Multimagic Cubes

The previous record ( and only other bimagic cube?) was an
order 25 by John Hendricks in June, 2000. On Jan. 23, Christian presented another bimagic cube. This
one contains 54 order 16 magic squares, the most possible for an order 16 cube! I have a similar page that illustrates multimagic squares. Featured are Christian Boyer’s Quadra and Pentamagic squares and Walter Trumps amazing (for its small size) order 12 trimagic square. Christian Boyer has an excellent site on multimagic squares and cubes.
Virtually nothing is known about Pfeffermann except that he published a number of magic squares, mostly between 1890 and 1896. The first bimagic square believed published, was by him in Les Tablettes du Chercheur  Journal de Jeux d'Esprit et de Combinaisons, number 2 of January 15, 1891. This Pfeffermann bimagic square
[1] was converted into a magic cube by A.
Huber [2]. If the parts 1, 2, 3 and 4 (or 2, 1, 4, 3, and so on) are
stacked, then we get a magic CUBE (but not bimagic).
[1]The Pfeffermann bimagic square above
was a problem published in "Revue des Jeux", June 26th, 1891. Collison 1990 Simple, associated This order 5 cube by David Collison [1] of Anaheim, CA is ALMOST bimagic! The cube is associated so the three central orthogonal planes and 1 oblique plane are simple magic squares. Pantriagonals in 3 of the 4 directions are correct. The magic sum (S_{1}) is 315 as expected. The squares of the numbers do not form a magic cube
because most lines do not sum to 26355. However, the total of the 25 cells in
each orthogonal plane (and 3 of the 6 oblique planes) sum to 5 times S_{2} or
131775. This cube is associated. It is not pantriagonal because only the pantriagonals in 3 of the 4 directions are correct. Three central orthogonal planes and 1 oblique plane are simple associated magic squares (because the cube is associated odd order.
V 
Top IV III [1] John R. Hendricks, Magic Square Course, selfpublished, 1991, page 411. Hendricks Order25 Bimagic The 25 x 25 square shown here is the top horizontal
layer of John Hendricks 25 x 25 x 25 bimagic cube. It is probably the first
bimagic cube to be published. Each of the 25 horizontal planes in Hendricks bimagic cube
is a bimagic square. The 25 vertical planes parallel to the front, and the 25
vertical planes parallel to the side are simple magic squares. (One or both
diagonals of the degree 2 squares are incorrect.) John used a set of 14 equations to construct this bimagic cube. The cube is displayed using the decimal numbers from 1 to 15,625_{10 }(25^{3}) but the construction used the quinary number system with numbers from 000,000 to 444,444_{5}. The coordinate equations also used the the quinary system with numbers from 00 to 44_{5} instead of decimal numbers 1 to 25_{10}. 5590 6570 10675 15380 860 8861 10466 14571 526 4631 9512 14367 2847 3802 8532 13413 1893 3623 7703 12433 1689 5794 7399 11604 12584 4049 8129 9859 13964 3069 7950 12030 13635 2240 3220 11846 12801 1281 6011 7116 15122 1077 5182 6787 10892 148 4978 9083 10063 14793 5733 7338 11443 13048 1503 6384 11239 15344 699 5404 10285 14390 495 4600 9305 14181 2661 4266 8496 9451 2457 3437 7542 12272 13352 4817 8922 10502 14732 87 8718 9698 13778 2883 3988 11994 13599 2054 3659 7764 12645 1875 5955 6935 11665 916 5021 6726 10831 15561 3251 8106 12211 13191 2296 7152 11257 12987 1467 6197 11053 15158 1138 5368 6348 14954 309 4414 9144 10249 2605 4210 8315 9920 14025 14619 599 4679 8784 10389 2770 3875 8580 9560 14290 3541 7646 12476 13456 1936 7442 11547 12502 1732 5837 10718 15448 778 5508 6613 13678 2158 3138 7993 12098 1329 6059 7039 11769 12874 5230 6835 10940 15045 1025 9001 10106 14836 191 4921 9777 13882 3112 4092 8197 15262 742 5472 6427 11157 413 4518 9373 10328 14433 4314 8419 9399 14229 2709 7590 12320 13300 2380 3485 11486 13091 1571 5651 7256 13846 2926 3906 8636 9741 2122 3702 7807 11912 13517 5898 6978 11708 12688 1793 6674 10754 15609 964 5069 10575 14655 10 4865 8970 12910 1390 6245 7225 11305 1181 5286 6266 11121 15201 4457 9187 10167 14897 352 8358 9963 14068 2548 4128 12134 13239 2344 3324 8029 8523 9603 14333 2813 3793 12424 13379 1984 3589 7694 12575 1655 5760 7490 11595 846 5551 6531 10636 15491 4747 8827 10432 14537 517 7082 11812 12792 1272 6102 10983 15088 1068 5173 6753 14759 239 4969 9074 10029 3035 4015 8245 9850 13930 3181 7911 12016 13746 2201 9291 10271 14476 456 4561 9442 14172 2627 4357 8462 13343 2448 3403 7508 12363 1619 5724 7304 11409 13014 5395 6500 11205 15310 665 7855 11960 13565 2045 3650 11626 12731 1836 5941 6921 15527 882 5112 6717 10822 53 4783 8888 10618 14723 3954 8684 9664 13769 2999 6314 11044 15149 1229 5334 10215 14945 300 4380 9235 14111 2591 4196 8276 9881 2262 3367 8097 12177 13157 6163 7143 11373 12953 1433 1902 3507 7737 12467 13447 5803 7408 11513 12618 1723 6579 10684 15414 769 5624 10480 14585 565 4670 8775 14251 2856 3836 8566 9546 1111 5216 6821 10901 15006 4887 9117 10097 14802 157 8163 9768 13998 3078 4058 12064 13669 2149 3229 7959 12840 1320 6050 7005 11860 2700 4280 8385 9490 14220 3471 7551 12281 13261 2491 7372 11452 13057 1537 5642 11148 15353 708 5438 6418 14424 379 4609 9339 10319 1759 5989 6969 11699 12654 5035 6640 10870 15600 930 8931 10536 14641 121 4826 9707 13812 2917 3897 8727 13608 2088 3693 7798 11878 343 4448 9153 10133 14988 4244 8349 9929 14034 2514 8020 12250 13205 2310 3290 11291 12896 1476 6206 7186 15192 1172 5252 6357 11087 11556 12536 1641 5871 7451 15457 812 5542 6522 10727 608 4713 8818 10423 14503 3759 8614 9594 14324 2779 7660 12390 13495 1975 3555 10020 14875 205 4935 9040 13916 3021 4101 8206 9811 2192 3172 7877 12107 13712 6093 7073 11778 12758 1363 6869 10974 15154 1034 5039 12329 13309 2414 3394 7624 13105 1585 5690 7295 11400 626 5481 6461 11191 15296 4527 9257 10362 14467 447 8428 9408 14138 2743 4348 10788 15518 998 5078 6683 14689 44 4774 8979 10584 2965 3945 8675 9630 13860 3736 7841 11946 13526 2006 6887 11742 12722 1802 5907 9997 14077 2557 4162 8267 13148 2353 3333 8063 12168 1424 6129 7234 11339 12944 5325 6280 11010 15240 1220 9221 10176 14906 261 4491 This cube is presented with construction details in a
booklet by John Hendricks published in June, 2000.
[1] Included is the listing for a
short Basic program for displaying any of the 13 lines passing through any
selected cell. The program also lists the coordinates of a number you input. [1] J. R. Hendricks, A Bimagic Cube Order
25, selfpublished, 0968470076, 2000 (now outofprint) On Jan. 20, 2003, Christian Boyer announced by email a bimagic cube of order 16. Here are some facts about his cube:
This is the top horizontal plane of Christian’s first order 16 bimagic cube. I will not take the time (or space) to list the entire cube. 2003 3079 2551 547 1652 3488 2128 900 1181 3913 2745 365 1338 3822 2846 202 2288 804 1748 3328 2391 643 1907 3239 3006 106 1434 3662 2585 461 1085 4073 3648 1428 100 2992 4071 1075 451 2583 3342 1754 810 2302 3241 1917 653 2393 355 2743 3911 1171 196 2832 3808 1332 557 2553 3081 2013 906 2142 3502 1658 972 2072 3560 1596 619 2495 3151 1947 130 2902 3750 1394 293 2801 3841 1237 3311 1851 715 2335 3400 1692 876 2232 4001 1141 389 2641 3590 1490 34 3062 2655 395 1147 4015 3064 44 1500 3592 2321 709 1845 3297 2230 866 1682 3398 1404 3752 2904 140 1243 3855 2815 299 1586 3558 2070 962 1941 3137 2481 613 2858 254 1294 3802 2701 345 1193 3965 2148 944 1600 3476 2499 535 2023 3123 1033 4061 2605 505 1454 3706 2954 94 1863 3219 2403 695 1760 3380 2244 784 697 2413 3229 1865 798 2250 3386 1774 503 2595 4051 1031 80 2948 3700 1440 3482 1614 958 2154 3133 2025 537 2509 3796 1280 240 2852 3955 1191 343 2691 3893 1249 273 2757 3730 1350 182 2914 3195 1967 607 2443 3548 1544 1016 2092 22 3010 3634 1510 433 2661 3989 1089 856 2188 3452 1704 767 2347 3291 1807 1702 3442 2178 854 1793 3285 2341 753 1512 3644 3020 24 1103 3995 2667 447 2437 593 1953 3189 2082 1014 1542 3538 2763 287 1263 3899 2924 184 1352 3740 This cube was announced via email by Christian Boyer on Jan. 23, 2003 It is an improvement on the cube of 3 days previous because it contains 54 order 16 magic squares instead of the 36 contained in the previous cube. It also was confirmed bimagic by Aale de Winkel, Walter Trump, and myself, all using different methods. The cube uses the same number series and thus has the same
magic sums as the previous cube. The 256 lines, 256 columns, 256 pillars, and 4 triagonals
are bimagic, making this a true bimagic cube (as is the previous one). The 96
planar diagonals equal S_{1} so there are 48 (plus 6 oblique) order 16
magic squares. Because this cube (in degree 1) contains all magic squares possible, but they are simple magic, the cube falls into a gap in John Hendricks simple, concise, and universal set of classifications for magic hypercubes. John and I originally decided to call this type of cube “Myers”, after the cube by Richard Myers in 1970 that started the whole discussion of “perfect” magic cubes. Later, Aale de Winkel suggested the term diagonal. This is suggestive of the fact that both main diagonals of all 3m planar squares sum correctly to S. This is the term we will use from now on, for this type of magic cube. Of course, Christian’s cube has the additional, and much more powerful feature, that it is bimagic. The planar broken diagonals are incorrect except for the S_{1} cube where the diagonal pair starting at 0,8 (and 8,0) are correct. This permits swapping the left and right 8 columns of each horizontal plane to obtain a different order 16 magic cube. Likewise you could swap the top and bottom 8 rows (or perform both these transformations). In all cases you get a different magic cube and you get 54 different order 16 magic squares. Be aware however, that these cubes are no longer bimagic. Rows, columns and pillars will be correct on the S_{2} cube but triagonals will be incorrect. Another feature is that the eight corners of many of the
subcubes within the main cube sum to 8/16 of S1. There are no correct 5x5x5 or 9x9x9 subcubes starting on the top 2 horizontal planes (so none in the cube?). The complete cube listing is here. Some email announcements from Christian Boyer! This email from Christian Boyer Jan. 27, 2003 19:38:01
+0100 (CET) Because all planar main diagonals are bimagic, there are 96 bimagic squares contained in this cube. Because it is too big for my test spreadsheets, I have not been able to determine what additional properties this cube possesses. (hh) A few days later  Feb. 2, 2003 13:01:13 +0100 (CET) Another email from Christian on Feb. 5/03 The cube is saved in a file having a size of 170Mb.
Really too big to send by email... ... and thanks again to Walter, he has been forced
to modify his test program in order to handle this huge file and his large
numbers. The complete listing for Christian Boyer's Jan 23, 2003 bimagic cube
is here. Christian Boyer has an excellent site on multimagic squares and cubes.
