Problem 51820. Count unique orderings of vertices of a polygon

Created by ChrisR

Solve Later

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If the corners are numbered as shown below, then they can be input in 24 ways.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>However, for the rectangle problem, many of the 24 ways are essentially the same. For example, 2341, 3214, and 4123 are effectively the same as 1234 because the numbers of the corners could be shifted around the rectangle. In fact, only three of the 24 ways are different (1234, 1243, and 1324). </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function to determine the unique orderings of vertices of a polygon with </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> sides. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"image\"><w:customXmlPr><w:attr w:name=\"height\" w:val=\"177\"/><w:attr w:name=\"width\" w:val=\"262\"/><w:attr w:name=\"verticalAlign\" w:val=\"baseline\"/><w:attr w:name=\"altText\" w:val=\"\"/><w:attr w:name=\"relationshipId\" w:val=\"rId1\"/></w:customXmlPr></w:customXml></w:p></w:body></w:document>","relationship":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/image","target":"/media/image1.png","relationshipId":"rId1"}]},{"partUri":"/media/image1.png","contentType":"image/png","content":"data:image/png;base64,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<div 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; "><div style="block-size: 356.917px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 178.458px; transform-origin: 407px 178.458px; vertical-align: baseline; "><div 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: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/2671"><span style=""><span style="text-decoration: underline; text-decoration-line: underline; ">Cody Problem 2671</span></span></a><span 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: 294.433px 7.91667px; transform-origin: 294.433px 7.91667px; unicode-bidi: normal; "><span style=""> asks us to determine whether four points can be the corners of a rectangle. The points are not necessarily input in order (either clockwise or counterclockwise). My initial attempt at the problem involved determining how the points might be presented. If the corners are numbered as shown below, then they can be input in 24 ways.</span></span></div><div 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: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span 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: 383.133px 7.91667px; transform-origin: 383.133px 7.91667px; unicode-bidi: normal; "><span style="">However, for the rectangle problem, many of the 24 ways are essentially the same. For example, 2341, 3214, and 4123 are effectively the same as 1234 because the numbers of the corners could be shifted around the rectangle. In fact, only three of the 24 ways are different (1234, 1243, and 1324). </span></span></div><div 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; "><span 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: 245.317px 7.91667px; transform-origin: 245.317px 7.91667px; unicode-bidi: normal; "><span style="">Write a function to determine the unique orderings of vertices of a polygon with </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><span 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: 22.1667px 7.91667px; transform-origin: 22.1667px 7.91667px; unicode-bidi: normal; "><span style=""> sides. </span></span></div><div style="block-size: 182.917px; 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 91.4583px; text-align: left; transform-origin: 384px 91.4583px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img class="imageNode" style="vertical-align: baseline;width: 262px;height: 177px" 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Last Solution submitted on May 10, 2022

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combinatorics permutations sequence

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Problem 51820. Count unique orderings of vertices of a polygon

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Problem 51820. Count unique orderings of vertices of a polygon

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