Examples of Solutions found by Simulated Annealing Hugo Pfoertner 2024-02-01 a(2) = 290 Trace(A) = 5: [1, 2; | 3 3, 4] | 7 _____ 4 6 Sums ----------- a(3)= 12761 Trace(A) = 16: [1, 2, 5; | 8 3, 6, 7; | 16 4, 8, 9] | 21 ________ 8 16 21 Sums ------------- a(4) = 203336 Trace(A) = 38: [1, 2, 5, 6; | 14 3, 8, 10, 11; | 32 4, 9, 13, 15; | 41 7, 12, 14, 16] | 49 ______________ 15 31 42 48 Sums [1, 2, 4, 7; | 14 3, 8, 9, 12; | 32 5, 10, 13, 14; | 42 6, 11, 15, 16] | 48 ______________ 15 31 41 49 Sums -------------- a(5) = 1797702 Trace(A) = 71: [ 1, 2, 5, 8, 9; | 25 3, 6, 11, 13, 16; | 49 4, 12, 17, 18, 21; | 72 7, 14, 19, 22, 23; | 85 10, 15, 20, 24, 25] | 94 ___________________ 25 49 72 85 94 Sums --------------- a(6) = 10816592 Trace(A) = 121: [ 1, 2, 4, 8, 9, 12; | 36 3, 6, 13, 16, 17, 21; | 76 5, 14, 19, 23, 24, 27; | 112 7, 15, 22, 26, 30, 32; | 132 10, 18, 25, 29, 33, 34; | 149 11, 20, 28, 31, 35, 36] | 161 _______________________ 37 75 111 133 148 162 Sums [ 1, 2, 5, 7, 9, 12; | 36 3, 6, 14, 15, 17, 21; | 76 4, 13, 19, 22, 25, 28; | 111 8, 16, 23, 26, 29, 31; | 133 10, 18, 24, 30, 33, 34; | 149 11, 20, 27, 32, 35, 36] | 161 _______________________ 37 75 112 132 148 162 Sums --------------- a(7) = 49651016 Trace(A) = 192 [ 1, 2, 5, 7, 10, 14, 15; | 54 3, 6, 12, 17, 21, 22, 24; | 105 4, 11, 19, 27, 28, 31, 34; | 154 8, 18, 26, 32, 35, 38, 40; | 197 9, 20, 29, 36, 39, 42, 45; | 220 13, 23, 30, 37, 43, 46, 48; | 240 16, 25, 33, 41, 44, 47, 49] | 255 ___________________________ 54 105 154 197 220 240 255 Sums [ 1, 2, 5, 8, 9, 13, 16; | 54 3, 6, 12, 18, 20, 22, 24; | 105 4, 11, 19, 27, 29, 30, 34; | 154 7, 17, 26, 32, 36, 38, 41; | 197 10, 21, 28, 35, 39, 43, 44; | 220 14, 23, 31, 37, 42, 46, 47; | 240 15, 25, 33, 40, 45, 48, 49] | 255 ___________________________ 54 105 154 197 220 240 255 Sums ---------------- a(8) = 186570817 Trace(A) = 286: [ 1, 2, 4, 8, 10, 13, 16, 17; | 71 3, 6, 11, 20, 23, 24, 29, 30; | 146 5, 12, 21, 27, 33, 35, 37, 39; | 209 7, 19, 26, 34, 42, 43, 47, 49; | 267 9, 22, 32, 41, 45, 51, 53, 56; | 309 14, 25, 36, 44, 50, 54, 57, 59; | 339 15, 28, 38, 46, 52, 58, 61, 63; | 361 18, 31, 40, 48, 55, 60, 62, 64] | 378 _______________________________ 72 145 208 268 310 338 362 377 Sums ---------------- a(9) = 601081822 Trace(A) = 409: [ 1, 2, 4, 8, 12, 13, 17, 21, 23; | 101 3, 6, 9, 15, 25, 26, 30, 33, 35; | 182 5, 10, 19, 28, 37, 39, 41, 46, 47; | 272 7, 16, 29, 38, 44, 49, 51, 55, 56; | 345 11, 24, 36, 43, 53, 58, 61, 64, 68; | 418 * 14, 27, 40, 50, 59, 62, 65, 69, 72; | 458 * 18, 31, 42, 52, 60, 66, 71, 74, 76; | 490 * 20, 32, 45, 54, 63, 70, 75, 78, 80; | 517 * 22, 34, 48, 57, 67, 73, 77, 79, 81] | 538 ___________________________________ 101 182 272 345 420 456 488 519 538 Sums * * * * [ 1, 2, 4, 8, 11, 14, 18, 20, 23; | 101 3, 6, 10, 15, 24, 27, 31, 32, 34; | 182 5, 9, 19, 28, 36, 40, 42, 45, 48; | 272 7, 16, 29, 38, 43, 50, 52, 54, 56; | 345 12, 25, 37, 44, 53, 59, 60, 63, 67; | 420 * 13, 26, 39, 49, 58, 62, 66, 70, 73; | 456 * 17, 30, 41, 51, 61, 65, 71, 75, 77; | 488 * 21, 33, 46, 55, 64, 69, 74, 78, 79; | 519 * 22, 35, 47, 57, 68, 72, 76, 80, 81] | 538 ___________________________________ 101 182 272 345 418 458 490 517 538 Sums * * * * [ 1, 2, 5, 7, 11, 14, 18, 20, 23; | 101 3, 6, 9, 15, 24, 27, 31, 32, 35; | 182 4, 10, 19, 29, 36, 40, 42, 45, 47; | 272 8, 16, 28, 38, 43, 50, 52, 54, 56; | 345 12, 25, 37, 44, 53, 59, 60, 63, 67; | 420 * 13, 26, 39, 49, 58, 62, 66, 70, 73; | 456 * 17, 30, 41, 51, 61, 65, 71, 75, 77; | 488 * 21, 33, 46, 55, 64, 69, 74, 78, 79; | 519 * 22, 34, 48, 57, 68, 72, 76, 80, 81] | 538 ___________________________________ 101 182 272 345 418 458 490 517 538 Sums * * * * ------------------- a(10) >= 1714111353 Trace(A) = 557: [ 1, 2, 5, 7, 11, 14, 19, 20, 24, 27; | 130 3, 6, 10, 15, 22, 31, 33, 36, 39, 42; | 237 4, 9, 17, 28, 34, 44, 46, 50, 51, 56; | 339 8, 16, 29, 38, 47, 54, 59, 60, 64, 66; | 441 12, 23, 35, 48, 57, 63, 68, 71, 73, 77; | 527 13, 30, 43, 53, 62, 70, 76, 80, 83, 87; | 597 18, 32, 45, 58, 69, 75, 81, 85, 89, 92; | 644 21, 37, 49, 61, 72, 79, 84, 90, 94, 95; | 682 25, 40, 52, 65, 74, 82, 88, 93, 97, 98; | 714 26, 41, 55, 67, 78, 86, 91, 96, 99,100] | 739 ______________________________________ 131 236 340 440 526 598 645 681 713 740 Sums