{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-06T14:01:22.000Z","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":"2026-04-06T00:00:00.000Z","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":54074,"title":"Determining if a Degree Sequence is Potentially a Graph","description":"A degree sequence is a list of numbers representing the degrees of vertices in a graph. While it is difficult to tell if a graph can be made from a degree sequence, there are some ways to tell for certain that a graph does not exist with a given degree sequence. One easy first check is the following: \r\nFirst, sort the degree sequence in descending order. Next, pop the first degree off the list and subtract one from the next N elements, where N is the degree you popped off. Repeat until the list is empty. If at any point a degree in the list is less than 0 or if there are not N elements left in the list to subtract from, there is no graph that exists with that degree sequence.\r\nWrite a function is_graph that returns true if this algorithm results in an empty list or false if it fails at any point.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(232, 230, 227); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(232, 230, 227); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 166.5px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 83.25px; transform-origin: 407px 83.25px; vertical-align: baseline; \"\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: 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; \"\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: 376.783px 8px; transform-origin: 376.783px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA degree sequence is a list of numbers representing the degrees of vertices in a graph. While it is difficult to tell if a graph can be made from a degree sequence, there are some ways to tell for certain that a graph does not exist with a given degree sequence. One easy first check is the following: \u003c/span\u003e\u003c/span\u003e\u003c/div\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: 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; \"\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: 379.867px 8px; transform-origin: 379.867px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFirst, sort the degree sequence in descending order. Next, pop the first degree off the list and subtract one from the next N elements, where N is the degree you popped off. Repeat until the list is empty. If at any point a degree in the list is less than 0 or if there are not N elements left in the list to subtract from, there is no graph that exists with that degree sequence.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22.5px; 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: 384px 11.25px; text-align: left; transform-origin: 384px 11.25px; 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: 50.4333px 8px; transform-origin: 50.4333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function \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: 37.8px 8px; transform-origin: 37.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 37.8px 8.5px; transform-origin: 37.8px 8.5px; \"\u003eis_graph \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: 259.8px 8px; transform-origin: 259.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethat returns true if this algorithm results in an empty list or false if it fails at any point.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = is_graph(x)\r\n  % Run algorithm\r\n  y = x;\r\nend","test_suite":"%%\r\nx = [1 1];\r\ny_correct = 1;\r\nassert(isequal(is_graph(x),y_correct))\r\n\r\n%%\r\nx = [1 1 3];\r\ny_correct = 0;\r\nassert(isequal(is_graph(x),y_correct))\r\n\r\n%%\r\nx = [15 3 6];\r\ny_correct = 0;\r\nassert(isequal(is_graph(x),y_correct))\r\n\r\n%%\r\nx = [5 4 1 1 1 1];\r\ny_correct = 0;\r\nassert(isequal(is_graph(x),y_correct))\r\n\r\n%%\r\nx = [5 7 2 2 3 3 2 2];\r\ny_correct = 1;\r\nassert(isequal(is_graph(x),y_correct))\r\n\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":2052130,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":9,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-03-03T20:12:17.000Z","updated_at":"2025-06-25T20:01:28.000Z","published_at":"2022-03-03T20:12:17.000Z","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\u003eA degree sequence is a list of numbers representing the degrees of vertices in a graph. While it is difficult to tell if a graph can be made from a degree sequence, there are some ways to tell for certain that a graph does not exist with a given degree sequence. One easy first check is the following: \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\u003eFirst, sort the degree sequence in descending order. Next, pop the first degree off the list and subtract one from the next N elements, where N is the degree you popped off. Repeat until the list is empty. If at any point a degree in the list is less than 0 or if there are not N elements left in the list to subtract from, there is no graph that exists with that degree sequence.\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 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eis_graph \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethat returns true if this algorithm results in an empty list or false if it fails at any point.\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\"}]}"},{"id":2732,"title":"Construct a precedence graph from a code segment","description":"A hypothetical MATLAB code segment containing n lines is given in the form of a cell array. The i-th cell contains the i-th line of the code. Each of the lines contains simple arithmetic expressions.\r\n\r\nNow, construct an adjacency matrix of a graph containing n-vertices. The i-th vertex will represent the i-th line of the code. There should be a directed edge from i-th vertex to j-th vertex only if the values generated at i-th line are used in the j-th line.\r\n\r\nAll the variables in the code will have single letter names (e.g.: a,b,x,y etc).\r\n\r\nExample:\r\n\r\n  C = {'a=1;'\r\n       'b=1;'\r\n       'c=a+b;'\r\n       'c=c+1;'};\r\n\r\nHere, the cell array C contains a code segment. The first two lines are independent in the sense that they do not use values generated at any other lines. The third line uses information generated at line 1 and 2. The fourth line uses information generated at line 1,2 and 3.\r\n\r\nThus the resulting adjacency matrix will be as follows:\r\n\r\n  \r\n  mat = [0 0 1 1;\r\n         0 0 1 1;\r\n         0 0 0 1;\r\n         0 0 0 0];\r\n\r\n\r\n\r\n\r\nDefinition of adjacency matrix:\r\n\u003chttp://en.wikipedia.org/wiki/Adjacency_matrix\u003e","description_html":"\u003cp\u003eA hypothetical MATLAB code segment containing n lines is given in the form of a cell array. The i-th cell contains the i-th line of the code. Each of the lines contains simple arithmetic expressions.\u003c/p\u003e\u003cp\u003eNow, construct an adjacency matrix of a graph containing n-vertices. The i-th vertex will represent the i-th line of the code. There should be a directed edge from i-th vertex to j-th vertex only if the values generated at i-th line are used in the j-th line.\u003c/p\u003e\u003cp\u003eAll the variables in the code will have single letter names (e.g.: a,b,x,y etc).\u003c/p\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eC = {'a=1;'\r\n     'b=1;'\r\n     'c=a+b;'\r\n     'c=c+1;'};\r\n\u003c/pre\u003e\u003cp\u003eHere, the cell array C contains a code segment. The first two lines are independent in the sense that they do not use values generated at any other lines. The third line uses information generated at line 1 and 2. The fourth line uses information generated at line 1,2 and 3.\u003c/p\u003e\u003cp\u003eThus the resulting adjacency matrix will be as follows:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003emat = [0 0 1 1;\r\n       0 0 1 1;\r\n       0 0 0 1;\r\n       0 0 0 0];\r\n\u003c/pre\u003e\u003cp\u003eDefinition of adjacency matrix: \u003ca href = \"http://en.wikipedia.org/wiki/Adjacency_matrix\"\u003ehttp://en.wikipedia.org/wiki/Adjacency_matrix\u003c/a\u003e\u003c/p\u003e","function_template":"function y = pGraph(x)\r\n\r\n\r\n\r\n\r\nend","test_suite":"%%\r\nC = {'a=1;'\r\n     'b=1;'\r\n     'c=a+b;'\r\n     'c=c+1;'};\r\nmat = [0 0 1 1;\r\n       0 0 1 1;\r\n       0 0 0 1;\r\n       0 0 0 0];\r\nassert(isequal(pGraph(C),mat))\r\n\r\n\r\n\r\n%%\r\nC = {'a=1;'\r\n     'a=1;'\r\n     'c=1;'\r\n     'c=1;'};\r\nmat = [0 0 0 0;\r\n       0 0 0 0;\r\n       0 0 0 0;\r\n       0 0 0 0];\r\nassert(isequal(pGraph(C),mat))\r\n\r\n%%\r\nC = {'a=1;'\r\n     'a=1;'\r\n     'c=a+1;'\r\n     'c=a+1;'};\r\nmat = [0 0 0 0;\r\n       0 0 1 1;\r\n       0 0 0 0;\r\n       0 0 0 0];\r\nassert(isequal(pGraph(C),mat))\r\n\r\n%%\r\nC = {'a=1;'\r\n     'b=a+2;'\r\n     'c=b+1;'\r\n     'd=c+1;'};\r\nmat = double(~tril(ones(4)))\r\nassert(isequal(pGraph(C),mat))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":17203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":"2014-12-06T07:56:27.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-12-06T07:54:40.000Z","updated_at":"2024-11-02T13:28:43.000Z","published_at":"2014-12-06T07:54:40.000Z","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\u003eA hypothetical MATLAB code segment containing n lines is given in the form of a cell array. The i-th cell contains the i-th line of the code. Each of the lines contains simple arithmetic expressions.\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\u003eNow, construct an adjacency matrix of a graph containing n-vertices. The i-th vertex will represent the i-th line of the code. There should be a directed edge from i-th vertex to j-th vertex only if the values generated at i-th line are used in the j-th line.\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\u003eAll the variables in the code will have single letter names (e.g.: a,b,x,y 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\u003c/w:pPr\u003e\u003cw:r\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=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[C = {'a=1;'\\n     'b=1;'\\n     'c=a+b;'\\n     'c=c+1;'};]]\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\u003eHere, the cell array C contains a code segment. The first two lines are independent in the sense that they do not use values generated at any other lines. The third line uses information generated at line 1 and 2. The fourth line uses information generated at line 1,2 and 3.\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\u003eThus the resulting adjacency matrix will be as follows:\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[mat = [0 0 1 1;\\n       0 0 1 1;\\n       0 0 0 1;\\n       0 0 0 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\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDefinition of adjacency matrix:\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/Adjacency_matrix\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehttp://en.wikipedia.org/wiki/Adjacency_matrix\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\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":1886,"title":"Graceful Double Wheel Graph","description":"\u003chttp://en.wikipedia.org/wiki/Graceful_labeling Graceful Graphs\u003e are the topic of the \u003chttp://www.azspcs.net/Contest/GracefulGraphs Primes Graceful Graph Contest\u003e , 21 September 2013 thru 21 December 2013.\r\n\r\nThis Challenge is to create \u003chttp://www.comp.leeds.ac.uk/bms/Graceful/doublewheel.html Graceful Double Wheel Graphs\u003e for various N. A \u003chttp://www.cs.cornell.edu/~lebras/publications/LeBras2013Double.pdf General Algorithm by Le Bras of Cornell\u003e may be helpful, Section 3 for Even/Odd Rings. The Double Wheel Graph produces valid but not Maximum Edge Graceful Graph solutions based upon \u003chttp://oeis.org/A004137 OEIS A004137\u003e.\r\n\r\n*Example:*\r\nOne solution for N=11:\r\n\r\n\u003c\u003chttp://www.comp.leeds.ac.uk/bms/Graceful/Images/2C5+K1.gif\u003e\u003e\r\n\r\nwhich could be answered as [1 3 14 6 19;20 5 17 7 16].\r\n\r\nThere are 20 links and thus the absolute differences between connected nodes must produce values 1 thru 20.  The max node value is equal to the number of links and the min is zero, at the center of the Double Wheel.\r\n\r\n*Input:* N [Total number of Nodes (odd) and N\u003e10 ]\r\n\r\n*Output:* M [ Matrix size [(N-1)/2, 2] of node values where row-1 is outer and row-2 is inner ring ]","description_html":"\u003cp\u003e\u003ca href = \"http://en.wikipedia.org/wiki/Graceful_labeling\"\u003eGraceful Graphs\u003c/a\u003e are the topic of the \u003ca href = \"http://www.azspcs.net/Contest/GracefulGraphs\"\u003ePrimes Graceful Graph Contest\u003c/a\u003e , 21 September 2013 thru 21 December 2013.\u003c/p\u003e\u003cp\u003eThis Challenge is to create \u003ca href = \"http://www.comp.leeds.ac.uk/bms/Graceful/doublewheel.html\"\u003eGraceful Double Wheel Graphs\u003c/a\u003e for various N. A \u003ca href = \"http://www.cs.cornell.edu/~lebras/publications/LeBras2013Double.pdf\"\u003eGeneral Algorithm by Le Bras of Cornell\u003c/a\u003e may be helpful, Section 3 for Even/Odd Rings. The Double Wheel Graph produces valid but not Maximum Edge Graceful Graph solutions based upon \u003ca href = \"http://oeis.org/A004137\"\u003eOEIS A004137\u003c/a\u003e.\u003c/p\u003e\u003cp\u003e\u003cb\u003eExample:\u003c/b\u003e\r\nOne solution for N=11:\u003c/p\u003e\u003cimg src = \"http://www.comp.leeds.ac.uk/bms/Graceful/Images/2C5+K1.gif\"\u003e\u003cp\u003ewhich could be answered as [1 3 14 6 19;20 5 17 7 16].\u003c/p\u003e\u003cp\u003eThere are 20 links and thus the absolute differences between connected nodes must produce values 1 thru 20.  The max node value is equal to the number of links and the min is zero, at the center of the Double Wheel.\u003c/p\u003e\u003cp\u003e\u003cb\u003eInput:\u003c/b\u003e N [Total number of Nodes (odd) and N\u003e10 ]\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput:\u003c/b\u003e M [ Matrix size [(N-1)/2, 2] of node values where row-1 is outer and row-2 is inner ring ]\u003c/p\u003e","function_template":"function m=double_wheel(n)\r\n  m=[];\r\nend","test_suite":"%%\r\ntic\r\nn=11;\r\nm=double_wheel(n);\r\nms=circshift(m,[0 -1]);\r\ndm=m-ms;\r\nd=unique([m(:) abs(dm(:))]);\r\nassert(all(diff(d)==1))\r\nassert(length(d)==2*(n-1))\r\nassert(max(d)==2*(n-1))\r\ntoc\r\n%%\r\nn=13;\r\nm=double_wheel(n);\r\nms=circshift(m,[0 -1]);\r\ndm=m-ms;\r\nd=unique([m(:) abs(dm(:))]);\r\nassert(all(diff(d)==1))\r\nassert(length(d)==2*(n-1))\r\nassert(max(d)==2*(n-1))\r\ntoc\r\n%%\r\nn=17;\r\nm=double_wheel(n);\r\nms=circshift(m,[0 -1]);\r\ndm=m-ms;\r\nd=unique([m(:) abs(dm(:))]);\r\nassert(all(diff(d)==1))\r\nassert(length(d)==2*(n-1))\r\nassert(max(d)==2*(n-1))\r\ntoc\r\n%%\r\nn=19;\r\nm=double_wheel(n);\r\nms=circshift(m,[0 -1]);\r\ndm=m-ms;\r\nd=unique([m(:) abs(dm(:))]);\r\nassert(all(diff(d)==1))\r\nassert(length(d)==2*(n-1))\r\nassert(max(d)==2*(n-1))\r\ntoc\r\n%%\r\nn=71;\r\nm=double_wheel(n);\r\nms=circshift(m,[0 -1]);\r\ndm=m-ms;\r\nd=unique([m(:) abs(dm(:))]);\r\nassert(all(diff(d)==1))\r\nassert(length(d)==2*(n-1))\r\nassert(max(d)==2*(n-1))\r\ntoc\r\n%%\r\nn=97;\r\nm=double_wheel(n);\r\nms=circshift(m,[0 -1]);\r\ndm=m-ms;\r\nd=unique([m(:) abs(dm(:))]);\r\nassert(all(diff(d)==1))\r\nassert(length(d)==2*(n-1))\r\nassert(max(d)==2*(n-1))\r\ntoc\r\n\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":3097,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-09-21T23:15:03.000Z","updated_at":"2013-09-22T01:16:42.000Z","published_at":"2013-09-22T01:16:42.000Z","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\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.gif\"}],\"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:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Graceful_labeling\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eGraceful Graphs\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e are the topic of the\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://www.azspcs.net/Contest/GracefulGraphs\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrimes Graceful Graph Contest\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e , 21 September 2013 thru 21 December 2013.\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\u003eThis Challenge is to create\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://www.comp.leeds.ac.uk/bms/Graceful/doublewheel.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eGraceful Double Wheel Graphs\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e for various N. 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://www.cs.cornell.edu/~lebras/publications/LeBras2013Double.pdf\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eGeneral Algorithm by Le Bras of Cornell\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e may be helpful, Section 3 for Even/Odd Rings. The Double Wheel Graph produces valid but not Maximum Edge Graceful Graph solutions based upon\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://oeis.org/A004137\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eOEIS A004137\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\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:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e One solution for N=11:\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: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=\\\"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\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhich could be answered as [1 3 14 6 19;20 5 17 7 16].\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\u003eThere are 20 links and thus the absolute differences between connected nodes must produce values 1 thru 20. The max node value is equal to the number of links and the min is zero, at the center of the Double Wheel.\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:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e N [Total number of Nodes (odd) and N\u0026gt;10 ]\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:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e M [ Matrix size [(N-1)/2, 2] of node values where row-1 is outer and row-2 is inner ring ]\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\"},{\"partUri\":\"/media/image1.gif\",\"contentType\":\"image/gif\",\"content\":\"data:image/gif;base64,<!DOCTYPE html>
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\"}]}"},{"id":1887,"title":"Graceful Graph: Wichmann Rulers","description":"This Challenge is to find maximum size Graceful Graphs via Wichmann Rulers for P\u003e13.  This Challenge is related to the \u003chttp://www.azspcs.net/Contest/GracefulGraphs Graceful Graph Contest\u003e which Rokicki completed in 97 minutes. The Wichmann Conjecture is that no larger solutions exist for P\u003e13.\r\n\r\nAn Optimal ruler is defined as having end points at 0 and Max with P-2 integer points between [0,Max] such that the distances 1 thru Max exist by deltas between points.\r\nAn \u003chttp://oeis.org/A193802 Optimal Wichmann Ruler\u003e readily creates solutions that can be tested for number of points and existence of all expected deltas.\r\n\r\nThe Wichmann difference vector is [Q(1,r), r+1, Q(2r+1,r), Q(4r+3,s), Q(2r+2,r+1), Q(1,r)] where Q(a,b) is b a's, e.g. Q(2,3) is [2 2 2]. The max value is L=4r(r+s+2)+3(s+1) for Points P=4r+s+3, (r and s \u003e=0 and integer).\r\n\r\nFor W(r,s), W(2,3) creates the difference sequence [1 1 3 5 5 11 11 11 6 6 6 1 1]. The points on the ruler are the cumsum of W with a zero pre-pended to produce S=[0 1 2 5 10 15 26 37 48 54 60 66 67 68], P=14. All deltas from 1 thru 68 can be realized.\r\n\r\n*Input:* P  (Number of Points on the ruler)\r\n\r\n*Output:* S (Vector of length P of locations on the ruler, 0 thru Max Value and can generate all deltas 1:S(end))\r\n\r\n*Notes:*\r\n\r\n  1) A W(r,s) does not guarantee all deltas can be generated\r\n  2) For any P there are multiple W(r,s) solutions \r\n  3) P=5 solution is 9, readily solved by brute force\r\n  4) P=13 Wichmann is 57 but the best solution is 58. Too big for brute force\r\n  5) Create Connectivity Graph for Cases, like Final Matlab Competition, for Fun ","description_html":"\u003cp\u003eThis Challenge is to find maximum size Graceful Graphs via Wichmann Rulers for P\u003e13.  This Challenge is related to the \u003ca href = \"http://www.azspcs.net/Contest/GracefulGraphs\"\u003eGraceful Graph Contest\u003c/a\u003e which Rokicki completed in 97 minutes. The Wichmann Conjecture is that no larger solutions exist for P\u003e13.\u003c/p\u003e\u003cp\u003eAn Optimal ruler is defined as having end points at 0 and Max with P-2 integer points between [0,Max] such that the distances 1 thru Max exist by deltas between points.\r\nAn \u003ca href = \"http://oeis.org/A193802\"\u003eOptimal Wichmann Ruler\u003c/a\u003e readily creates solutions that can be tested for number of points and existence of all expected deltas.\u003c/p\u003e\u003cp\u003eThe Wichmann difference vector is [Q(1,r), r+1, Q(2r+1,r), Q(4r+3,s), Q(2r+2,r+1), Q(1,r)] where Q(a,b) is b a's, e.g. Q(2,3) is [2 2 2]. The max value is L=4r(r+s+2)+3(s+1) for Points P=4r+s+3, (r and s \u003e=0 and integer).\u003c/p\u003e\u003cp\u003eFor W(r,s), W(2,3) creates the difference sequence [1 1 3 5 5 11 11 11 6 6 6 1 1]. The points on the ruler are the cumsum of W with a zero pre-pended to produce S=[0 1 2 5 10 15 26 37 48 54 60 66 67 68], P=14. All deltas from 1 thru 68 can be realized.\u003c/p\u003e\u003cp\u003e\u003cb\u003eInput:\u003c/b\u003e P  (Number of Points on the ruler)\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput:\u003c/b\u003e S (Vector of length P of locations on the ruler, 0 thru Max Value and can generate all deltas 1:S(end))\u003c/p\u003e\u003cp\u003e\u003cb\u003eNotes:\u003c/b\u003e\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003e1) A W(r,s) does not guarantee all deltas can be generated\r\n2) For any P there are multiple W(r,s) solutions \r\n3) P=5 solution is 9, readily solved by brute force\r\n4) P=13 Wichmann is 57 but the best solution is 58. Too big for brute force\r\n5) Create Connectivity Graph for Cases, like Final Matlab Competition, for Fun \r\n\u003c/pre\u003e","function_template":"function s=Graceful_Wichmann(n)\r\n  s=0;\r\nend","test_suite":"%%\r\ntic\r\nn=17;\r\nexp=101;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=19;\r\nexp=123;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=23;\r\nexp=183;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=29;\r\nexp=289;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=31;\r\nexp=327;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=37;\r\nexp=465;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=41;\r\nexp=573;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=43;\r\nexp=627;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=47;\r\nexp=751;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=53;\r\nexp=953;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=59;\r\nexp=1179;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=61;\r\nexp=1257;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=67;\r\nexp=1515;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=71;\r\nexp=1703;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=73;\r\nexp=1797;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=79;\r\nexp=2103;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=83;\r\nexp=2323;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=89;\r\nexp=2669;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=97;\r\nexp=3165;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":3097,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-09-23T01:30:25.000Z","updated_at":"2013-09-23T13:04:40.000Z","published_at":"2013-09-23T04:00:18.000Z","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\u003eThis Challenge is to find maximum size Graceful Graphs via Wichmann Rulers for P\u0026gt;13. This Challenge is related to the\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://www.azspcs.net/Contest/GracefulGraphs\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eGraceful Graph Contest\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e which Rokicki completed in 97 minutes. The Wichmann Conjecture is that no larger solutions exist for P\u0026gt;13.\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\u003eAn Optimal ruler is defined as having end points at 0 and Max with P-2 integer points between [0,Max] such that the distances 1 thru Max exist by deltas between points. An\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://oeis.org/A193802\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eOptimal Wichmann Ruler\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e readily creates solutions that can be tested for number of points and existence of all expected deltas.\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\u003eThe Wichmann difference vector is [Q(1,r), r+1, Q(2r+1,r), Q(4r+3,s), Q(2r+2,r+1), Q(1,r)] where Q(a,b) is b a's, e.g. Q(2,3) is [2 2 2]. The max value is L=4r(r+s+2)+3(s+1) for Points P=4r+s+3, (r and s \u0026gt;=0 and integer).\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 W(r,s), W(2,3) creates the difference sequence [1 1 3 5 5 11 11 11 6 6 6 1 1]. The points on the ruler are the cumsum of W with a zero pre-pended to produce S=[0 1 2 5 10 15 26 37 48 54 60 66 67 68], P=14. All deltas from 1 thru 68 can be realized.\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:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e P (Number of Points on the ruler)\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:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e S (Vector of length P of locations on the ruler, 0 thru Max Value and can generate all deltas 1:S(end))\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:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNotes:\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[1) A W(r,s) does not guarantee all deltas can be generated\\n2) For any P there are multiple W(r,s) solutions \\n3) P=5 solution is 9, readily solved by brute force\\n4) P=13 Wichmann is 57 but the best solution is 58. Too big for brute force\\n5) Create Connectivity Graph for Cases, like Final Matlab Competition, for Fun]]\u003e\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\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":54074,"title":"Determining if a Degree Sequence is Potentially a Graph","description":"A degree sequence is a list of numbers representing the degrees of vertices in a graph. While it is difficult to tell if a graph can be made from a degree sequence, there are some ways to tell for certain that a graph does not exist with a given degree sequence. One easy first check is the following: \r\nFirst, sort the degree sequence in descending order. Next, pop the first degree off the list and subtract one from the next N elements, where N is the degree you popped off. Repeat until the list is empty. If at any point a degree in the list is less than 0 or if there are not N elements left in the list to subtract from, there is no graph that exists with that degree sequence.\r\nWrite a function is_graph that returns true if this algorithm results in an empty list or false if it fails at any point.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(232, 230, 227); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(232, 230, 227); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 166.5px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 83.25px; transform-origin: 407px 83.25px; vertical-align: baseline; \"\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: 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; \"\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: 376.783px 8px; transform-origin: 376.783px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA degree sequence is a list of numbers representing the degrees of vertices in a graph. While it is difficult to tell if a graph can be made from a degree sequence, there are some ways to tell for certain that a graph does not exist with a given degree sequence. One easy first check is the following: \u003c/span\u003e\u003c/span\u003e\u003c/div\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: 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; \"\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: 379.867px 8px; transform-origin: 379.867px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFirst, sort the degree sequence in descending order. Next, pop the first degree off the list and subtract one from the next N elements, where N is the degree you popped off. Repeat until the list is empty. If at any point a degree in the list is less than 0 or if there are not N elements left in the list to subtract from, there is no graph that exists with that degree sequence.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22.5px; 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: 384px 11.25px; text-align: left; transform-origin: 384px 11.25px; 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: 50.4333px 8px; transform-origin: 50.4333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function \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: 37.8px 8px; transform-origin: 37.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 37.8px 8.5px; transform-origin: 37.8px 8.5px; \"\u003eis_graph \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: 259.8px 8px; transform-origin: 259.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ethat returns true if this algorithm results in an empty list or false if it fails at any point.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = is_graph(x)\r\n  % Run algorithm\r\n  y = x;\r\nend","test_suite":"%%\r\nx = [1 1];\r\ny_correct = 1;\r\nassert(isequal(is_graph(x),y_correct))\r\n\r\n%%\r\nx = [1 1 3];\r\ny_correct = 0;\r\nassert(isequal(is_graph(x),y_correct))\r\n\r\n%%\r\nx = [15 3 6];\r\ny_correct = 0;\r\nassert(isequal(is_graph(x),y_correct))\r\n\r\n%%\r\nx = [5 4 1 1 1 1];\r\ny_correct = 0;\r\nassert(isequal(is_graph(x),y_correct))\r\n\r\n%%\r\nx = [5 7 2 2 3 3 2 2];\r\ny_correct = 1;\r\nassert(isequal(is_graph(x),y_correct))\r\n\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":2052130,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":9,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-03-03T20:12:17.000Z","updated_at":"2025-06-25T20:01:28.000Z","published_at":"2022-03-03T20:12:17.000Z","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\u003eA degree sequence is a list of numbers representing the degrees of vertices in a graph. While it is difficult to tell if a graph can be made from a degree sequence, there are some ways to tell for certain that a graph does not exist with a given degree sequence. One easy first check is the following: \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\u003eFirst, sort the degree sequence in descending order. Next, pop the first degree off the list and subtract one from the next N elements, where N is the degree you popped off. Repeat until the list is empty. If at any point a degree in the list is less than 0 or if there are not N elements left in the list to subtract from, there is no graph that exists with that degree sequence.\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 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eis_graph \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ethat returns true if this algorithm results in an empty list or false if it fails at any point.\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\"}]}"},{"id":2732,"title":"Construct a precedence graph from a code segment","description":"A hypothetical MATLAB code segment containing n lines is given in the form of a cell array. The i-th cell contains the i-th line of the code. Each of the lines contains simple arithmetic expressions.\r\n\r\nNow, construct an adjacency matrix of a graph containing n-vertices. The i-th vertex will represent the i-th line of the code. There should be a directed edge from i-th vertex to j-th vertex only if the values generated at i-th line are used in the j-th line.\r\n\r\nAll the variables in the code will have single letter names (e.g.: a,b,x,y etc).\r\n\r\nExample:\r\n\r\n  C = {'a=1;'\r\n       'b=1;'\r\n       'c=a+b;'\r\n       'c=c+1;'};\r\n\r\nHere, the cell array C contains a code segment. The first two lines are independent in the sense that they do not use values generated at any other lines. The third line uses information generated at line 1 and 2. The fourth line uses information generated at line 1,2 and 3.\r\n\r\nThus the resulting adjacency matrix will be as follows:\r\n\r\n  \r\n  mat = [0 0 1 1;\r\n         0 0 1 1;\r\n         0 0 0 1;\r\n         0 0 0 0];\r\n\r\n\r\n\r\n\r\nDefinition of adjacency matrix:\r\n\u003chttp://en.wikipedia.org/wiki/Adjacency_matrix\u003e","description_html":"\u003cp\u003eA hypothetical MATLAB code segment containing n lines is given in the form of a cell array. The i-th cell contains the i-th line of the code. Each of the lines contains simple arithmetic expressions.\u003c/p\u003e\u003cp\u003eNow, construct an adjacency matrix of a graph containing n-vertices. The i-th vertex will represent the i-th line of the code. There should be a directed edge from i-th vertex to j-th vertex only if the values generated at i-th line are used in the j-th line.\u003c/p\u003e\u003cp\u003eAll the variables in the code will have single letter names (e.g.: a,b,x,y etc).\u003c/p\u003e\u003cp\u003eExample:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eC = {'a=1;'\r\n     'b=1;'\r\n     'c=a+b;'\r\n     'c=c+1;'};\r\n\u003c/pre\u003e\u003cp\u003eHere, the cell array C contains a code segment. The first two lines are independent in the sense that they do not use values generated at any other lines. The third line uses information generated at line 1 and 2. The fourth line uses information generated at line 1,2 and 3.\u003c/p\u003e\u003cp\u003eThus the resulting adjacency matrix will be as follows:\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003emat = [0 0 1 1;\r\n       0 0 1 1;\r\n       0 0 0 1;\r\n       0 0 0 0];\r\n\u003c/pre\u003e\u003cp\u003eDefinition of adjacency matrix: \u003ca href = \"http://en.wikipedia.org/wiki/Adjacency_matrix\"\u003ehttp://en.wikipedia.org/wiki/Adjacency_matrix\u003c/a\u003e\u003c/p\u003e","function_template":"function y = pGraph(x)\r\n\r\n\r\n\r\n\r\nend","test_suite":"%%\r\nC = {'a=1;'\r\n     'b=1;'\r\n     'c=a+b;'\r\n     'c=c+1;'};\r\nmat = [0 0 1 1;\r\n       0 0 1 1;\r\n       0 0 0 1;\r\n       0 0 0 0];\r\nassert(isequal(pGraph(C),mat))\r\n\r\n\r\n\r\n%%\r\nC = {'a=1;'\r\n     'a=1;'\r\n     'c=1;'\r\n     'c=1;'};\r\nmat = [0 0 0 0;\r\n       0 0 0 0;\r\n       0 0 0 0;\r\n       0 0 0 0];\r\nassert(isequal(pGraph(C),mat))\r\n\r\n%%\r\nC = {'a=1;'\r\n     'a=1;'\r\n     'c=a+1;'\r\n     'c=a+1;'};\r\nmat = [0 0 0 0;\r\n       0 0 1 1;\r\n       0 0 0 0;\r\n       0 0 0 0];\r\nassert(isequal(pGraph(C),mat))\r\n\r\n%%\r\nC = {'a=1;'\r\n     'b=a+2;'\r\n     'c=b+1;'\r\n     'd=c+1;'};\r\nmat = double(~tril(ones(4)))\r\nassert(isequal(pGraph(C),mat))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":17203,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":"2014-12-06T07:56:27.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-12-06T07:54:40.000Z","updated_at":"2024-11-02T13:28:43.000Z","published_at":"2014-12-06T07:54:40.000Z","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\u003eA hypothetical MATLAB code segment containing n lines is given in the form of a cell array. The i-th cell contains the i-th line of the code. Each of the lines contains simple arithmetic expressions.\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\u003eNow, construct an adjacency matrix of a graph containing n-vertices. The i-th vertex will represent the i-th line of the code. There should be a directed edge from i-th vertex to j-th vertex only if the values generated at i-th line are used in the j-th line.\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\u003eAll the variables in the code will have single letter names (e.g.: a,b,x,y 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\u003c/w:pPr\u003e\u003cw:r\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=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[C = {'a=1;'\\n     'b=1;'\\n     'c=a+b;'\\n     'c=c+1;'};]]\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\u003eHere, the cell array C contains a code segment. The first two lines are independent in the sense that they do not use values generated at any other lines. The third line uses information generated at line 1 and 2. The fourth line uses information generated at line 1,2 and 3.\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\u003eThus the resulting adjacency matrix will be as follows:\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[mat = [0 0 1 1;\\n       0 0 1 1;\\n       0 0 0 1;\\n       0 0 0 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\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDefinition of adjacency matrix:\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/Adjacency_matrix\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehttp://en.wikipedia.org/wiki/Adjacency_matrix\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\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":1886,"title":"Graceful Double Wheel Graph","description":"\u003chttp://en.wikipedia.org/wiki/Graceful_labeling Graceful Graphs\u003e are the topic of the \u003chttp://www.azspcs.net/Contest/GracefulGraphs Primes Graceful Graph Contest\u003e , 21 September 2013 thru 21 December 2013.\r\n\r\nThis Challenge is to create \u003chttp://www.comp.leeds.ac.uk/bms/Graceful/doublewheel.html Graceful Double Wheel Graphs\u003e for various N. A \u003chttp://www.cs.cornell.edu/~lebras/publications/LeBras2013Double.pdf General Algorithm by Le Bras of Cornell\u003e may be helpful, Section 3 for Even/Odd Rings. The Double Wheel Graph produces valid but not Maximum Edge Graceful Graph solutions based upon \u003chttp://oeis.org/A004137 OEIS A004137\u003e.\r\n\r\n*Example:*\r\nOne solution for N=11:\r\n\r\n\u003c\u003chttp://www.comp.leeds.ac.uk/bms/Graceful/Images/2C5+K1.gif\u003e\u003e\r\n\r\nwhich could be answered as [1 3 14 6 19;20 5 17 7 16].\r\n\r\nThere are 20 links and thus the absolute differences between connected nodes must produce values 1 thru 20.  The max node value is equal to the number of links and the min is zero, at the center of the Double Wheel.\r\n\r\n*Input:* N [Total number of Nodes (odd) and N\u003e10 ]\r\n\r\n*Output:* M [ Matrix size [(N-1)/2, 2] of node values where row-1 is outer and row-2 is inner ring ]","description_html":"\u003cp\u003e\u003ca href = \"http://en.wikipedia.org/wiki/Graceful_labeling\"\u003eGraceful Graphs\u003c/a\u003e are the topic of the \u003ca href = \"http://www.azspcs.net/Contest/GracefulGraphs\"\u003ePrimes Graceful Graph Contest\u003c/a\u003e , 21 September 2013 thru 21 December 2013.\u003c/p\u003e\u003cp\u003eThis Challenge is to create \u003ca href = \"http://www.comp.leeds.ac.uk/bms/Graceful/doublewheel.html\"\u003eGraceful Double Wheel Graphs\u003c/a\u003e for various N. A \u003ca href = \"http://www.cs.cornell.edu/~lebras/publications/LeBras2013Double.pdf\"\u003eGeneral Algorithm by Le Bras of Cornell\u003c/a\u003e may be helpful, Section 3 for Even/Odd Rings. The Double Wheel Graph produces valid but not Maximum Edge Graceful Graph solutions based upon \u003ca href = \"http://oeis.org/A004137\"\u003eOEIS A004137\u003c/a\u003e.\u003c/p\u003e\u003cp\u003e\u003cb\u003eExample:\u003c/b\u003e\r\nOne solution for N=11:\u003c/p\u003e\u003cimg src = \"http://www.comp.leeds.ac.uk/bms/Graceful/Images/2C5+K1.gif\"\u003e\u003cp\u003ewhich could be answered as [1 3 14 6 19;20 5 17 7 16].\u003c/p\u003e\u003cp\u003eThere are 20 links and thus the absolute differences between connected nodes must produce values 1 thru 20.  The max node value is equal to the number of links and the min is zero, at the center of the Double Wheel.\u003c/p\u003e\u003cp\u003e\u003cb\u003eInput:\u003c/b\u003e N [Total number of Nodes (odd) and N\u003e10 ]\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput:\u003c/b\u003e M [ Matrix size [(N-1)/2, 2] of node values where row-1 is outer and row-2 is inner ring ]\u003c/p\u003e","function_template":"function m=double_wheel(n)\r\n  m=[];\r\nend","test_suite":"%%\r\ntic\r\nn=11;\r\nm=double_wheel(n);\r\nms=circshift(m,[0 -1]);\r\ndm=m-ms;\r\nd=unique([m(:) abs(dm(:))]);\r\nassert(all(diff(d)==1))\r\nassert(length(d)==2*(n-1))\r\nassert(max(d)==2*(n-1))\r\ntoc\r\n%%\r\nn=13;\r\nm=double_wheel(n);\r\nms=circshift(m,[0 -1]);\r\ndm=m-ms;\r\nd=unique([m(:) abs(dm(:))]);\r\nassert(all(diff(d)==1))\r\nassert(length(d)==2*(n-1))\r\nassert(max(d)==2*(n-1))\r\ntoc\r\n%%\r\nn=17;\r\nm=double_wheel(n);\r\nms=circshift(m,[0 -1]);\r\ndm=m-ms;\r\nd=unique([m(:) abs(dm(:))]);\r\nassert(all(diff(d)==1))\r\nassert(length(d)==2*(n-1))\r\nassert(max(d)==2*(n-1))\r\ntoc\r\n%%\r\nn=19;\r\nm=double_wheel(n);\r\nms=circshift(m,[0 -1]);\r\ndm=m-ms;\r\nd=unique([m(:) abs(dm(:))]);\r\nassert(all(diff(d)==1))\r\nassert(length(d)==2*(n-1))\r\nassert(max(d)==2*(n-1))\r\ntoc\r\n%%\r\nn=71;\r\nm=double_wheel(n);\r\nms=circshift(m,[0 -1]);\r\ndm=m-ms;\r\nd=unique([m(:) abs(dm(:))]);\r\nassert(all(diff(d)==1))\r\nassert(length(d)==2*(n-1))\r\nassert(max(d)==2*(n-1))\r\ntoc\r\n%%\r\nn=97;\r\nm=double_wheel(n);\r\nms=circshift(m,[0 -1]);\r\ndm=m-ms;\r\nd=unique([m(:) abs(dm(:))]);\r\nassert(all(diff(d)==1))\r\nassert(length(d)==2*(n-1))\r\nassert(max(d)==2*(n-1))\r\ntoc\r\n\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":3097,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-09-21T23:15:03.000Z","updated_at":"2013-09-22T01:16:42.000Z","published_at":"2013-09-22T01:16:42.000Z","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\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.gif\"}],\"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:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Graceful_labeling\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eGraceful Graphs\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e are the topic of the\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://www.azspcs.net/Contest/GracefulGraphs\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrimes Graceful Graph Contest\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e , 21 September 2013 thru 21 December 2013.\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\u003eThis Challenge is to create\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://www.comp.leeds.ac.uk/bms/Graceful/doublewheel.html\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eGraceful Double Wheel Graphs\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e for various N. 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://www.cs.cornell.edu/~lebras/publications/LeBras2013Double.pdf\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eGeneral Algorithm by Le Bras of Cornell\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e may be helpful, Section 3 for Even/Odd Rings. The Double Wheel Graph produces valid but not Maximum Edge Graceful Graph solutions based upon\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://oeis.org/A004137\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eOEIS A004137\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\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:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e One solution for N=11:\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: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=\\\"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\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhich could be answered as [1 3 14 6 19;20 5 17 7 16].\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\u003eThere are 20 links and thus the absolute differences between connected nodes must produce values 1 thru 20. The max node value is equal to the number of links and the min is zero, at the center of the Double Wheel.\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:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e N [Total number of Nodes (odd) and N\u0026gt;10 ]\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:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e M [ Matrix size [(N-1)/2, 2] of node values where row-1 is outer and row-2 is inner ring ]\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\"},{\"partUri\":\"/media/image1.gif\",\"contentType\":\"image/gif\",\"content\":\"data:image/gif;base64,<!DOCTYPE html>
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Saturday 10 October 2020, 10:00 - 16:00                                            </p>

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\"}]}"},{"id":1887,"title":"Graceful Graph: Wichmann Rulers","description":"This Challenge is to find maximum size Graceful Graphs via Wichmann Rulers for P\u003e13.  This Challenge is related to the \u003chttp://www.azspcs.net/Contest/GracefulGraphs Graceful Graph Contest\u003e which Rokicki completed in 97 minutes. The Wichmann Conjecture is that no larger solutions exist for P\u003e13.\r\n\r\nAn Optimal ruler is defined as having end points at 0 and Max with P-2 integer points between [0,Max] such that the distances 1 thru Max exist by deltas between points.\r\nAn \u003chttp://oeis.org/A193802 Optimal Wichmann Ruler\u003e readily creates solutions that can be tested for number of points and existence of all expected deltas.\r\n\r\nThe Wichmann difference vector is [Q(1,r), r+1, Q(2r+1,r), Q(4r+3,s), Q(2r+2,r+1), Q(1,r)] where Q(a,b) is b a's, e.g. Q(2,3) is [2 2 2]. The max value is L=4r(r+s+2)+3(s+1) for Points P=4r+s+3, (r and s \u003e=0 and integer).\r\n\r\nFor W(r,s), W(2,3) creates the difference sequence [1 1 3 5 5 11 11 11 6 6 6 1 1]. The points on the ruler are the cumsum of W with a zero pre-pended to produce S=[0 1 2 5 10 15 26 37 48 54 60 66 67 68], P=14. All deltas from 1 thru 68 can be realized.\r\n\r\n*Input:* P  (Number of Points on the ruler)\r\n\r\n*Output:* S (Vector of length P of locations on the ruler, 0 thru Max Value and can generate all deltas 1:S(end))\r\n\r\n*Notes:*\r\n\r\n  1) A W(r,s) does not guarantee all deltas can be generated\r\n  2) For any P there are multiple W(r,s) solutions \r\n  3) P=5 solution is 9, readily solved by brute force\r\n  4) P=13 Wichmann is 57 but the best solution is 58. Too big for brute force\r\n  5) Create Connectivity Graph for Cases, like Final Matlab Competition, for Fun ","description_html":"\u003cp\u003eThis Challenge is to find maximum size Graceful Graphs via Wichmann Rulers for P\u003e13.  This Challenge is related to the \u003ca href = \"http://www.azspcs.net/Contest/GracefulGraphs\"\u003eGraceful Graph Contest\u003c/a\u003e which Rokicki completed in 97 minutes. The Wichmann Conjecture is that no larger solutions exist for P\u003e13.\u003c/p\u003e\u003cp\u003eAn Optimal ruler is defined as having end points at 0 and Max with P-2 integer points between [0,Max] such that the distances 1 thru Max exist by deltas between points.\r\nAn \u003ca href = \"http://oeis.org/A193802\"\u003eOptimal Wichmann Ruler\u003c/a\u003e readily creates solutions that can be tested for number of points and existence of all expected deltas.\u003c/p\u003e\u003cp\u003eThe Wichmann difference vector is [Q(1,r), r+1, Q(2r+1,r), Q(4r+3,s), Q(2r+2,r+1), Q(1,r)] where Q(a,b) is b a's, e.g. Q(2,3) is [2 2 2]. The max value is L=4r(r+s+2)+3(s+1) for Points P=4r+s+3, (r and s \u003e=0 and integer).\u003c/p\u003e\u003cp\u003eFor W(r,s), W(2,3) creates the difference sequence [1 1 3 5 5 11 11 11 6 6 6 1 1]. The points on the ruler are the cumsum of W with a zero pre-pended to produce S=[0 1 2 5 10 15 26 37 48 54 60 66 67 68], P=14. All deltas from 1 thru 68 can be realized.\u003c/p\u003e\u003cp\u003e\u003cb\u003eInput:\u003c/b\u003e P  (Number of Points on the ruler)\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput:\u003c/b\u003e S (Vector of length P of locations on the ruler, 0 thru Max Value and can generate all deltas 1:S(end))\u003c/p\u003e\u003cp\u003e\u003cb\u003eNotes:\u003c/b\u003e\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003e1) A W(r,s) does not guarantee all deltas can be generated\r\n2) For any P there are multiple W(r,s) solutions \r\n3) P=5 solution is 9, readily solved by brute force\r\n4) P=13 Wichmann is 57 but the best solution is 58. Too big for brute force\r\n5) Create Connectivity Graph for Cases, like Final Matlab Competition, for Fun \r\n\u003c/pre\u003e","function_template":"function s=Graceful_Wichmann(n)\r\n  s=0;\r\nend","test_suite":"%%\r\ntic\r\nn=17;\r\nexp=101;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=19;\r\nexp=123;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=23;\r\nexp=183;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=29;\r\nexp=289;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=31;\r\nexp=327;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=37;\r\nexp=465;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=41;\r\nexp=573;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=43;\r\nexp=627;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=47;\r\nexp=751;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=53;\r\nexp=953;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=59;\r\nexp=1179;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=61;\r\nexp=1257;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=67;\r\nexp=1515;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=71;\r\nexp=1703;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=73;\r\nexp=1797;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=79;\r\nexp=2103;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta 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unique\r\ntoc","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":3097,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-09-23T01:30:25.000Z","updated_at":"2013-09-23T13:04:40.000Z","published_at":"2013-09-23T04:00:18.000Z","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\u003eThis Challenge is to find maximum size Graceful Graphs via Wichmann Rulers for P\u0026gt;13. This Challenge is related to the\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://www.azspcs.net/Contest/GracefulGraphs\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eGraceful Graph Contest\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e which Rokicki completed in 97 minutes. The Wichmann Conjecture is that no larger solutions exist for P\u0026gt;13.\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\u003eAn Optimal ruler is defined as having end points at 0 and Max with P-2 integer points between [0,Max] such that the distances 1 thru Max exist by deltas between points. An\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://oeis.org/A193802\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eOptimal Wichmann Ruler\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e readily creates solutions that can be tested for number of points and existence of all expected deltas.\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\u003eThe Wichmann difference vector is [Q(1,r), r+1, Q(2r+1,r), Q(4r+3,s), Q(2r+2,r+1), Q(1,r)] where Q(a,b) is b a's, e.g. Q(2,3) is [2 2 2]. The max value is L=4r(r+s+2)+3(s+1) for Points P=4r+s+3, (r and s \u0026gt;=0 and integer).\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 W(r,s), W(2,3) creates the difference sequence [1 1 3 5 5 11 11 11 6 6 6 1 1]. The points on the ruler are the cumsum of W with a zero pre-pended to produce S=[0 1 2 5 10 15 26 37 48 54 60 66 67 68], P=14. All deltas from 1 thru 68 can be realized.\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:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e P (Number of Points on the ruler)\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:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e S (Vector of length P of locations on the ruler, 0 thru Max Value and can generate all deltas 1:S(end))\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:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNotes:\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[1) A W(r,s) does not guarantee all deltas can be generated\\n2) For any P there are multiple W(r,s) solutions \\n3) P=5 solution is 9, readily solved by brute force\\n4) P=13 Wichmann is 57 but the best solution is 58. 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