Problems & Puzzles: Puzzles

 Puzzle 717 Associative squares Stanley Natalia Makarova sent the following puzzle Definition 1 Stanley antimagic square of index d and order n is an n X n matrix where the sum of any n matrix elements in pairwise distinct rows and pairwise distinct columns equals d. (see http://www.primepuzzles.net/puzzles/puzz_681.htm ) Definition 2 Stanley square of order n to the index d is associative, if the sum of any two elements symmetrical with respect to the center of the square is the same number k. The number k is called the constant associativity. The index d of the square and constant associativity k related as follows: k = 2d/n. Here shows minimal associative squares Stanley orders n = 2 - 8: n=2, d=16, k=16 3 5 11 13 n=3, d=177, k=118 5 17 29 47 59 71 89 101 113 n=4, d=240, k=120 7 13 31 37 17 23 41 47 73 79 97 103 83 89 107 113 n=5,  d=3505, k=1402 41 101 491 881 941 113 173 563 953 1013 251 311 701 1091 1151 389 449 839 1229 1289 461 521 911 1301 1361 n=6, d=2520, k=840 17 53 83 101 131 167 113 149 179 197 227 263 317 353 383 401 431 467 373 409 439 457 487 523 577 613 643 661 691 727 673 709 739 757 787 823 n=7, d=135023, k=38578 389 1181 3701 5009 6317 8837 9629 3449 4241 6761 8069 9377 11897 12689 13109 13901 16421 17729 19037 21557 22349 14669 15461 17981 19289 20597 23117 23909 16229 17021 19541 20849 22157 24677 25469 25889 26681 29201 30509 31817 34337 35129 28949 29741 32261 33569 34877 37397 38189 n=8, d=24024, k=6006 19 83 1019 1583 3229 3793 4729 4793 103 167 1103 1667 3313 3877 4813 4877 499 563 1499 2063 3709 4273 5209 5273 523 587 1523 2087 3733 4297 5233 5297 709 773 1709 2273 3919 4483 5419 5483 733 797 1733 2297 3943 4507 5443 5507 1129 1193 2129 2693 4339 4903 5839 5903 1213 1277 2213 2777 4423 4987 5923 5987 Q1. Find the minimal square of order 8. Q2. Find the minimal or best solutions for n>8.

Natalia Makarova sent the following correction

A colleague told me about a error in the decision:

n=7, d=135023, k=38578

389 1181 3701 5009 6317 8837 9629
3449 4241 6761 8069 9377 11897 12689
13109 13901 16421 17729 19037 21557 22349
14669 15461 17981 19289 20597 23117 23909
16229 17021 19541 20849 22157 24677 25469
25889 26681 29201 30509 31817 34337 35129
28949 29741 32261 33569 34877 37397 38189

Here the number 20597 is not prime. In my program was a mistake. Sorry.
I found another solution:

n=7, d=181321, k=51806

5857  6793  7717  7753  7789  8713  9649
11827  12763  13687  13723  13759  14683  15619
17377  18313  19237  19273  19309  20233  21169
24007  24943  25867  25903  25939  26863  27799
30637  31573  32497  32533  32569  33493  34429
36187  37123  38047  38083  38119  39043  39979
42157  43093  44017  44053  44089  45013  45949

I'm not sure that this solution is minimal.

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