ coordinates column vectors of a pentagon included in the unit circle.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 102.167px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 405px 51.0833px; transform-origin: 405px 51.0833px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 405px 10.2167px; text-wrap-mode: nowrap; transform-origin: 405px 10.2167px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 123.2px 8.5px; tab-size: 4; transform-origin: 123.2px 8.5px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eV = [1           0            0;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 405px 10.2167px; text-wrap-mode: nowrap; transform-origin: 405px 10.2167px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 123.2px 8.5px; tab-size: 4; transform-origin: 123.2px 8.5px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     cos(2*pi/5) sin(2*pi/5)  0;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 405px 10.2167px; text-wrap-mode: nowrap; transform-origin: 405px 10.2167px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 123.2px 8.5px; tab-size: 4; transform-origin: 123.2px 8.5px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     cos(4*pi/5) sin(4*pi/5)  0;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 405px 10.2167px; text-wrap-mode: nowrap; transform-origin: 405px 10.2167px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 123.2px 8.5px; tab-size: 4; transform-origin: 123.2px 8.5px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     cos(4*pi/5) sin(-4*pi/5) 0;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 405px 10.2167px; text-wrap-mode: nowrap; transform-origin: 405px 10.2167px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 127.05px 8.5px; tab-size: 4; transform-origin: 127.05px 8.5px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     cos(2*pi/5) sin(-2*pi/5) 0];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.1833px 8px; transform-origin: 64.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA triangulated mesh \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4.275px 8px; transform-origin: 4.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eT\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 176.983px 8px; transform-origin: 176.983px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (stands for triangles here) -or a triangulation- is simply a \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.05833px 8px; transform-origin: 5.05833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eN\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.633px 8px; transform-origin: 132.633px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e x 3 matrix of positive integers where each row contains the vertex indices of a triangle, and where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.05833px 8px; transform-origin: 5.05833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eN\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.4px 8px; transform-origin: 84.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the number of triangles. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384.442px 8px; transform-origin: 384.442px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eYour task here is to mesh this pentagon with the minimum possible number of triangles. To do so, you will list the pentagons/rows in a matrix of faces, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.25833px 8px; transform-origin: 7.25833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eF. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.575px 8px; transform-origin: 168.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe row order of the triangles in the list doesn't matter.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.7833px 8px; transform-origin: 28.7833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExample\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 92.9583px 8px; transform-origin: 92.9583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe first triangle here can be [\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.3833px 8px; transform-origin: 21.3833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e1, 2, 3]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 89.8583px 8px; transform-origin: 89.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e if counterclockwise oriented.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 339.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 169.75px; text-align: left; transform-origin: 385px 169.75px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"445\" height=\"334\" style=\"vertical-align: baseline;width: 445px;height: 334px\" 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AhAUx7lAlow8mJbOyhW3AARL6AL5GWr/DkyPxAUgZPs/pyb5+YgAaVG5AAB5ELpAHlQuAEDSrB0BmVG5AACps7cM5ETlAgDkROgCqVO5AAAZsLcMZEPlAgDkwd4ykAeVCwCQDaELZEDlAgBkSegCiVK5AAA5cUEXSJ3KBQDIjL1lIGkqFwAgP0IXSJfKBQDIm9AF0qJyAQCy5IIukCiVCwCQK3vLQIpULgAAQhdIh8oFAMiYvWUgOSoXACBv9paBtKhcAIDsCV0gISoXAIAOQheInMoFAMAFXSAdKhcAgFH2loEkqFwAAPYRukD8VC4AABMRukCcVC4AAB1c0AUip3IBABjL3jIQM5ULAEB3QheIjcoFAGAce8tAtFQuAAATsbcMxEnlAgDQhdAFIqRyAQDogdAFIqFyAQDozgVdIDYqFwCASdlbBqKicgEAmIrQBeKhcgEA6IfQBcKmcgEA6IELukAkVC4AAL2xtwzEQOUCANA/oQuESuUCANAze8tA8FQuAAD9sLcMhE3lAgDQJ6ELBEzlAgAwAKELBEblAgDQPxd0gVCpXAAApsXeMhAklQsAwHQJXSA8KhcAgDIIXSAMKhcAgAG4oAsERuUCADAYe8tASFQuAADlEbpA01QuAAADs7cMBEPlAgBQBnvLQBhULgAAJRG6QABULgAAFRC6QENULgAA5XFBF2iaygUAoFT2loFGqVwAAMomdIHmqFwAAKokdIF6qVwAACrggi7QEJULAEA17C0DTVC5AABUT+gCdVG5AABUxt4yUDuVCwBAlewtA/VSuQAAVEzoAjVSuQAA1EjoAhVTuQAAVM8FXaAuKhcAgFrYWwZqoXIBAKid0AUqo3IBAKhL596y0AWqoXIBAKiRC7pAxVQuAAD1ckEXqJLKBQCgOUIXKJvKBQCgdvaWgcqoXAAAmmBvGaiGygUAoCFCF6iAygUAIABCFyiJygUAoDku6AJlU7kAADTK3jJQKpULAEAwhC4wMJULAEDTOveWhS4wGJULAEAAXNAFSqJyAQAIgwu6QBlULgAA4RG6wHSpXAAAgmFvGRiYygUAICT2loHBqFwAAAIjdIEBqFwAAAImdIE+qVwAAMLjgi4wXSoXAIAg2VsGpkXlAgAQPKEL9EzlAgAQqs69ZaEL9EblAgAQMBd0gT6pXAAAwuaCLtAPlQsAQDyELjAVlQsAQPDsLQM9U7kAAMTA3jLQG5ULAEAkhC7QA5ULAECEhC7QhcoFACAeLugCU1G5AABExd4yMCmVCwBAtIQuMI7KBQAgNvaWge5ULgAAEbK3DHShcgEAiJPQBSaicgEAiJ/QBfZRuQAARMsFXWAclQsAQMzsLQNjqVwAACIndIEOKhcAgIQIXcieygUAIH4u6AL7qFwAAJJgbxkYpXIBAEiO0IWMqVwAAFJhbxlQuQAAJMXeMmRP5QIAkBahC3lTuQAApEvoQn5ULgAAyXFBFzKmcgEASJG9ZciVygUAIFFCF7KkcgEAyIDQhWyoXAAA0uWCLuRH5QIAkDR7y5AZlQsAQDaELmRA5QIAkDp7y5ATlQsAQAbsLUM2VC4AAHkQupAHlQsAQH6ELqRL5QIAkA0XdCEDKhcAgJzYW4bUqVwAADIjdCFpKhcAgIwJXUiOygUAID8u6EK6VC4AAFmytwyJUrkAAGRP6EJCVC4AALmytwwpUrkAAGTM3jIkR+UCAJA3oQtpUbkAALCP0IX4qVwAALLngi4kROUCAIC9ZUiHygUAgFFCF5KgcgEAYByhC9FSuQAAsI8LuhA/lQsAAB3sLUPkVC4AAHQhdCFCKhcAAMaytwwxU7kAADCOvWWIlsoFAICJCF2Ik8oFAICpCF2Ih8oFAIAuXNCFCKlcAADozt4yxEblAgDApIQuREXlAgBAz4QuBE/lAgDAVFzQhXioXAAA6IG9ZYiEygUAgD4JXQiYygUAgN7YW4YYqFwAAOiZvWUInsoFAIB+CF0Im8oFAIDpEroQHpULAAB9ckEXAqZyAQCgf/aWIVQqFwAApkXoQpBULgAADEzoQjBULgAATJcLuhAelQsAAAOwtwyBUbkAAFASoQsBULkAADAYe8sQEpULAAADs7cMwVC5AABQBqELYVC5AABQNqELzVG5AABQEhd0IQAqFwAAymNvGZqmcgEAoFRCFxqlcgEAoDJCF2qncgEAoGwu6EJzVC4AAFTA3jI0ROUCAEDFhC7USOUCAEA17C1DE1QuAABUxt4y1E7lAgBAlYQu1EvlAgBAXYQuVE/lAgBAxVzQhRqpXAAAqJ69ZaiLygUAgFoIXaiFygUAgNoJXaiMygUAgLq4oAvVU7kAAFAje8tQMZULAAANEbpQAZULAAD1srcMVVK5AABQO3vLUBmVCwAATRC6UA2VCwAATRO6UB6VCwAADXFBFyqgcgEAoDn2lqFsKhcAAMIgdKEMKhcAABrVubcsdGFgKhcAAJrmgi6UR+UCAEAAXNCFkqhcAAAIjNCFAahcAAAIg71lKIPKBQCAYNhbhoGpXAAACInQhcGoXAAACJXQhf6pXAAACIwLujAAlQsAAOGxtwzTpXIBACBsQhf6oXIBACBInXvLQhd6pnIBACBULuhC/1QuAAAEzAVd6JPKBQCASAhd6IHKBQCAsNlbhn6oXAAACJ69ZeiZygUAgBgIXeiNygUAgNgIXehO5QIAQCRc0IUeqFwAAIiHvWWYisoFAIA4CV2YiMoFAICodO4tC10YR+UCAEBsXNCF7lQuAABEyAVd6ELlAgBA5IQudFC5AAAQJ3vLMBGVCwAA0bK3DOOoXAAAiJnQhbFULgAApELogsoFAIDouaALHVQuAADEz94y7KNyAQAgLUKXvKlcAABIgr1lGKVyAQAgFfaWQeUCAEBSKg7d97///Xv27Nm+fXv7awiPygUAgESVHbojIyM33XTTjBkz2l9DkFQuAACkpZoLurNnz77rrruOP/749tcQKpULAADJKXtvef78+ffdd98ZZ5zR/hoCpnIBACBF5YXuihUr1q9ff8opp7S/hrCpXAAASN10Q/eEE05Yu3btTTfdNHfu3NaX11133X//+9/iWxAslQsAAIka+ILuqlWrTj311NbDSy+9dM4551xyySXFewiZygUAgHQNvLf82muvfetb3zruuONuvfXW9isIm8oFAIA89B+6v/nNbxYuXHjFFVfs2LGj/QqCp3IBACBpA+wtP/TQQ9u2bWt/AZFQuQAAkLqy/2EhCJnKBQCADAhdsqFyAQAgA1de2X44+OD2AyRK5QIAQOokLjlRuQAAkLT9iQt5ULkAAJAHB7nkQeUCAEC67CqTH5ULAACJsqtMllQuAACkzkEuOVG5AACQIrvK5ErlAgBAciQuGVO5AACQFtdxyZvKBQCARDnIJUsqFwAAElLxrvK8efMOPvjgww8/vP01hEflAgBAKlzHBZULAACJcB0XRqlcAABIi4Nc8qZyAQAgfnaVYR+VCwAAkbOrDB1ULgAApMJBLqhcAACIm11lGEvlAgBAtCQujKNyAQAgTq7jwkRULgAARM5BLnRQuQAAECG7ytCFygUAgNjYVYbuVC4AAESlM3Ed5MI4KhcAAOIkcWEiKhcAAOLhOi5MReUCAEAkXMeFHqhcAACIjYNc6E7lAgBADOwqQ29ULgAABE/iQs9ULgAAhM11XOiHygUAgEg4yIUeqFwAAAiYXWXok8oFAIBQ2VWG/qlcAAAIUmfiOsiFnqlcAAAIm8SFfqhcAAAIj+u4MF0qFwAAAuM6LgxA5QIAQKgc5EL/VC4AAITErjIMRuUCAEAwJC4MTOUCAEAYXMeFMqhcAAAIjINcGIDKBQCAANhVhpKoXAAAaJpdZSiPygUAgEZ1Jq6DXBiYygUAgDBIXCiDygUAgOa4jgtlU7kAANAQ13GhAioXAACa5iAXyqNyAQCgCXaVoRoqFwAAaidxoTIqFwAA6uU6LlRJ5QIAQEMc5EIFVC4AANTIrjJUTOUCAEBd7CpD9VQuAADUzkEuVEblAgBALewqQy1ULgAAVE/iQl1ULgAAVMx1XKiRygUAgLo4yIXqqVwAAKiSXWWol8oFAIDKSFyoncoFAIBquI4LTVC5AABQMQe5UCOVCwAAFbCrDA1RuQAAUDa7ytAclQsAAJVxkAu1U7kAAFAqu8rQKJULAADlkbjQNJULAAAlcR0XAqByAQCgbA5yoTkqFwAAymBXGcKgcgEAYGASF4KhcgEAYDCu40JIVC4AAJTEQS4EQOUCAMAA7CpDYFQuAABMl11lCI/KBQCAgTnIhWCoXAAAmBa7yhAklQsAAP2TuBAqlQsAAH1yHRcCpnIBAGC6HORCeFQuAAD0w64yhE3lAgBAzyQuBE/lAgBAb1zHhRioXAAA6JODXAiYygUAgB7YVYZIqFwAAJiKXWWIh8oFAICeOciF4KlcAACYlF1liIrKBQCA7iQuxEblAgBAF67jQoRULgAATMVBLsRD5QIAwETsKkOcVC4AAIwjcSFaKhcAAMZyHRdipnIBAKALB7kQIZULAAAd7CpD5FQuAADsY1cZ4qdyAQBgHAe5EC2VCwAAo+wqQxJULgAASFxIh8oFACB7ruNCQlQuAADs4yAX4qdyAQDIm11lSIvKBQAgYxIXkqNyAQDIleu4kCKVCwBA9hzkQkJULgAAWbKrDIlSuQAA5MeuMqRL5QIAkDEHuZAclQsAQGbsKkPSVC4AADmRuJA6lQsAQDZcx4UMqFwAAPLjIBfSpXIBAMiDXWXIg8oFACADEheyoXIBAEid67iQE5ULAEA2HORCBlQuAABJs6sMmVG5AACky64y5EflAgCQAQe5kA2VCwBAouwqQ5ZULgAAKZK4kCuVCwBAclzHhYypXAAA0uUgF/KjcgEASItdZcibygUAICESF7KncgEASIXruIDKBQAgQQ5yIWMqFwCAJNhVBkapXAAA4mdXGdhH5QIAkBAHuZA9lQsAQOTsKgMdVC4AADGTuMBYKhcA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data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 131.092px 8px; transform-origin: 131.092px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBeware to avoid self intersecting triangles.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/57483\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eMesh processing\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/fileexchange/85173-mesh-generation-toolbox?s_tid=prof_contriblnk\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eMesh generation toolbox\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function T = mesh_the_pentagon()\r\n  T = 1;\r\nend","test_suite":"%%\r\nT_correct1 = [1 2 3;\r\n              1 3 4;\r\n              1 4 5];\r\n\r\nT_correct2 = [2 3 4;\r\n              2 4 5;\r\n              2 5 1];\r\n\r\nT_correct3 = [3 4 5;\r\n              3 5 1;\r\n              3 1 2];\r\n\r\nT_correct4 = [3 4 5;\r\n              3 5 1;\r\n              3 1 2];\r\n\r\nT_correct5 = [5 1 2;\r\n              5 2 3;\r\n              5 3 4];\r\n\r\n% Check every possible solutions\r\nassert(isequal(sortrows(sort(mesh_the_pentagon(),2)),sortrows(sort(T_correct1,2)))...\r\n     | isequal(sortrows(sort(mesh_the_pentagon(),2)),sortrows(sort(T_correct2,2)))...\r\n     | isequal(sortrows(sort(mesh_the_pentagon(),2)),sortrows(sort(T_correct3,2)))...\r\n     | isequal(sortrows(sort(mesh_the_pentagon(),2)),sortrows(sort(T_correct4,2)))...\r\n     | isequal(sortrows(sort(mesh_the_pentagon(),2)),sortrows(sort(T_correct5,2))))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('mesh_the_pentagon.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":1,"comments_count":6,"created_by":149128,"edited_by":149128,"edited_at":"2025-08-13T05:29:08.000Z","deleted_by":null,"deleted_at":null,"solvers_count":31,"test_suite_updated_at":"2025-08-13T05:29:08.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-23T12:59:46.000Z","updated_at":"2026-02-10T17:07:57.000Z","published_at":"2025-07-23T15:54:36.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn pentagon is a regular polygon with 5 vertices and 5 edges. Here below is an example of the vertex set \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eV,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e corresponding to the XYZ coordinates column vectors of a pentagon included in the unit circle.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[V = [1           0            0;\\n     cos(2*pi/5) sin(2*pi/5)  0;\\n     cos(4*pi/5) sin(4*pi/5)  0;\\n     cos(4*pi/5) sin(-4*pi/5) 0;\\n     cos(2*pi/5) sin(-2*pi/5) 0];]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA triangulated mesh \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eT\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (stands for triangles here) -or a triangulation- is simply a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e x 3 matrix of positive integers where each row contains the vertex indices of a triangle, and where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the number of triangles. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour task here is to mesh this pentagon with the minimum possible number of triangles. To do so, you will list the pentagons/rows in a matrix of faces, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eF. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eThe row order of the triangles in the list doesn't matter.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe first triangle here can be [\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1, 2, 3]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e if counterclockwise oriented.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"334\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"445\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTip\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBeware to avoid self intersecting triangles.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/57483\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eMesh 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margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eConsider a hexagon sitting in Quadrant I as depicted in an example below:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 315px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 157.5px; text-align: left; transform-origin: 384px 157.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"297\" 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data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThis hexagon is to be split into two regions (e.g., red and blue). Given the ratio of the two regions and the side of the hexagon, determine x coordinate of the line that splits the regions. The ratio between the regions (red to blue) is presented through the first two entries in the input. For example, if the ratio is 2 to 1, then these two numbers will be the first two numbers in the input. The last entry is the side of the hexagon.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = ratio_polygon(s)\r\n  y = s;\r\nend","test_suite":"%%\r\ns=[1 1 1];\r\ny=ratio_polygon(s)\r\ny_correct=0.8660;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[1 0 1];\r\ny=ratio_polygon(s);\r\ny_correct=sqrt(3);\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[1 3 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.5032;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[4 3 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.9613;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[7 1 1];\r\ny=ratio_polygon(s);\r\ny_correct=1.4523;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[1 7 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.2795;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":180632,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":25,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-01-22T01:27:38.000Z","updated_at":"2025-06-25T14:22:53.000Z","published_at":"2021-01-22T19:33:09.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider a hexagon sitting in Quadrant I as depicted in an example below:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"309\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"297\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis hexagon is to be split into two regions (e.g., red and blue). Given the ratio of the two regions and the side of the hexagon, determine x coordinate of the line that splits the regions. The ratio between the regions (red to blue) is presented through the first two entries in the input. For example, if the ratio is 2 to 1, then these two numbers will be the first two numbers in the input. 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Hexagon - Problem the second","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 415px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 207.5px; transform-origin: 407px 207.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eConsider a hexagon sitting in Quadrant I as depicted in an example below:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 292px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 146px; text-align: left; transform-origin: 384px 146px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" 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data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThis hexagon is to be split into two regions (e.g., red and blue). Given the ratio between the two regions and the side of the hexagon, determine x coordinate of the line that splits the regions. The ratio between the regions (red to blue) is presented through the first two entries in the input. For example, if the ratio is 3 to 7, then these two numbers will be the first two numbers in the input. The last entry is the side of the hexagon.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = ratio_polygon(s)\r\n  y = s;\r\nend","test_suite":"%%\r\ns=[0 1 1];\r\ny=ratio_polygon(s);\r\ny_correct=0;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[1 1 1];\r\ny=ratio_polygon(s);\r\ny_correct=1;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[2 3 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.85;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[4 3 1];\r\ny=ratio_polygon(s);\r\ny_correct=1.1071;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[5 1 1];\r\ny=ratio_polygon(s)\r\ny_correct=1.5;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[1 0 1];\r\ny=ratio_polygon(s)\r\ny_correct=2;\r\nassert(abs(y-y_correct)\u003c5e-4)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":180632,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":26,"test_suite_updated_at":"2021-01-22T21:58:02.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-01-22T01:31:35.000Z","updated_at":"2025-10-02T21:01:31.000Z","published_at":"2021-01-22T21:58:02.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider a hexagon sitting in Quadrant I as depicted in an example below:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis hexagon is to be split into two regions (e.g., red and blue). Given the ratio between the two regions and the side of the hexagon, determine x coordinate of the line that splits the regions. The ratio between the regions (red to blue) is presented through the first two entries in the input. For example, if the ratio is 3 to 7, then these two numbers will be the first two numbers in the input. 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Tiling Dots in a Circle","description":"Return how many \u003chttp://en.wikipedia.org/wiki/Triangular_tiling Triangular Tiling\u003e grid points there are inside a circle of radius _r_ centred at (0,0) (including points on the edge).  \r\n\r\nAssume that a Triangular Tiling grid is a \u003chttp://en.wikipedia.org/wiki/Bravais_lattice 2D Hexagonal Bravais lattice\u003e with | _a1_ | = | _a2_ | = 1 and _\u0026phi;_ = 120\u0026deg;.\r\n\r\nNeither *string operations* nor *interpolations* are allowed!","description_html":"\u003cp\u003eReturn how many \u003ca href = \"http://en.wikipedia.org/wiki/Triangular_tiling\"\u003eTriangular Tiling\u003c/a\u003e grid points there are inside a circle of radius \u003ci\u003er\u003c/i\u003e centred at (0,0) (including points on the edge).\u003c/p\u003e\u003cp\u003eAssume that a Triangular Tiling grid is a \u003ca href = \"http://en.wikipedia.org/wiki/Bravais_lattice\"\u003e2D Hexagonal Bravais lattice\u003c/a\u003e with | \u003ci\u003ea1\u003c/i\u003e | = | \u003ci\u003ea2\u003c/i\u003e | = 1 and \u003ci\u003e\u0026phi;\u003c/i\u003e = 120\u0026deg;.\u003c/p\u003e\u003cp\u003eNeither \u003cb\u003estring operations\u003c/b\u003e nor \u003cb\u003einterpolations\u003c/b\u003e are allowed!\u003c/p\u003e","function_template":"function n = hexagonal_dots_in_circle(r)\r\n  n = r;\r\nend","test_suite":"%%\r\nuser_solution = fileread('hexagonal_dots_in_circle.m');\r\nassert(isempty(strfind(user_solution,'regexp')));\r\nassert(isempty(strfind(user_solution,'2str')));\r\nassert(isempty(strfind(user_solution,'str2')));\r\nassert(isempty(strfind(user_solution,'interp')));\r\nassert(isempty(strfind(user_solution,'printf')));\r\nassert(isempty(strfind(user_solution,'assert')));\r\n\r\n%%\r\nr = 0;\r\nn_correct = 1;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 0.5;\r\nn_correct = 1;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 1;\r\nn_correct = 7;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 1.5;\r\nn_correct = 7;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 2;\r\nn_correct = 19;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 2.5;\r\nn_correct = 19;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 3;\r\nn_correct = 37;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 5;\r\nn_correct = 91;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 7.5;\r\nn_correct = 199;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 10;\r\nn_correct = 367;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 15;\r\nn_correct = 823;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 20;\r\nn_correct = 1459;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 25;\r\nn_correct = 2263;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 50;\r\nn_correct = 9061;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 100;\r\nn_correct = 36295;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n ","published":true,"deleted":false,"likes_count":0,"comments_count":1,"created_by":10352,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":32,"test_suite_updated_at":"2013-05-05T10:49:55.000Z","rescore_all_solutions":false,"group_id":20,"created_at":"2013-04-25T18:57:26.000Z","updated_at":"2026-02-16T10:52:57.000Z","published_at":"2013-04-25T18:57:26.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eReturn how many\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Triangular_tiling\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eTriangular Tiling\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e grid points there are inside a circle of radius\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e centred at (0,0) (including points on the edge).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAssume that a Triangular Tiling grid is a\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Bravais_lattice\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e2D Hexagonal Bravais lattice\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e with |\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ea1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e | = |\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ea2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e | = 1 and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eφ\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e = 120°.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNeither\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estring operations\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e nor\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003einterpolations\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e are allowed!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60925,"title":"Intersect three sequences","description":"Most numbers have interesting properties, if you look hard enough and interpret “interesting” liberally. Let’s choose a number at random—300, say—and list some of its properties:\r\nIt is divisible by the square of its largest prime factor. \r\nIt is the area of a triangle with integer sides and integer area\r\nIt is a folding point of the non-negative integers written in a hexagonal spiral (see below), as are 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, etc. \r\nA number that shares these properties is 108: (a) It is divisible by 9, or the square of its largest prime factor (3), (b) it is the area of a triangle with sides 15, 15, and 18, and (c) it is a folding point of the hexagon (it would be to the right of 75 in the hexagon below). \r\nWrite a function to determine whether a number has the three properties listed above. \r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 677.233px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 338.617px; transform-origin: 408px 338.617px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 361.508px 8px; transform-origin: 361.508px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eMost numbers have interesting properties, if you look hard enough and interpret “interesting” liberally. Let’s choose a number at random—300, say—and list some of its properties:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 164.125px 8px; transform-origin: 164.125px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIt is divisible by the square of its largest prime factor. \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 185.933px 8px; transform-origin: 185.933px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIt is the area of a triangle with integer sides and integer area\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 363.283px 8px; transform-origin: 363.283px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIt is a folding point of the non-negative integers written in a hexagonal spiral (see below), as are 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, etc. \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 379.592px 8px; transform-origin: 379.592px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA number that shares these properties is 108: (a) It is divisible by 9, or the square of its largest prime factor (3), (b) it is the area of a triangle with sides 15, 15, and 18, and (c) it is a folding point of the hexagon (it would be to the right of 75 in the hexagon below). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 266.708px 8px; transform-origin: 266.708px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to determine whether a number has the three properties listed above. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 421.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 210.75px; text-align: left; transform-origin: 385px 210.75px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"487\" height=\"416\" style=\"vertical-align: baseline;width: 487px;height: 416px\" 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data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function tf = isX3Seq(x)\r\n  tf = false;\r\nend","test_suite":"%%\r\nassert(isX3Seq(108))\r\n\r\n%%\r\nassert(~isX3Seq(144))\r\n\r\n%%\r\nassert(~isX3Seq(200))\r\n\r\n%%\r\nassert(~isX3Seq(256))\r\n\r\n%%\r\nassert(isX3Seq(300))\r\n\r\n%%\r\nassert(isX3Seq(432))\r\n\r\n%%\r\nassert(~isX3Seq(500))\r\n\r\n%%\r\nassert(~isX3Seq(648))\r\n\r\n%%\r\nassert(~isX3Seq(660))\r\n\r\n%%\r\nassert(~isX3Seq(768))\r\n\r\n%%\r\nassert(isX3Seq(1200))\r\n\r\n%%\r\nassert(~isX3Seq(1553))\r\n\r\n%%\r\nassert(isX3Seq(1728))\r\n\r\n%%\r\nassert(~isX3Seq(1875))\r\n\r\n%%\r\nassert(~isX3Seq(3072))\r\n\r\n%%\r\nassert(~isX3Seq(3924))\r\n\r\n%%\r\nassert(isX3Seq(4332))\r\n\r\n%%\r\nassert(~isX3Seq(6084))\r\n\r\n%%\r\nassert(~isX3Seq(8118))\r\n\r\n%%\r\nassert(~isX3Seq(10806))\r\n\r\n%%\r\nassert(~isX3Seq(14283))\r\n\r\n%%\r\nassert(isX3Seq(18252))\r\n\r\n%%\r\nassert(~isX3Seq(26010))\r\n\r\n%%\r\nassert(~isX3Seq(31236))\r\n\r\n%%\r\nassert(~isX3Seq(42336))\r\n\r\n%%\r\nassert(~isX3Seq(49152))\r\n\r\n%%\r\nassert(isX3Seq(65712))\r\n\r\n%%\r\nassert(isX3Seq(99372))\r\n\r\n%%\r\nassert(~isX3Seq(116592))\r\n\r\n%%\r\nassert(~isX3Seq(140000))\r\n\r\n%%\r\nassert(~isX3Seq(152172))\r\n\r\n%%\r\nassert(~isX3Seq(160314))\r\n\r\n%%\r\nassert(~isX3Seq(170008))\r\n\r\n%%\r\nassert(isX3Seq(212268))\r\n\r\n%%\r\nassert(isX3Seq(248832))\r\n\r\n%%\r\nassert(isX3Seq(280908))\r\n\r\n%%\r\nassert(~isX3Seq(296274))\r\n\r\n%%\r\nassert(isX3Seq(303372))\r\n\r\n%%\r\nassert(isX3Seq(314928))\r\n\r\n%%\r\nassert(~isX3Seq(340707))\r\n\r\n%%\r\nassert(isX3Seq(489648))\r\n\r\n%%\r\nassert(~isX3Seq(3145728))\r\n\r\n%%\r\nassert(isX3Seq(7114800))\r\n\r\n%%\r\nassert(~isX3Seq(12582912))\r\n\r\n%%\r\nassert(isX3Seq(14865228))","published":true,"deleted":false,"likes_count":0,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2025-06-02T16:45:27.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2025-06-02T03:01:11.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2025-05-27T13:26:30.000Z","updated_at":"2025-06-06T14:09:48.000Z","published_at":"2025-05-27T13:26:30.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMost numbers have interesting properties, if you look hard enough and interpret “interesting” liberally. Let’s choose a number at random—300, say—and list some of its properties:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt is divisible by the square of its largest prime factor. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt is the area of a triangle with integer sides and integer area\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt is a folding point of the non-negative integers written in a hexagonal spiral (see below), as are 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, etc. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA number that shares these properties is 108: (a) It is divisible by 9, or the square of its largest prime factor (3), (b) it is the area of a triangle with sides 15, 15, and 18, and (c) it is a folding point of the hexagon (it would be to the right of 75 in the hexagon below). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to determine whether a number has the three properties listed above. 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\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":1490,"title":"Shifted Hexagonal Tiling Dots in a Circle","description":"Return how many \u003chttp://en.wikipedia.org/wiki/Hexagonal_grid Hexagonal Tiling\u003e grid points there are inside a circle of radius _r_ centred at (0,0) (including points on the edge).  Assume that a Hexagonal Tiling grid is a \u003chttp://en.wikipedia.org/wiki/Regular_tiling 2D Regular Hexagonal Tessellation\u003e with equal edges of size _e_=1.  \r\n\r\nFor _shifted_ symmetry purposes, assume that (0,0) is a _grid point_.\r\n\r\nNeither *string operations* nor *interpolations* are allowed!","description_html":"\u003cp\u003eReturn how many \u003ca href = \"http://en.wikipedia.org/wiki/Hexagonal_grid\"\u003eHexagonal Tiling\u003c/a\u003e grid points there are inside a circle of radius \u003ci\u003er\u003c/i\u003e centred at (0,0) (including points on the edge).  Assume that a Hexagonal Tiling grid is a \u003ca href = \"http://en.wikipedia.org/wiki/Regular_tiling\"\u003e2D Regular Hexagonal Tessellation\u003c/a\u003e with equal edges of size \u003ci\u003ee\u003c/i\u003e=1.\u003c/p\u003e\u003cp\u003eFor \u003ci\u003eshifted\u003c/i\u003e symmetry purposes, assume that (0,0) is a \u003ci\u003egrid point\u003c/i\u003e.\u003c/p\u003e\u003cp\u003eNeither \u003cb\u003estring operations\u003c/b\u003e nor \u003cb\u003einterpolations\u003c/b\u003e are allowed!\u003c/p\u003e","function_template":"function n = shifted_hexagonal_tiling_dots_in_circle(r)\r\n  n = r;\r\nend","test_suite":"%%\r\nuser_solution = fileread('shifted_hexagonal_tiling_dots_in_circle.m');\r\nassert(isempty(strfind(user_solution,'regexp')));\r\nassert(isempty(strfind(user_solution,'2str')));\r\nassert(isempty(strfind(user_solution,'str2')));\r\nassert(isempty(strfind(user_solution,'interp')));\r\nassert(isempty(strfind(user_solution,'printf')));\r\nassert(isempty(strfind(user_solution,'assert')));\r\n\r\n%%\r\nr = 0;\r\nN_correct = 1;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 0.5;\r\nN_correct = 1;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 1;\r\nN_correct = 4;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 1.5;\r\nN_correct = 4;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 2;\r\nN_correct = 13;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 2.5;\r\nN_correct = 13;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 3;\r\nN_correct = 25;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 5;\r\nN_correct = 61;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 7.5;\r\nN_correct = 130;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 10;\r\nN_correct = 244;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 15;\r\nN_correct = 547;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 20;\r\nN_correct = 979;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 25;\r\nN_correct = 1510;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 50;\r\nN_correct = 6049;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 100;\r\nN_correct = 24202;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":10352,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":25,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":20,"created_at":"2013-05-05T11:12:35.000Z","updated_at":"2026-02-16T10:46:10.000Z","published_at":"2013-05-05T11:13:56.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eReturn how many\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Hexagonal_grid\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eHexagonal Tiling\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e grid points there are inside a circle of radius\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e centred at (0,0) (including points on the edge). Assume that a Hexagonal Tiling grid is a\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Regular_tiling\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e2D Regular Hexagonal Tessellation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e with equal edges of size\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ee\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=1.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eshifted\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e symmetry purposes, assume that (0,0) is a\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egrid point\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNeither\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estring operations\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e nor\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003einterpolations\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e are allowed!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1489,"title":"Hexagonal Tiling Dots in a Circle","description":"Return how many \u003chttp://en.wikipedia.org/wiki/Hexagonal_grid Hexagonal Tiling\u003e grid points there are inside a circle of radius _r_ centred at (0,0) (including points on the edge).  Assume that a Hexagonal Tiling grid is a \u003chttp://en.wikipedia.org/wiki/Regular_tiling 2D Regular Hexagonal Tessellation\u003e with equal edges of size _e_=1.  \r\n\r\nFor symmetry purposes, assume that (0,0) point is a _vacancy_; i.e., there _are_ points at (\u0026plusmn;1,0), (\u0026plusmn;1/2,\u0026plusmn;\u0026radic;3/2), etcetera.\r\n\r\nNeither *string operations* nor *interpolations* are allowed!","description_html":"\u003cp\u003eReturn how many \u003ca href = \"http://en.wikipedia.org/wiki/Hexagonal_grid\"\u003eHexagonal Tiling\u003c/a\u003e grid points there are inside a circle of radius \u003ci\u003er\u003c/i\u003e centred at (0,0) (including points on the edge).  Assume that a Hexagonal Tiling grid is a \u003ca href = \"http://en.wikipedia.org/wiki/Regular_tiling\"\u003e2D Regular Hexagonal Tessellation\u003c/a\u003e with equal edges of size \u003ci\u003ee\u003c/i\u003e=1.\u003c/p\u003e\u003cp\u003eFor symmetry purposes, assume that (0,0) point is a \u003ci\u003evacancy\u003c/i\u003e; i.e., there \u003ci\u003eare\u003c/i\u003e points at (\u0026plusmn;1,0), (\u0026plusmn;1/2,\u0026plusmn;\u0026radic;3/2), etcetera.\u003c/p\u003e\u003cp\u003eNeither \u003cb\u003estring operations\u003c/b\u003e nor \u003cb\u003einterpolations\u003c/b\u003e are allowed!\u003c/p\u003e","function_template":"function n = hexagonal_tiling_dots_in_circle(r)\r\n  n = r;\r\nend","test_suite":"%%\r\nuser_solution = fileread('hexagonal_tiling_dots_in_circle.m');\r\nassert(isempty(strfind(user_solution,'regexp')));\r\nassert(isempty(strfind(user_solution,'2str')));\r\nassert(isempty(strfind(user_solution,'str2')));\r\nassert(isempty(strfind(user_solution,'interp')));\r\nassert(isempty(strfind(user_solution,'printf')));\r\nassert(isempty(strfind(user_solution,'assert')));\r\n\r\n%%\r\nr = 0;\r\nN_correct = 0;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 0.5;\r\nN_correct = 0;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 1;\r\nN_correct = 6;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 1.5;\r\nN_correct = 6;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 2;\r\nN_correct = 12;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 2.5;\r\nN_correct = 12;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 3;\r\nN_correct = 24;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 5;\r\nN_correct = 60;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 7.5;\r\nN_correct = 138;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 10;\r\nN_correct = 246;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 15;\r\nN_correct = 552;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 20;\r\nN_correct = 960;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 25;\r\nN_correct = 1506;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 50;\r\nN_correct = 6024;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 100;\r\nN_correct = 24186;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":1,"created_by":10352,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":29,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":20,"created_at":"2013-05-05T10:39:46.000Z","updated_at":"2026-03-25T00:01:03.000Z","published_at":"2013-05-05T10:54:39.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eReturn how many\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Hexagonal_grid\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eHexagonal Tiling\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e grid points there are inside a circle of radius\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e centred at (0,0) (including points on the edge). Assume that a Hexagonal Tiling grid is a\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Regular_tiling\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e2D Regular Hexagonal Tessellation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e with equal edges of size\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ee\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=1.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor symmetry purposes, assume that (0,0) point is a\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003evacancy\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e; i.e., there\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eare\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e points at (±1,0), (±1/2,±√3/2), etcetera.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNeither\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estring operations\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e nor\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003einterpolations\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e are allowed!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60227,"title":"Rotate matrix 60 degrees","description":"Given a 2*m-1 by 4*m-3 matrix in which certain elements form a hexagon of side length m, rotate the hexagon 60 degrees counterclockwise, as shown below for m=3:\r\nA=[\r\n'--A-A-A--'\r\n'-B-B-B-B-'\r\n'C-C-C-C-C'\r\n'-D-D-D-D-'\r\n'--E-E-E--']\r\n\r\nrot60(A)=[\r\n'--A-B-C--'\r\n'-A-B-C-D-'\r\n'A-B-C-D-E'\r\n'-B-C-D-E-'\r\n'--C-D-E--']\r\nA=rot60(A,n) should rotate the hexagon 60*n degrees counterclockwise (and thus clockwise if n\u003c0).","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 348.75px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 331.989px 174.375px; transform-origin: 331.996px 174.375px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42.017px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 308.991px 21.0085px; text-align: left; transform-origin: 308.999px 21.0085px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven a 2*m-1 by 4*m-3 matrix in which certain elements form a hexagon of side length m, rotate the hexagon 60 degrees counterclockwise, as shown below for m=3:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 265.724px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 328.991px 132.855px; transform-origin: 328.999px 132.862px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eA=[\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'--A-A-A--'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'-B-B-B-B-'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'C-C-C-C-C'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'-D-D-D-D-'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'--E-E-E--'\u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003erot60(A)=[\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'--A-B-C--'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'-A-B-C-D-'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'A-B-C-D-E'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'-B-C-D-E-'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'--C-D-E--'\u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.0085px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 308.991px 10.4972px; text-align: left; transform-origin: 308.999px 10.5043px; white-space-collapse: preserve; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA=rot60(A,n) should rotate the hexagon 60*n degrees counterclockwise (and thus clockwise if n\u0026lt;0).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function A=rot60(A,n)\r\nA=rot90(A,2*n/3);\r\n","test_suite":"%%\r\nA=['--A-A-A--'\r\n   '-B-B-B-B-'\r\n   'C-C-C-C-C'\r\n   '-D-D-D-D-'\r\n   '--E-E-E--'];\r\nB=['--A-B-C--'\r\n   '-A-B-C-D-'\r\n   'A-B-C-D-E'\r\n   '-B-C-D-E-'\r\n   '--C-D-E--'];\r\nassert(isequal(rot60(A),B))\r\n%%\r\nA=['  A A A  '\r\n   ' B B B B '\r\n   'C C C C C'\r\n   ' D D D D '\r\n   '  E E E  '];\r\nB=['  C B A  '\r\n   ' D C B A '\r\n   'E D C B A'\r\n   ' E D C B '\r\n   '  E D C  '];\r\nassert(isequal(rot60(A,-1),B))\r\n%%\r\nA=['   3 1 4 1   '\r\n   '  5 9 2 6 5  '\r\n   ' 3 5 8 9 7 9 '\r\n   '3 2 3 8 4 6 2'\r\n   ' 6 4 3 3 8 3 '\r\n   '  2 7 9 5 0  '\r\n   '   2 8 8 4   '];\r\nB=['   1 5 9 2   '\r\n   '  4 6 7 6 3  '\r\n   ' 1 2 9 4 8 0 '\r\n   '3 9 8 8 3 5 4'\r\n   ' 5 5 3 3 9 8 '\r\n   '  3 2 4 7 8  '\r\n   '   3 6 2 2   '];\r\nassert(isequal(rot60(A),B))\r\n%%\r\nA=['    A B C D E    '\r\n   '   F G H I J K   '\r\n   '  L M N O P Q R  '\r\n   ' S T U V W X Y Z '\r\n   '1 2 3 4 5 6 7 8 9'\r\n   ' a b c d e f g h '\r\n   '  i j k l m n o  '\r\n   '   p q r s t u   '\r\n   '    v w x y z    '];\r\nB=['    E K R Z 9    '\r\n   '   D J Q Y 8 h   '\r\n   '  C I P X 7 g o  '\r\n   ' B H O W 6 f n u '\r\n   'A G N V 5 e m t z'\r\n   ' F M U 4 d l s y '\r\n   '  L T 3 c k r x  '\r\n   '   S 2 b j q w   '\r\n   '    1 a i p v    '];\r\nassert(isequal(rot60(A),B))\r\nn=randi([-100 100],1);\r\nassert(isequal(rot60(A,3*n),rot90(A,2*n)))\r\n%%\r\nA=[nan nan 120 nan  90 nan  60 nan nan\r\n   nan 150 nan 120 nan  60 nan  30 nan\r\n   180 nan 180 nan nan nan   0 nan   0\r\n   nan 210 nan 240 nan 300 nan 330 nan\r\n   nan nan 240 nan 270 nan 300 nan nan];\r\nfor n=1:6\r\n   assert(isequaln(rot60(A,n),mod(A-60*n,360)))\r\nend\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":245,"edited_by":245,"edited_at":"2024-05-10T20:49:04.000Z","deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-05-08T01:12:38.000Z","updated_at":"2026-04-16T10:54:35.000Z","published_at":"2024-05-10T20:49:04.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a 2*m-1 by 4*m-3 matrix in which certain elements form a hexagon of side length m, rotate the hexagon 60 degrees counterclockwise, as shown below for m=3:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[A=[\\n'--A-A-A--'\\n'-B-B-B-B-'\\n'C-C-C-C-C'\\n'-D-D-D-D-'\\n'--E-E-E--']\\n\\nrot60(A)=[\\n'--A-B-C--'\\n'-A-B-C-D-'\\n'A-B-C-D-E'\\n'-B-C-D-E-'\\n'--C-D-E--']]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA=rot60(A,n) should rotate the hexagon 60*n degrees counterclockwise (and thus clockwise if n\u0026lt;0).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":49943,"title":"Splitting Hexagon - Problem the third","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 402px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 201px; transform-origin: 407px 201px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eConsider a hexagon sitting in Quadrant I as depicted in an example below:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 279px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; 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data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThis hexagon is to be split into two regions (e.g., red and blue). Given the ratio between the two regions and the side of the hexagon, determine the radius of the circle that splits the region. The ratio between the regions (red to blue) is presented through the first two entries in the input. For example, if the ratio is 1 to 2, then these two numbers will be the first two entries in the input. The last entry is the side of the hexagon.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = ratio_polygon(s)\r\n  y = s;\r\nend","test_suite":"%%\r\ns=[1 2 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.5250;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[0 1 1];\r\ny=ratio_polygon(s);\r\ny_correct=0;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[1 7 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.3215;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[3 7 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.4981;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[4 1 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.8134;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[3 5 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.5569;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":5,"created_by":180632,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":33,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-01-22T22:23:48.000Z","updated_at":"2025-12-27T03:32:18.000Z","published_at":"2021-01-22T22:24:53.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider a hexagon sitting in Quadrant I as depicted in an example below:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"273\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"307\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis hexagon is to be split into two regions (e.g., red and blue). Given the ratio between the two regions and the side of the hexagon, determine the radius of the circle that splits the region. The ratio between the regions (red to blue) is presented through the first two entries in the input. For example, if the ratio is 1 to 2, then these two numbers will be the first two entries in the input. The last entry is the side of the 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the pentagon (with the minimum number of triangles)","description":"Problem statement\r\n\r\nAn pentagon is a regular polygon with 5 vertices and 5 edges. Here below is an example of the vertex set V, corresponding to the XYZ coordinates column vectors of a pentagon included in the unit circle.\r\n\r\nV = [1           0            0;\r\n     cos(2*pi/5) sin(2*pi/5)  0;\r\n     cos(4*pi/5) sin(4*pi/5)  0;\r\n     cos(4*pi/5) sin(-4*pi/5) 0;\r\n     cos(2*pi/5) sin(-2*pi/5) 0];\r\n\r\nA triangulated mesh T (stands for triangles here) -or a triangulation- is simply a N x 3 matrix of positive integers where each row contains the vertex indices of a triangle, and where N is the number of triangles. \r\n\r\nYour task here is to mesh this pentagon with the minimum possible number of triangles. To do so, you will list the pentagons/rows in a matrix of faces, F. The row order of the triangles in the list doesn't matter.\r\n\r\nExample\r\nThe first triangle here can be [1, 2, 3] if counterclockwise oriented.\r\n\r\n\r\n\r\n\r\nTip\r\nBeware to avoid self intersecting triangles.\r\n\r\nForbidden functions / expressions\r\nregexp\r\nassignin\r\nstr2num\r\necho\r\n\r\nSee also\r\nMesh processing\r\nMesh generation toolbox","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1278.4px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 639.2px; transform-origin: 408px 639.2px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 327.525px 8px; transform-origin: 327.525px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn pentagon is a regular polygon with 5 vertices and 5 edges. Here below is an example of the vertex set \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.09167px 8px; transform-origin: 6.09167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eV,\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.2417px 8px; transform-origin: 48.2417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e corresponding to the XYZ coordinates column vectors of a pentagon included in the unit circle.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 102.167px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 405px 51.0833px; transform-origin: 405px 51.0833px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 405px 10.2167px; text-wrap-mode: nowrap; transform-origin: 405px 10.2167px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 123.2px 8.5px; tab-size: 4; transform-origin: 123.2px 8.5px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eV = [1           0            0;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 405px 10.2167px; text-wrap-mode: nowrap; transform-origin: 405px 10.2167px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 123.2px 8.5px; tab-size: 4; transform-origin: 123.2px 8.5px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     cos(2*pi/5) sin(2*pi/5)  0;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 405px 10.2167px; text-wrap-mode: nowrap; transform-origin: 405px 10.2167px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 123.2px 8.5px; tab-size: 4; transform-origin: 123.2px 8.5px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     cos(4*pi/5) sin(4*pi/5)  0;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 405px 10.2167px; text-wrap-mode: nowrap; transform-origin: 405px 10.2167px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 123.2px 8.5px; tab-size: 4; transform-origin: 123.2px 8.5px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     cos(4*pi/5) sin(-4*pi/5) 0;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 405px 10.2167px; text-wrap-mode: nowrap; transform-origin: 405px 10.2167px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 127.05px 8.5px; tab-size: 4; transform-origin: 127.05px 8.5px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     cos(2*pi/5) sin(-2*pi/5) 0];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.1833px 8px; transform-origin: 64.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA triangulated mesh \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4.275px 8px; transform-origin: 4.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eT\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 176.983px 8px; transform-origin: 176.983px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (stands for triangles here) -or a triangulation- is simply a \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.05833px 8px; transform-origin: 5.05833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eN\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.633px 8px; transform-origin: 132.633px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e x 3 matrix of positive integers where each row contains the vertex indices of a triangle, and where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.05833px 8px; transform-origin: 5.05833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eN\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.4px 8px; transform-origin: 84.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the number of triangles. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384.442px 8px; transform-origin: 384.442px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eYour task here is to mesh this pentagon with the minimum possible number of triangles. To do so, you will list the pentagons/rows in a matrix of faces, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.25833px 8px; transform-origin: 7.25833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eF. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.575px 8px; transform-origin: 168.575px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe row order of the triangles in the list doesn't matter.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.7833px 8px; transform-origin: 28.7833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExample\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 92.9583px 8px; transform-origin: 92.9583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe first triangle here can be [\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.3833px 8px; transform-origin: 21.3833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e1, 2, 3]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 89.8583px 8px; transform-origin: 89.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e if counterclockwise oriented.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 339.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 169.75px; text-align: left; transform-origin: 385px 169.75px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"445\" height=\"334\" style=\"vertical-align: baseline;width: 445px;height: 334px\" 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data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 131.092px 8px; transform-origin: 131.092px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBeware to avoid self intersecting triangles.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 114.308px 8px; transform-origin: 114.308px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions / expressions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/57483\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eMesh processing\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/fileexchange/85173-mesh-generation-toolbox?s_tid=prof_contriblnk\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eMesh generation toolbox\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function T = mesh_the_pentagon()\r\n  T = 1;\r\nend","test_suite":"%%\r\nT_correct1 = [1 2 3;\r\n              1 3 4;\r\n              1 4 5];\r\n\r\nT_correct2 = [2 3 4;\r\n              2 4 5;\r\n              2 5 1];\r\n\r\nT_correct3 = [3 4 5;\r\n              3 5 1;\r\n              3 1 2];\r\n\r\nT_correct4 = [3 4 5;\r\n              3 5 1;\r\n              3 1 2];\r\n\r\nT_correct5 = [5 1 2;\r\n              5 2 3;\r\n              5 3 4];\r\n\r\n% Check every possible solutions\r\nassert(isequal(sortrows(sort(mesh_the_pentagon(),2)),sortrows(sort(T_correct1,2)))...\r\n     | isequal(sortrows(sort(mesh_the_pentagon(),2)),sortrows(sort(T_correct2,2)))...\r\n     | isequal(sortrows(sort(mesh_the_pentagon(),2)),sortrows(sort(T_correct3,2)))...\r\n     | isequal(sortrows(sort(mesh_the_pentagon(),2)),sortrows(sort(T_correct4,2)))...\r\n     | isequal(sortrows(sort(mesh_the_pentagon(),2)),sortrows(sort(T_correct5,2))))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('mesh_the_pentagon.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":1,"comments_count":6,"created_by":149128,"edited_by":149128,"edited_at":"2025-08-13T05:29:08.000Z","deleted_by":null,"deleted_at":null,"solvers_count":31,"test_suite_updated_at":"2025-08-13T05:29:08.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-23T12:59:46.000Z","updated_at":"2026-02-10T17:07:57.000Z","published_at":"2025-07-23T15:54:36.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn pentagon is a regular polygon with 5 vertices and 5 edges. Here below is an example of the vertex set \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eV,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e corresponding to the XYZ coordinates column vectors of a pentagon included in the unit circle.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[V = [1           0            0;\\n     cos(2*pi/5) sin(2*pi/5)  0;\\n     cos(4*pi/5) sin(4*pi/5)  0;\\n     cos(4*pi/5) sin(-4*pi/5) 0;\\n     cos(2*pi/5) sin(-2*pi/5) 0];]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA triangulated mesh \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eT\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (stands for triangles here) -or a triangulation- is simply a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e x 3 matrix of positive integers where each row contains the vertex indices of a triangle, and where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eN\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the number of triangles. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour task here is to mesh this pentagon with the minimum possible number of triangles. To do so, you will list the pentagons/rows in a matrix of faces, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eF. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eThe row order of the triangles in the list doesn't matter.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe first triangle here can be [\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1, 2, 3]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e if counterclockwise oriented.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"334\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"445\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTip\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBeware to avoid self intersecting triangles.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions / expressions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/57483\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eMesh 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margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eConsider a hexagon sitting in Quadrant I as depicted in an example below:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 315px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 157.5px; text-align: left; transform-origin: 384px 157.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"297\" 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data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThis hexagon is to be split into two regions (e.g., red and blue). Given the ratio of the two regions and the side of the hexagon, determine x coordinate of the line that splits the regions. The ratio between the regions (red to blue) is presented through the first two entries in the input. For example, if the ratio is 2 to 1, then these two numbers will be the first two numbers in the input. The last entry is the side of the hexagon.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = ratio_polygon(s)\r\n  y = s;\r\nend","test_suite":"%%\r\ns=[1 1 1];\r\ny=ratio_polygon(s)\r\ny_correct=0.8660;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[1 0 1];\r\ny=ratio_polygon(s);\r\ny_correct=sqrt(3);\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[1 3 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.5032;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[4 3 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.9613;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[7 1 1];\r\ny=ratio_polygon(s);\r\ny_correct=1.4523;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[1 7 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.2795;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":180632,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":25,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2021-01-22T01:27:38.000Z","updated_at":"2025-06-25T14:22:53.000Z","published_at":"2021-01-22T19:33:09.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider a hexagon sitting in Quadrant I as depicted in an example below:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"309\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"297\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis hexagon is to be split into two regions (e.g., red and blue). Given the ratio of the two regions and the side of the hexagon, determine x coordinate of the line that splits the regions. The ratio between the regions (red to blue) is presented through the first two entries in the input. For example, if the ratio is 2 to 1, then these two numbers will be the first two numbers in the input. 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Hexagon - Problem the second","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 415px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 207.5px; transform-origin: 407px 207.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eConsider a hexagon sitting in Quadrant I as depicted in an example below:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 292px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 146px; text-align: left; transform-origin: 384px 146px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" 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data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThis hexagon is to be split into two regions (e.g., red and blue). Given the ratio between the two regions and the side of the hexagon, determine x coordinate of the line that splits the regions. The ratio between the regions (red to blue) is presented through the first two entries in the input. For example, if the ratio is 3 to 7, then these two numbers will be the first two numbers in the input. The last entry is the side of the hexagon.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = ratio_polygon(s)\r\n  y = s;\r\nend","test_suite":"%%\r\ns=[0 1 1];\r\ny=ratio_polygon(s);\r\ny_correct=0;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[1 1 1];\r\ny=ratio_polygon(s);\r\ny_correct=1;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[2 3 1];\r\ny=ratio_polygon(s);\r\ny_correct=0.85;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[4 3 1];\r\ny=ratio_polygon(s);\r\ny_correct=1.1071;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[5 1 1];\r\ny=ratio_polygon(s)\r\ny_correct=1.5;\r\nassert(abs(y-y_correct)\u003c5e-4)\r\n%%\r\ns=[1 0 1];\r\ny=ratio_polygon(s)\r\ny_correct=2;\r\nassert(abs(y-y_correct)\u003c5e-4)","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":180632,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":26,"test_suite_updated_at":"2021-01-22T21:58:02.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-01-22T01:31:35.000Z","updated_at":"2025-10-02T21:01:31.000Z","published_at":"2021-01-22T21:58:02.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider a hexagon sitting in Quadrant I as depicted in an example below:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis hexagon is to be split into two regions (e.g., red and blue). Given the ratio between the two regions and the side of the hexagon, determine x coordinate of the line that splits the regions. The ratio between the regions (red to blue) is presented through the first two entries in the input. For example, if the ratio is 3 to 7, then these two numbers will be the first two numbers in the input. 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Tiling Dots in a Circle","description":"Return how many \u003chttp://en.wikipedia.org/wiki/Triangular_tiling Triangular Tiling\u003e grid points there are inside a circle of radius _r_ centred at (0,0) (including points on the edge).  \r\n\r\nAssume that a Triangular Tiling grid is a \u003chttp://en.wikipedia.org/wiki/Bravais_lattice 2D Hexagonal Bravais lattice\u003e with | _a1_ | = | _a2_ | = 1 and _\u0026phi;_ = 120\u0026deg;.\r\n\r\nNeither *string operations* nor *interpolations* are allowed!","description_html":"\u003cp\u003eReturn how many \u003ca href = \"http://en.wikipedia.org/wiki/Triangular_tiling\"\u003eTriangular Tiling\u003c/a\u003e grid points there are inside a circle of radius \u003ci\u003er\u003c/i\u003e centred at (0,0) (including points on the edge).\u003c/p\u003e\u003cp\u003eAssume that a Triangular Tiling grid is a \u003ca href = \"http://en.wikipedia.org/wiki/Bravais_lattice\"\u003e2D Hexagonal Bravais lattice\u003c/a\u003e with | \u003ci\u003ea1\u003c/i\u003e | = | \u003ci\u003ea2\u003c/i\u003e | = 1 and \u003ci\u003e\u0026phi;\u003c/i\u003e = 120\u0026deg;.\u003c/p\u003e\u003cp\u003eNeither \u003cb\u003estring operations\u003c/b\u003e nor \u003cb\u003einterpolations\u003c/b\u003e are allowed!\u003c/p\u003e","function_template":"function n = hexagonal_dots_in_circle(r)\r\n  n = r;\r\nend","test_suite":"%%\r\nuser_solution = fileread('hexagonal_dots_in_circle.m');\r\nassert(isempty(strfind(user_solution,'regexp')));\r\nassert(isempty(strfind(user_solution,'2str')));\r\nassert(isempty(strfind(user_solution,'str2')));\r\nassert(isempty(strfind(user_solution,'interp')));\r\nassert(isempty(strfind(user_solution,'printf')));\r\nassert(isempty(strfind(user_solution,'assert')));\r\n\r\n%%\r\nr = 0;\r\nn_correct = 1;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 0.5;\r\nn_correct = 1;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 1;\r\nn_correct = 7;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 1.5;\r\nn_correct = 7;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 2;\r\nn_correct = 19;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 2.5;\r\nn_correct = 19;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 3;\r\nn_correct = 37;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 5;\r\nn_correct = 91;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 7.5;\r\nn_correct = 199;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 10;\r\nn_correct = 367;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 15;\r\nn_correct = 823;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 20;\r\nn_correct = 1459;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 25;\r\nn_correct = 2263;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 50;\r\nn_correct = 9061;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n\r\n%%\r\nr = 100;\r\nn_correct = 36295;\r\nassert(isequal(hexagonal_dots_in_circle(r),n_correct));\r\n ","published":true,"deleted":false,"likes_count":0,"comments_count":1,"created_by":10352,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":32,"test_suite_updated_at":"2013-05-05T10:49:55.000Z","rescore_all_solutions":false,"group_id":20,"created_at":"2013-04-25T18:57:26.000Z","updated_at":"2026-02-16T10:52:57.000Z","published_at":"2013-04-25T18:57:26.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eReturn how many\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Triangular_tiling\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eTriangular Tiling\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e grid points there are inside a circle of radius\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e centred at (0,0) (including points on the edge).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAssume that a Triangular Tiling grid is a\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Bravais_lattice\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e2D Hexagonal Bravais lattice\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e with |\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ea1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e | = |\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ea2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e | = 1 and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eφ\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e = 120°.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNeither\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estring operations\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e nor\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003einterpolations\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e are allowed!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60925,"title":"Intersect three sequences","description":"Most numbers have interesting properties, if you look hard enough and interpret “interesting” liberally. Let’s choose a number at random—300, say—and list some of its properties:\r\nIt is divisible by the square of its largest prime factor. \r\nIt is the area of a triangle with integer sides and integer area\r\nIt is a folding point of the non-negative integers written in a hexagonal spiral (see below), as are 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, etc. \r\nA number that shares these properties is 108: (a) It is divisible by 9, or the square of its largest prime factor (3), (b) it is the area of a triangle with sides 15, 15, and 18, and (c) it is a folding point of the hexagon (it would be to the right of 75 in the hexagon below). \r\nWrite a function to determine whether a number has the three properties listed above. \r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 677.233px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 338.617px; transform-origin: 408px 338.617px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 361.508px 8px; transform-origin: 361.508px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eMost numbers have interesting properties, if you look hard enough and interpret “interesting” liberally. Let’s choose a number at random—300, say—and list some of its properties:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 40.8667px; transform-origin: 392px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 164.125px 8px; transform-origin: 164.125px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIt is divisible by the square of its largest prime factor. \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 185.933px 8px; transform-origin: 185.933px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIt is the area of a triangle with integer sides and integer area\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 363.283px 8px; transform-origin: 363.283px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIt is a folding point of the non-negative integers written in a hexagonal spiral (see below), as are 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, etc. \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 379.592px 8px; transform-origin: 379.592px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA number that shares these properties is 108: (a) It is divisible by 9, or the square of its largest prime factor (3), (b) it is the area of a triangle with sides 15, 15, and 18, and (c) it is a folding point of the hexagon (it would be to the right of 75 in the hexagon below). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 266.708px 8px; transform-origin: 266.708px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to determine whether a number has the three properties listed above. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 421.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 210.75px; text-align: left; transform-origin: 385px 210.75px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"487\" height=\"416\" style=\"vertical-align: baseline;width: 487px;height: 416px\" 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data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function tf = isX3Seq(x)\r\n  tf = false;\r\nend","test_suite":"%%\r\nassert(isX3Seq(108))\r\n\r\n%%\r\nassert(~isX3Seq(144))\r\n\r\n%%\r\nassert(~isX3Seq(200))\r\n\r\n%%\r\nassert(~isX3Seq(256))\r\n\r\n%%\r\nassert(isX3Seq(300))\r\n\r\n%%\r\nassert(isX3Seq(432))\r\n\r\n%%\r\nassert(~isX3Seq(500))\r\n\r\n%%\r\nassert(~isX3Seq(648))\r\n\r\n%%\r\nassert(~isX3Seq(660))\r\n\r\n%%\r\nassert(~isX3Seq(768))\r\n\r\n%%\r\nassert(isX3Seq(1200))\r\n\r\n%%\r\nassert(~isX3Seq(1553))\r\n\r\n%%\r\nassert(isX3Seq(1728))\r\n\r\n%%\r\nassert(~isX3Seq(1875))\r\n\r\n%%\r\nassert(~isX3Seq(3072))\r\n\r\n%%\r\nassert(~isX3Seq(3924))\r\n\r\n%%\r\nassert(isX3Seq(4332))\r\n\r\n%%\r\nassert(~isX3Seq(6084))\r\n\r\n%%\r\nassert(~isX3Seq(8118))\r\n\r\n%%\r\nassert(~isX3Seq(10806))\r\n\r\n%%\r\nassert(~isX3Seq(14283))\r\n\r\n%%\r\nassert(isX3Seq(18252))\r\n\r\n%%\r\nassert(~isX3Seq(26010))\r\n\r\n%%\r\nassert(~isX3Seq(31236))\r\n\r\n%%\r\nassert(~isX3Seq(42336))\r\n\r\n%%\r\nassert(~isX3Seq(49152))\r\n\r\n%%\r\nassert(isX3Seq(65712))\r\n\r\n%%\r\nassert(isX3Seq(99372))\r\n\r\n%%\r\nassert(~isX3Seq(116592))\r\n\r\n%%\r\nassert(~isX3Seq(140000))\r\n\r\n%%\r\nassert(~isX3Seq(152172))\r\n\r\n%%\r\nassert(~isX3Seq(160314))\r\n\r\n%%\r\nassert(~isX3Seq(170008))\r\n\r\n%%\r\nassert(isX3Seq(212268))\r\n\r\n%%\r\nassert(isX3Seq(248832))\r\n\r\n%%\r\nassert(isX3Seq(280908))\r\n\r\n%%\r\nassert(~isX3Seq(296274))\r\n\r\n%%\r\nassert(isX3Seq(303372))\r\n\r\n%%\r\nassert(isX3Seq(314928))\r\n\r\n%%\r\nassert(~isX3Seq(340707))\r\n\r\n%%\r\nassert(isX3Seq(489648))\r\n\r\n%%\r\nassert(~isX3Seq(3145728))\r\n\r\n%%\r\nassert(isX3Seq(7114800))\r\n\r\n%%\r\nassert(~isX3Seq(12582912))\r\n\r\n%%\r\nassert(isX3Seq(14865228))","published":true,"deleted":false,"likes_count":0,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2025-06-02T16:45:27.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2025-06-02T03:01:11.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2025-05-27T13:26:30.000Z","updated_at":"2025-06-06T14:09:48.000Z","published_at":"2025-05-27T13:26:30.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMost numbers have interesting properties, if you look hard enough and interpret “interesting” liberally. Let’s choose a number at random—300, say—and list some of its properties:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt is divisible by the square of its largest prime factor. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt is the area of a triangle with integer sides and integer area\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIt is a folding point of the non-negative integers written in a hexagonal spiral (see below), as are 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, etc. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA number that shares these properties is 108: (a) It is divisible by 9, or the square of its largest prime factor (3), (b) it is the area of a triangle with sides 15, 15, and 18, and (c) it is a folding point of the hexagon (it would be to the right of 75 in the hexagon below). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to determine whether a number has the three properties listed above. 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\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":1490,"title":"Shifted Hexagonal Tiling Dots in a Circle","description":"Return how many \u003chttp://en.wikipedia.org/wiki/Hexagonal_grid Hexagonal Tiling\u003e grid points there are inside a circle of radius _r_ centred at (0,0) (including points on the edge).  Assume that a Hexagonal Tiling grid is a \u003chttp://en.wikipedia.org/wiki/Regular_tiling 2D Regular Hexagonal Tessellation\u003e with equal edges of size _e_=1.  \r\n\r\nFor _shifted_ symmetry purposes, assume that (0,0) is a _grid point_.\r\n\r\nNeither *string operations* nor *interpolations* are allowed!","description_html":"\u003cp\u003eReturn how many \u003ca href = \"http://en.wikipedia.org/wiki/Hexagonal_grid\"\u003eHexagonal Tiling\u003c/a\u003e grid points there are inside a circle of radius \u003ci\u003er\u003c/i\u003e centred at (0,0) (including points on the edge).  Assume that a Hexagonal Tiling grid is a \u003ca href = \"http://en.wikipedia.org/wiki/Regular_tiling\"\u003e2D Regular Hexagonal Tessellation\u003c/a\u003e with equal edges of size \u003ci\u003ee\u003c/i\u003e=1.\u003c/p\u003e\u003cp\u003eFor \u003ci\u003eshifted\u003c/i\u003e symmetry purposes, assume that (0,0) is a \u003ci\u003egrid point\u003c/i\u003e.\u003c/p\u003e\u003cp\u003eNeither \u003cb\u003estring operations\u003c/b\u003e nor \u003cb\u003einterpolations\u003c/b\u003e are allowed!\u003c/p\u003e","function_template":"function n = shifted_hexagonal_tiling_dots_in_circle(r)\r\n  n = r;\r\nend","test_suite":"%%\r\nuser_solution = fileread('shifted_hexagonal_tiling_dots_in_circle.m');\r\nassert(isempty(strfind(user_solution,'regexp')));\r\nassert(isempty(strfind(user_solution,'2str')));\r\nassert(isempty(strfind(user_solution,'str2')));\r\nassert(isempty(strfind(user_solution,'interp')));\r\nassert(isempty(strfind(user_solution,'printf')));\r\nassert(isempty(strfind(user_solution,'assert')));\r\n\r\n%%\r\nr = 0;\r\nN_correct = 1;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 0.5;\r\nN_correct = 1;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 1;\r\nN_correct = 4;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 1.5;\r\nN_correct = 4;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 2;\r\nN_correct = 13;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 2.5;\r\nN_correct = 13;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 3;\r\nN_correct = 25;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 5;\r\nN_correct = 61;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 7.5;\r\nN_correct = 130;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 10;\r\nN_correct = 244;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 15;\r\nN_correct = 547;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 20;\r\nN_correct = 979;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 25;\r\nN_correct = 1510;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 50;\r\nN_correct = 6049;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 100;\r\nN_correct = 24202;\r\nassert(isequal(shifted_hexagonal_tiling_dots_in_circle(r),N_correct));\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":10352,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":25,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":20,"created_at":"2013-05-05T11:12:35.000Z","updated_at":"2026-02-16T10:46:10.000Z","published_at":"2013-05-05T11:13:56.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eReturn how many\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Hexagonal_grid\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eHexagonal Tiling\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e grid points there are inside a circle of radius\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e centred at (0,0) (including points on the edge). Assume that a Hexagonal Tiling grid is a\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Regular_tiling\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e2D Regular Hexagonal Tessellation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e with equal edges of size\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ee\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=1.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eshifted\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e symmetry purposes, assume that (0,0) is a\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egrid point\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNeither\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estring operations\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e nor\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003einterpolations\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e are allowed!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":1489,"title":"Hexagonal Tiling Dots in a Circle","description":"Return how many \u003chttp://en.wikipedia.org/wiki/Hexagonal_grid Hexagonal Tiling\u003e grid points there are inside a circle of radius _r_ centred at (0,0) (including points on the edge).  Assume that a Hexagonal Tiling grid is a \u003chttp://en.wikipedia.org/wiki/Regular_tiling 2D Regular Hexagonal Tessellation\u003e with equal edges of size _e_=1.  \r\n\r\nFor symmetry purposes, assume that (0,0) point is a _vacancy_; i.e., there _are_ points at (\u0026plusmn;1,0), (\u0026plusmn;1/2,\u0026plusmn;\u0026radic;3/2), etcetera.\r\n\r\nNeither *string operations* nor *interpolations* are allowed!","description_html":"\u003cp\u003eReturn how many \u003ca href = \"http://en.wikipedia.org/wiki/Hexagonal_grid\"\u003eHexagonal Tiling\u003c/a\u003e grid points there are inside a circle of radius \u003ci\u003er\u003c/i\u003e centred at (0,0) (including points on the edge).  Assume that a Hexagonal Tiling grid is a \u003ca href = \"http://en.wikipedia.org/wiki/Regular_tiling\"\u003e2D Regular Hexagonal Tessellation\u003c/a\u003e with equal edges of size \u003ci\u003ee\u003c/i\u003e=1.\u003c/p\u003e\u003cp\u003eFor symmetry purposes, assume that (0,0) point is a \u003ci\u003evacancy\u003c/i\u003e; i.e., there \u003ci\u003eare\u003c/i\u003e points at (\u0026plusmn;1,0), (\u0026plusmn;1/2,\u0026plusmn;\u0026radic;3/2), etcetera.\u003c/p\u003e\u003cp\u003eNeither \u003cb\u003estring operations\u003c/b\u003e nor \u003cb\u003einterpolations\u003c/b\u003e are allowed!\u003c/p\u003e","function_template":"function n = hexagonal_tiling_dots_in_circle(r)\r\n  n = r;\r\nend","test_suite":"%%\r\nuser_solution = fileread('hexagonal_tiling_dots_in_circle.m');\r\nassert(isempty(strfind(user_solution,'regexp')));\r\nassert(isempty(strfind(user_solution,'2str')));\r\nassert(isempty(strfind(user_solution,'str2')));\r\nassert(isempty(strfind(user_solution,'interp')));\r\nassert(isempty(strfind(user_solution,'printf')));\r\nassert(isempty(strfind(user_solution,'assert')));\r\n\r\n%%\r\nr = 0;\r\nN_correct = 0;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 0.5;\r\nN_correct = 0;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 1;\r\nN_correct = 6;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 1.5;\r\nN_correct = 6;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 2;\r\nN_correct = 12;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 2.5;\r\nN_correct = 12;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 3;\r\nN_correct = 24;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 5;\r\nN_correct = 60;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 7.5;\r\nN_correct = 138;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 10;\r\nN_correct = 246;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 15;\r\nN_correct = 552;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 20;\r\nN_correct = 960;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 25;\r\nN_correct = 1506;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 50;\r\nN_correct = 6024;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n\r\n%%\r\nr = 100;\r\nN_correct = 24186;\r\nassert(isequal(hexagonal_tiling_dots_in_circle(r),N_correct));\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":1,"created_by":10352,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":29,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":20,"created_at":"2013-05-05T10:39:46.000Z","updated_at":"2026-03-25T00:01:03.000Z","published_at":"2013-05-05T10:54:39.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eReturn how many\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Hexagonal_grid\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eHexagonal Tiling\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e grid points there are inside a circle of radius\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e centred at (0,0) (including points on the edge). Assume that a Hexagonal Tiling grid is a\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Regular_tiling\\\"\u003e\u003cw:r\u003e\u003cw:t\u003e2D Regular Hexagonal Tessellation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e with equal edges of size\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ee\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=1.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor symmetry purposes, assume that (0,0) point is a\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003evacancy\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e; i.e., there\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eare\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e points at (±1,0), (±1/2,±√3/2), etcetera.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNeither\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estring operations\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e nor\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003einterpolations\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e are allowed!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":60227,"title":"Rotate matrix 60 degrees","description":"Given a 2*m-1 by 4*m-3 matrix in which certain elements form a hexagon of side length m, rotate the hexagon 60 degrees counterclockwise, as shown below for m=3:\r\nA=[\r\n'--A-A-A--'\r\n'-B-B-B-B-'\r\n'C-C-C-C-C'\r\n'-D-D-D-D-'\r\n'--E-E-E--']\r\n\r\nrot60(A)=[\r\n'--A-B-C--'\r\n'-A-B-C-D-'\r\n'A-B-C-D-E'\r\n'-B-C-D-E-'\r\n'--C-D-E--']\r\nA=rot60(A,n) should rotate the hexagon 60*n degrees counterclockwise (and thus clockwise if n\u003c0).","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 348.75px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 331.989px 174.375px; transform-origin: 331.996px 174.375px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42.017px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 308.991px 21.0085px; text-align: left; transform-origin: 308.999px 21.0085px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven a 2*m-1 by 4*m-3 matrix in which certain elements form a hexagon of side length m, rotate the hexagon 60 degrees counterclockwise, as shown below for m=3:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 265.724px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 328.991px 132.855px; transform-origin: 328.999px 132.862px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eA=[\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'--A-A-A--'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'-B-B-B-B-'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'C-C-C-C-C'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'-D-D-D-D-'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'--E-E-E--'\u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003erot60(A)=[\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'--A-B-C--'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'-A-B-C-D-'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'A-B-C-D-E'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'-B-C-D-E-'\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4403px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.909091px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.909091px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.909091px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.909091px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 328.991px 10.2131px; text-wrap: nowrap; transform-origin: 328.999px 10.2202px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); \"\u003e'--C-D-E--'\u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.0085px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 308.991px 10.4972px; text-align: left; transform-origin: 308.999px 10.5043px; white-space-collapse: preserve; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA=rot60(A,n) should rotate the hexagon 60*n degrees counterclockwise (and thus clockwise if n\u0026lt;0).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function A=rot60(A,n)\r\nA=rot90(A,2*n/3);\r\n","test_suite":"%%\r\nA=['--A-A-A--'\r\n   '-B-B-B-B-'\r\n   'C-C-C-C-C'\r\n   '-D-D-D-D-'\r\n   '--E-E-E--'];\r\nB=['--A-B-C--'\r\n   '-A-B-C-D-'\r\n   'A-B-C-D-E'\r\n   '-B-C-D-E-'\r\n   '--C-D-E--'];\r\nassert(isequal(rot60(A),B))\r\n%%\r\nA=['  A A A  '\r\n   ' B B B B '\r\n   'C C C C C'\r\n   ' D D D D '\r\n   '  E E E  '];\r\nB=['  C B A  '\r\n   ' D C B A '\r\n   'E D C B A'\r\n   ' E D C B '\r\n   '  E D C  '];\r\nassert(isequal(rot60(A,-1),B))\r\n%%\r\nA=['   3 1 4 1   '\r\n   '  5 9 2 6 5  '\r\n   ' 3 5 8 9 7 9 '\r\n   '3 2 3 8 4 6 2'\r\n   ' 6 4 3 3 8 3 '\r\n   '  2 7 9 5 0  '\r\n   '   2 8 8 4   '];\r\nB=['   1 5 9 2   '\r\n   '  4 6 7 6 3  '\r\n   ' 1 2 9 4 8 0 '\r\n   '3 9 8 8 3 5 4'\r\n   ' 5 5 3 3 9 8 '\r\n   '  3 2 4 7 8  '\r\n   '   3 6 2 2   '];\r\nassert(isequal(rot60(A),B))\r\n%%\r\nA=['    A B C D E    '\r\n   '   F G H I J K   '\r\n   '  L M N O P Q R  '\r\n   ' S T U V W X Y Z '\r\n   '1 2 3 4 5 6 7 8 9'\r\n   ' a b c d e f g h '\r\n   '  i j k l m n o  '\r\n   '   p q r s t u   '\r\n   '    v w x y z    '];\r\nB=['    E K R Z 9    '\r\n   '   D J Q Y 8 h   '\r\n   '  C I P X 7 g o  '\r\n   ' B H O W 6 f n u '\r\n   'A G N V 5 e m t z'\r\n   ' F M U 4 d l s y '\r\n   '  L T 3 c k r x  '\r\n   '   S 2 b j q w   '\r\n   '    1 a i p v    '];\r\nassert(isequal(rot60(A),B))\r\nn=randi([-100 100],1);\r\nassert(isequal(rot60(A,3*n),rot90(A,2*n)))\r\n%%\r\nA=[nan nan 120 nan  90 nan  60 nan nan\r\n   nan 150 nan 120 nan  60 nan  30 nan\r\n   180 nan 180 nan nan nan   0 nan   0\r\n   nan 210 nan 240 nan 300 nan 330 nan\r\n   nan nan 240 nan 270 nan 300 nan nan];\r\nfor n=1:6\r\n   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