16th-order non-normal trimagic squares

In taking some freedom with the strict definition of a magic square, we have to point out the results of an American, David M. Collison, which communicates in 1991 to John R. Hendricks a trimagic square of order-16.

 1160 1189 539 496 672 695 57 10 11 58 631 654 515 558 1123 1152 531 560 675 632 43 66 1179 1132 1133 1180 2 25 651 694 494 523 1155 1089 422 379 831 767 92 45 91 44 790 808 403 360 1118 1126 832 766 99 56 1154 1090 415 368 414 367 1113 1131 80 37 795 803 1106 1135 411 454 716 739 27 74 75 28 757 780 473 430 1143 1172 409 438 717 760 19 42 1115 1162 1163 1116 60 83 779 736 446 475 999 1007 192 235 977 995 164 211 163 210 1018 954 173 216 1036 970 982 990 175 218 994 1012 181 228 180 227 1035 971 156 199 1019 953 183 191 991 1034 195 213 963 1010 962 1009 236 172 972 1015 220 154 200 208 974 1017 178 196 980 1027 979 1026 219 155 955 998 237 171 715 744 20 63 1107 1130 418 465 466 419 1148 1171 82 39 752 781 18 47 1108 1151 410 433 724 771 772 725 451 474 1170 1127 55 84 101 35 1153 1110 423 359 823 776 822 775 382 400 1134 1091 64 72 424 358 830 787 100 36 1146 1099 1145 1098 59 77 811 768 387 395 667 696 46 3 1165 1188 550 503 504 551 1124 1147 22 65 630 659 38 67 1168 1125 536 559 686 639 640 687 495 518 1144 1187 1 30

S1 = 9,520. S2 = 8,228,000. S3 = 7,946,344,000.

His freedom was to have not taken consecutive numbers. The 16x16 = 256 used numbers are between 1 and 1,189, so there are numerous gaps. For example, no number between 237 and 358 was used. In spite of that freedom, hat's off to the artist!

Download the 16th-order non-normal trimagic square of David Collison, Excel file 69Kb.

Jacques Guéron (France), has communicated us in May 2002 a 16th-order non-normal trimagic square that he had constructed in 1987. Like the David Collison's square, he has taken some freedoms with the definition of the magic squares, but different freedoms: the numbers from 0 to 63 are here placed 4 times in his square. In fact, it is the same 8th-order square, more some rotations, placed in the 4 quarters of the 16th-order square.

 4 11 29 18 56 55 33 46 17 30 8 7 45 34 52 59 40 39 49 62 20 27 13 2 61 50 36 43 1 14 24 23 15 0 22 25 51 60 42 37 26 21 3 12 38 41 63 48 35 44 58 53 31 16 6 9 54 57 47 32 10 5 19 28 54 57 47 32 10 5 19 28 35 44 58 53 31 16 6 9 26 21 3 12 38 41 63 48 15 0 22 25 51 60 42 37 61 50 36 43 1 14 24 23 40 39 49 62 20 27 13 2 17 30 8 7 45 34 52 59 4 11 29 18 56 55 33 46 46 33 55 56 18 29 11 4 59 52 34 45 7 8 30 17 2 13 27 20 62 49 39 40 23 24 14 1 43 36 50 61 37 42 60 51 25 22 0 15 48 63 41 38 12 3 21 26 9 6 16 31 53 58 44 35 28 19 5 10 32 47 57 54 28 19 5 10 32 47 57 54 9 6 16 31 53 58 44 35 48 63 41 38 12 3 21 26 37 42 60 51 25 22 0 15 23 24 14 1 43 36 50 61 2 13 27 20 62 49 39 40 59 52 34 45 7 8 30 17 46 33 55 56 18 29 11 4

S1 = 504, S2 = 21 336, S3 = 1 016 064.

Download the 16th-order non-normal trimagic square of Jacques Guéron, Excel file 44Kb.