{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","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":"2025-12-14T00:00:00.000Z","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":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":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. 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