六边形上19个黑点上不同数字求所有的解！

王守恩

wayne 发表于 2019-5-12 14:19
受到mathe的手工求解的启发。我也写了一个程序, 耗时的地方主要是整数的拆分，代码暂时不能上传，被论坛的 ...

谢谢wayne！ 可以改一下吗？改成这样：
六个顶点按从小到大排列：1号位是最小的，最小数的两边总有一个数是比较小的，摆2号位。或
六个顶点按从大到小排列：1号位是最大的，最大数的两边总有一个数是比较大的，摆2号位。
3,4,5,6号位随着1,2号位的方向走，这样就不存在 “逆(顺)时针”问题，答案就是唯一的了。

wayne

恩，稍微改改，六个顶点的最小数排在1号位，与之相邻的两个数中最小的排在2号位，最大的排在末位。

1. ans=Flatten[Table[Select[Flatten[Table[Table[Table[{tmp={s[[1]],k,S-2MovingAverage[PadRight[k,7,First[k]],2],S-s[[1]]-k};{S,s[[1]],k},Union[Flatten[tmp]]},{k,Flatten[Table[Flatten[{c[[1;;2]],cc,c[[3]]}],{c,Flatten[{First[t],#}]&/@Sort/@Subsets[Rest[t],{2}]},{cc,Permutations[Complement[t,c]]}],1]}],{t,s[[2]]}],{s,Table[{2i,Select[Sort/@IntegerPartitions[6S-Binomial[20,2]/2-5i,{6}],Length[Union[#]]==6 &&Not@MemberQ[#,2i]&&Max[#]<20&]},{i,(6S-Binomial[20,2]/2-Binomial[7,2])/5}]}],2],Length[Last[#]]==19&&Min[Last[#]]==1&][[All,1]],{S,22,31}],1]

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1. {22,4}
   
2. {23,2}
   
3. {24,6}
   
4. {25,6}
   
5. {26,4}
   
6. {27,11}
   
7. {28,5}
   
8. {29,8}
   
9. {31,8}
   
10. {32,5}
    
11. {33,11}
    
12. {34,4}
    
13. {35,6}
    
14. {36,6}
    
15. {37,2}
    
16. {38,4}

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1. {22,2,{1,3,5,7,4,12}}
   
2. {22,2,{1,3,5,11,4,8}}
   
3. {22,2,{1,3,4,8,9,7}}
   
4. {22,4,{1,2,6,5,10,3}}
   
5. {23,6,{1,3,8,10,2,4}}
   
6. {23,6,{1,3,2,4,8,10}}
   
7. {24,2,{3,5,9,4,8,15}}
   
8. {24,4,{2,3,11,12,5,6}}
   
9. {24,4,{2,3,5,6,11,12}}
   
10. {24,6,{1,4,2,13,3,11}}
    
11. {24,6,{1,11,4,2,3,13}}
    
12. {24,8,{1,5,12,3,2,6}}
    
13. {25,2,{4,8,7,17,5,9}}
    
14. {25,2,{4,8,16,6,5,11}}
    
15. {25,4,{2,6,10,3,8,16}}
    
16. {25,4,{2,5,8,14,10,6}}
    
17. {25,8,{1,5,7,14,2,6}}
    
18. {25,8,{1,6,2,4,12,10}}
    
19. {26,4,{3,5,10,9,16,8}}
    
20. {26,8,{1,6,2,13,4,15}}
    
21. {26,8,{2,5,11,14,3,6}}
    
22. {26,8,{2,5,3,6,11,14}}
    
23. {27,2,{6,7,17,9,13,10}}
    
24. {27,6,{3,8,4,11,7,19}}
    
25. {27,6,{3,9,10,4,7,19}}
    
26. {27,6,{4,5,19,7,9,8}}
    
27. {27,8,{2,6,18,5,7,9}}
    
28. {27,8,{3,5,7,9,17,6}}
    
29. {27,8,{1,7,6,16,2,15}}
    
30. {27,8,{1,15,7,6,2,16}}
    
31. {27,8,{1,14,6,2,9,15}}
    
32. {27,14,{1,7,4,10,2,8}}
    
33. {27,14,{1,7,2,8,4,10}}
    
34. {28,4,{5,6,15,10,16,11}}
    
35. {28,4,{5,6,10,15,11,16}}
    
36. {28,8,{1,14,4,17,2,15}}
    
37. {28,8,{1,14,2,15,4,17}}
    
38. {28,10,{2,7,3,17,5,14}}
    
39. {29,2,{8,14,11,17,9,15}}
    
40. {29,2,{8,14,9,15,11,17}}
    
41. {29,4,{6,10,18,8,16,11}}
    
42. {29,10,{2,13,11,3,7,18}}
    
43. {29,10,{1,12,14,6,4,17}}
    
44. {29,12,{3,7,4,8,16,11}}
    
45. {29,16,{2,10,5,6,4,12}}
    
46. {29,16,{1,9,7,8,3,11}}
    
47. {31,4,{9,17,12,13,11,19}}
    
48. {31,4,{8,16,14,15,10,18}}
    
49. {31,8,{4,9,17,13,16,12}}
    
50. {31,10,{3,16,14,6,8,19}}
    
51. {31,10,{2,13,17,9,7,18}}
    
52. {31,16,{2,10,14,9,4,12}}
    
53. {31,18,{3,9,6,12,5,11}}
    
54. {31,18,{3,9,5,11,6,12}}
    
55. {32,10,{3,15,6,18,13,17}}
    
56. {32,12,{3,16,6,19,5,18}}
    
57. {32,12,{3,16,5,18,6,19}}
    
58. {32,16,{4,9,15,14,5,10}}
    
59. {32,16,{4,9,5,10,14,15}}
    
60. {33,6,{10,16,13,19,12,18}}
    
61. {33,6,{10,16,12,18,13,19}}
    
62. {33,12,{4,14,13,19,5,18}}
    
63. {33,12,{4,18,14,13,5,19}}
    
64. {33,12,{5,11,18,14,6,19}}
    
65. {33,12,{2,14,18,11,13,15}}
    
66. {33,12,{3,11,13,15,17,14}}
    
67. {33,14,{1,13,9,16,12,17}}
    
68. {33,14,{1,13,16,10,11,17}}
    
69. {33,14,{1,13,11,12,16,15}}
    
70. {33,18,{3,11,7,10,14,13}}
    
71. {34,12,{5,16,7,18,14,19}}
    
72. {34,12,{6,9,15,18,14,17}}
    
73. {34,12,{6,9,14,17,15,18}}
    
74. {34,16,{4,11,10,15,17,12}}
    
75. {35,12,{8,10,19,14,18,16}}
    
76. {35,12,{6,13,15,19,14,18}}
    
77. {35,16,{4,12,17,10,14,18}}
    
78. {35,16,{6,10,14,18,15,12}}
    
79. {35,18,{4,12,16,9,15,14}}
    
80. {35,18,{3,13,12,16,11,15}}
    
81. {36,12,{8,15,19,14,18,17}}
    
82. {36,14,{7,17,9,19,16,18}}
    
83. {36,14,{7,17,18,16,9,19}}
    
84. {36,16,{8,9,17,18,14,15}}
    
85. {36,16,{8,9,14,15,17,18}}
    
86. {36,18,{5,12,16,11,15,17}}
    
87. {37,14,{10,12,17,19,16,18}}
    
88. {37,14,{10,12,16,18,17,19}}
    
89. {38,16,{10,15,14,18,19,17}}
    
90. {38,18,{8,16,13,15,17,19}}
    
91. {38,18,{9,15,17,19,12,16}}
    
92. {38,18,{11,12,16,17,19,13}}

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markfang2050

northwolves 发表于 2019-5-12 11:00
sum=[22-29,31-38]有解

大佬！这个比较漂亮！何种语言写的？

markfang2050

期待C或python写的代码

markfang2050

markfang2050 发表于 2019-5-12 18:35
期待C或python写的代码

zeroieme

wayne 发表于 2019-5-12 18:04
恩，稍微改改，六个顶点的最小数排在1号位，与之相邻的两个数中最小的排在2号位，最大的排在末位。

能直接输出图吗？我想了很久。

hujunhua

如果两数之和等于20，就称为彼此的互补数。
显然，互补替换保持这种异形幻六方的等和性，只是定和（幻方数）有变化，前后幻方数之和等于60。
可见幻方数之和为60的幻方是一一对应的，故实际上只需要搜索幻方数在30以内的幻方。

wayne

恩恩，很有道理，反正程序就只需改一个数字，，运算时间也只是线性的增加，计算结果刚好也确实是对称的，互相印证～

wayne

zeroieme 发表于 2019-5-12 19:16
能直接输出图吗？我想了很久。

先借用前面的代码，算出解

1. ans=Flatten[Table[Select[Flatten[Table[Table[Table[{tmp={s[[1]],k,S-2MovingAverage[PadRight[k,7,First[k]],2],S-s[[1]]-k};{S,s[[1]],k},Union[Flatten[tmp]]},{k,Flatten[Table[Flatten[{c[[1;;2]],cc,c[[3]]}],{c,Flatten[{First[t],#}]&/@Sort/@Subsets[Rest[t],{2}]},{cc,Permutations[Complement[t,c]]}],1]}],{t,s[[2]]}],{s,Table[{2i,Select[Sort/@IntegerPartitions[6S-Binomial[20,2]/2-5i,{6}],Length[Union[#]]==6 &&Not@MemberQ[#,2i]&&Max[#]<20&]},{i,(6S-Binomial[20,2]/2-Binomial[7,2])/5}]}],2],Length[Last[#]]==19&&Min[Last[#]]==1&][[All,1]],{S,22,31}],1]
   

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再画图：

1. Table[mid=Function[{a,b},{t[[1]]-a[[1]]-b[[1]],Mean[{a[[2]],b[[2]]}]}];
   
2. vtx=Table[{If[k==0,t[[3,-1]],t[[3,k]]],{Cos[(2Pi)/6 k],Sin[(2Pi)/6 k]}},{k,0,6}];
   
3. origin={t[[2]],{0,0}};
   
4. others=Join[Table[mid@@m,{m,Partition[vtx,2,1]}],Table[mid[origin,m],{m,Rest[vtx]}]];
   
5. all=Join[Rest[vtx],others,{origin}];
   
6. Graphics[{Thickness[0.01],Magenta,Line[vtx[[All,2]]],Line/@(Partition[Rest[vtx[[All,2]]],4,1][[All,{1,4}]]),Blue,PointSize[.1],Point[all[[All,2]]],Yellow,Text[Style[#[[1]],FontSize->20],#[[2]]]&/@all},ImageSize->500,PlotLabel->Style[Framed[t[[1]]],30,Red,Background->Yellow],Background->RGBColor["#eedddd"]],{t,ans}]

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markfang2050

Table[mid =
   Function[{a, b}, {t[[1]] - a[[1]] - b[[1]],
     Mean[{a[[2]], b[[2]]}]}];