### Bimagic 25x25x25 cube of John-R. Hendricks analyzed

John-R. Hendricks constructed in 2000 the first known bimagic cube (see website http://www.multimagie.com/English/Cube.htm). I have analyzed the bimagic 25x25x25 cube. It is possible to use Hendricks' method and to construct 15625 different bimagic 25x25x25 cubes. One result is even symmetric.

The first grid is based on the following 5x5 square:

 65 65 65 65 65 65 65 55 1 6 11 16 21 75 15 20 25 5 10 65 65 70 24 4 9 14 19 65 65 65 8 13 18 23 3 65 65 60 17 22 2 7 12 65 65

The 5x5 square vierkant is not fully magic, because not all rows give the magic sum of 65.

The first level of the first grid consists of the 25 shifted versions of the 5x5 squares on a 2x2 carpet. It has the following row-column coordinates (e.g. square 5 - 3 has 2 in the top left corner):

1 - 1
5 - 3   4 - 5   3 - 2
2 - 4
2 - 5   1 - 2   5 - 4   4 - 1
3 - 3
3 - 4   2 - 1   1 - 3   5 - 5
4 - 2
4 - 3   3 - 5   2 - 2   1 - 4
5 - 1
5 - 2   4 - 4   3 - 1   2 - 3
1 - 5

Level 1 up to 25 of the first grid are the shifted versions of the first level with the following row/column-coordinates on a 2x2 carpet of the first level:

1-1, 4-2, 2-3, 5-4, 3-5, 4-5, 2-1, 5-2, 3-3, 1-4, 2-4, 5-5, 3-1, 1-2, 4-3, 5-3, 3-4, 1-5, 4-1, 2-2, 3-2, 1-3, 4-4, 2-5 and 5-1.

The second grid is based on the diagonal shifted version of the 5x5 square which is used for the construction of the first grid:

5x5 square first grid      --> diagonal shift second grid

 1 6 11 16 21 1 20 9 23 12 15 20 25 5 10 19 8 22 11 5 24 4 9 14 19 7 21 15 4 18 8 13 18 23 3 25 14 3 17 6 17 22 2 7 12 13 2 16 10 24

The 5x5 square of the second grid is a shifted  version of a symmetric and panmagic (= ultra magic) 5x5 square! The first level of the second grid consists of the 25 shifted versions of the 5x5 square on a 2x2 carpet. It has the following row-column coordinates (e.g. square 3 - 1 has 7 in the top left corner):

1 - 1
3 - 1   5 - 1   2 - 1
4 - 1
1 - 5   3 - 5   5 - 5   2 - 5
4 - 5
1 - 4   3 - 4   5 - 4   2 - 4
4 - 4
1 - 3   3 - 3   5 - 3   2 - 3
4 - 3
1 - 2   3 - 2   5 - 2   2 - 2
4 - 2

Level 1 up to 25 of the second grid are the shifted versions of the first level with the following row/column-coordinates on a 2x2 carpet of the first level:

1-1, 3-2, 5-3, 2-4, 4-5, 4-3, 1-4, 3-5, 5-1, 2-2, 2-5, 4-1, 1-2, 3-3, 5-4, 5-2, 2-3, 4-4, 1-5, 3-1, 3-4, 5-5, 2-1, 4-2 en 1-3.

The third grid is based on the following 5x5 square:

 15 40 65 90 115 55 65 65 1 8 15 17 24 65 4 6 13 20 22 65 65 65 2 9 11 18 25 65 75 65 5 7 14 16 23 65 60 65 3 10 12 19 21 65 70

The square vierkant is not fully magic, not all rows and not all (pan)diagonals give the magic sum of 65.

The first level of the third grid consists of the 25 shifted versions of the 5x5 square on a 2x2 carpet. It has the following row-column coordinates (e.g. square 4 - 2 has 7 in the top left corner):

1 - 1
4 - 2   2 - 3   5 - 4
3 - 5
4 - 4   2 - 5   5 - 1   3 - 2
1 - 3
2 - 2   5 - 3   3 - 4   1 - 5
4 - 1
5 - 5   3 - 1   1 - 2   4 - 3
2 - 4
3 - 3   1 - 4   4 - 5   2 - 1
5 - 2

Level 1 up to 25 of the third grid are the shifted versions of the first level with the following row/column-coordinates on a 2x2 carpet of the first level:

1-1, 1-3, 1-5, 1-2, 1-4, 2-4, 2-1, 2-3, 2-5, 2-2, 3-2, 3-4, 3-1, 3-3, 3-5, 4-5, 4-2, 4-4, 4-1, 4-3, 5-3, 5-5, 5-2, 5-4 and 5-1.

It is possible to use in each grid one of the 25 shifted versions of the 5x5 squares. So there are 25x25x25 is 15625 different possibilities to construct a Hendricks' bimagic 25x25x25 cube. It is even possible to construct a symmetric version; see below the 13th (= middle) level:

13th (= middle) level of the symmetric version of Hendricks' bimagic 25x25x25 cube

 6875 10955 15060 1040 5145 10021 14851 206 4936 9041 13922 3002 4107 8212 9817 2198 3153 7883 12113 13718 6099 7054 11784 12764 1369 8434 9414 14144 2749 4329 12335 13315 2420 3400 7605 13106 1586 5691 7296 11376 632 5487 6467 11197 15277 4533 9263 10368 14473 428 6893 11748 12703 1808 5913 10794 15524 979 5084 6689 14695 50 4755 8985 10590 2966 3946 8651 9631 13861 3742 7847 11927 13532 2012 9202 10182 14912 267 4497 9978 14083 2563 4168 8273 13129 2359 3339 8069 12174 1405 6135 7240 11345 12950 5301 6281 11011 15241 1221 7661 12391 13496 1951 3556 11562 12542 1647 5852 7457 15463 818 5548 6503 10733 614 4719 8824 10404 14509 3765 8620 9600 14305 2785 129 4984 9089 10069 14799 4030 8135 9865 13970 3075 7926 12031 13636 2241 3221 11827 12807 1287 6017 7122 15103 1083 5188 6793 10898 2463 3443 7548 12253 13358 5739 7344 11449 13029 1509 6390 11245 15350 680 5410 10286 14391 496 4576 9306 14187 2667 4272 8477 9457 922 5002 6732 10837 15567 4823 8903 10508 14738 93 8724 9679 13784 2889 3994 12000 13580 2060 3665 7770 12646 1851 5956 6936 11666 2606 4211 8316 9921 14001 3257 8112 12217 13197 2277 7158 11263 12993 1473 6178 11059 15164 1144 5374 6329 14960 315 4420 9150 10230 1695 5800 7380 11610 12590 5591 6571 10651 15381 861 8867 10472 14552 532 4637 9518 14373 2828 3808 8538 13419 1899 3604 7709 12439 9783 13888 3118 4098 8178 13684 2164 3144 7999 12079 1335 6065 7045 11775 12855 5231 6836 10941 15046 1001 9007 10112 14842 197 4902 11492 13097 1552 5657 7262 15268 748 5453 6433 11163 419 4524 9354 10334 14439 4320 8425 9380 14235 2715 7591 12321 13276 2381 3486 10551 14656 11 4866 8971 13827 2932 3912 8642 9747 2103 3708 7813 11918 13523 5879 6984 11714 12694 1799 6655 10760 15615 970 5075 12140 13245 2350 3305 8035 12911 1391 6246 7201 11306 1187 5292 6272 11102 15207 4463 9193 10173 14878 358 8364 9969 14074 2529 4134 10724 15429 784 5514 6619 14625 580 4685 8790 10395 2771 3851 8581 9561 14291 3547 7627 12482 13462 1942 7448 11528 12508 1738 5843 3187 7917 12022 13727 2207 7088 11818 12798 1253 6108 10989 15094 1074 5154 6759 14765 245 4975 9055 10035 3036 4016 8246 9826 13931 5396 6476 11206 15311 666 9297 10252 14482 462 4567 9448 14153 2633 4363 8468 13349 2429 3409 7514 12369 1625 5705 7310 11415 13020 3960 8690 9670 13775 2980 7856 11961 13566 2046 3626 11632 12737 1842 5947 6902 15533 888 5118 6723 10803 59 4789 8894 10624 14704 6169 7149 11354 12959 1439 6320 11050 15130 1235 5340 10216 14946 276 4381 9236 14117 2597 4177 8282 9887 2268 3373 8078 12183 13163 4728 8833 10438 14543 523 8504 9609 14339 2819 3799 12405 13385 1990 3595 7700 12551 1656 5761 7491 11596 827 5557 6537 10642 15497 12841 1321 6026 7006 11861 1117 5222 6802 10907 15012 4893 9123 10078 14808 163 8169 9774 13979 3084 4064 12070 13675 2130 3235 7965 14405 385 4615 9345 10325 2676 4281 8386 9491 14221 3452 7557 12287 13267 2497 7353 11458 13063 1543 5648 11129 15359 714 5444 6424 13614 2094 3699 7779 11884 1765 5995 6975 11680 12660 5036 6641 10871 15576 931 8937 10542 14647 102 4832 9713 13818 2923 3878 8733 15198 1153 5258 6363 11093 349 4429 9159 10139 14994 4250 8330 9935 14040 2520 8021 12226 13206 2311 3291 11297 12877 1482 6212 7192 14257 2862 3842 8572 9527 1908 3513 7743 12473 13428 5809 7414 11519 12624 1704 6585 10690 15420 775 5605 10481 14586 566 4671 8751

See all grids and all levels of the 25x25x25 bimagic cube in the download below:

25x25x25, Bimagic 25x25x25 cube, grids.x