Multimagic Cubes

On Jan. 20, 2003, Christian Boyer (France) presented an order 16 Bi-magic cube. It uses the consecutive numbers from 0 to 4095. S₁ = 32,760; S₂ = 89,445,720. The S₁ cube contains 36 order 16 magic squares.
Within 24 hours, the cube had been confirmed as bimagic by Aale de Winkel (the Netherlands), Walter Trump (Germany), and myself (Canada), all using different methods to do so.

The previous record ( and only other bimagic cube?) was an order 25 by John Hendricks in June, 2000.
(But remember, smaller is better, when constructing these special cubes.)

On Jan. 23, Christian presented another bimagic cube. This one contains 54 order 16 magic squares, the most possible for an order 16 cube!
On Jan 27, he produced an order 32 bi-magic cube with all planar diagonals correct for both S₁ and S₂, which means there are 96 order 32 bi-magic squares. The triagonals are tri-magic!
On Feb. 1, 2003, Christian produced the first trimagic cube. It is order 64 so uses numbers 0 to 262143. As a bonus, the 192 orthogonal planes are order 64 bimagic squares! Congratulations Christian.
On Feb. 3, 2003 Christian produced an order 256 trimagic cube in which all 768 orthogonal planes are trimagic squares. This monster uses the numbers from 0 to 16,777,215. It was confirmed correct on Feb 5 by Walter Trump. When will this saga end?
On May 13, 2003 announcement of 7 more multimagic cubes with differing features, including 2 tetramagic cubes! These are on my Monster cubes page.

This page describes what I believe to be the first published multimagic cubes.

I have a similar page that illustrates multimagic squares. Featured are Christian Boyer’s Quadra and Pentamagic squares and Walter Trumps amazing (for its size) order 12 trimagic square.

Christian Boyer has an excellent site on multimagic squares and cubes.

G. Pfeffermann	An old bimagic square transforms to an old magic cube
Collison	This order 5 magic cube is an early attempt at a bimagic cube.
Hendricks	This order 25 cube is probably the first bimagic cube published.
Boyer order 16	This bimagic cube is order 16 and contains 36 simple magic squares
Boyer order 16-2	Also order 16 but contains the maximum of 54 simple magic squares
Boyer trimagic	e-mail messages announcing his progressively better bimagic and trimagic cubes
Monster cubes	This page is an update on Christian Boyer's multimagic hypercubes of May 13/03

G. Pfeffermann bimagic square

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].
This information was supplied to me by Christian Boyer in a Dec. 6, 2002 email.

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).
The sum is 130 in the 16 columns, 16 rows, 16 pillars, and in the 4 main diagonals of the cube.

The Huber Cube

The Huber Cube squared.

Compare the two cubes on the left with the two squares above.

The Huber cube is magic. All rows, columns, pillars and the 4 main triagonals sum to 130.

The degree 2 cube is not magic. None of the 4 sets of lines sum correctly.

[1]The Pfeffermann bimagic square above was a problem published in "Revue des Jeux", June 26th, 1891.
The numbers from 1 to 24 were missing, and so the square had to be finished to be filled.
[2] The solution was published by A. Huber, Revue des Jeux, July 10th, 1891, Paris, together with
Huber's instructions on how to construct a magic cube from it.

Collison - 1990 - simple - associated

This order 5 cube by David Collison 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 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 S₂ or 131775.
For a normal magic cube (degree 1), the total of all the cells in each oblique plane will always sum to mS. This feature is obviously true for all the orthogonal planes because it is the total of all the rows (or all the columns). It is true for the oblique planes because all the rows or all the columns are always the correct sum, and the incorrect columns (or rows) will always sum to mS.

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.

V - Top                    IV                         III
27   66   85 124   13     58   97 111    5   44     89 103   17   31   75
65   79 118    7   46     91 110   24   38   52    122   11   30   69   83
98 112    1   45   59    104   18   32   71   90     10   49   63   77 116
106   25   39   53   92     12   26   70   84 123     43   57   96 115    4
19   33   72   86 105     50   64   78 117    6     51   95 109   23   37
II                         I - Bottom
120    9   48   62   76     21   40   54   93 107
3   42   56 100 114     34   73   87 101   20
36   55   94 108   22     67   81 125   14   28
74   88 102   16   35     80 119    8   47   61
82 121   15   29   68    113    2   41   60   99

John R. Hendricks, Magic Square Course, self-published, 1991, page 411.

Hendricks Order-25 Bimagic Cube

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.
Note of Interest. David M. Collison (1937-1991) reported to John Hendricks in a telephone conversation just days before his untimely death, that he had constructed an order 25 bimagic cube. No details have since come to light regarding that cube.

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.)
S₁ = 195,325. S₂ = 2,034,700,525. On all 75 of the degree 1 magic squares, the 25 cells of each 5x5 sub-squares also sum to 195,325. The cube, of course, has the same S₁ and S₂ in each of its 625 rows, columns, pillars and 4 main triagonals.

John used a set of 14 equations to construct this bimagic cube. The cube is displayed using the decimal numbers from 1 to 15,625₁₀ (25³) but the construction used the quinary number system with numbers from 000,000 to 444,444₅. The coordinate equations also used the the quinary system with numbers from 00 to 44₅ instead of decimal numbers 1 to 25₁₀.

 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. 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.
Holger Danielsson has produced a beautifully typeset and printed booklet with graphic diagrams and the 25 horizontal planes. He also has a great spreadsheet (BimagicCube.xls) that shows each of the 25 horizontal bimagic squares (both degree 1 and degree 2).

J. R. Hendricks, A Bimagic Cube Order 25, self-published, 0-9684700-7-6, 2000 (now out-of-print)
Holger Danielsson, Printout of a Bimagic Cube Order 25, self-published, 2001.

Boyer order 16

On Jan. 20, 2003, Christian Boyer announced by email a bimagic cube of order 16.

Here are some facts about his cube:
This cube is using numbers from 0 to 4095. Notice that the number 2003 appears in the top left back cell of the cube.
Magic sums: (For rows, columns, pillars and the 4 triagonals)
S₁ = 32760
S₂ = 89445720

For the degree 1 cube (S₁) there are additional features. The diagonals are correct in all 16 horizontal planes and all 16 vertical planes parallel to the sides of the cubes. Also rows and columns are correct for 4 of the 6 diagonal planes. So there are a total of 36 order 16 simple magic squares in this cube.

Another feature found in this cube is the fact that the 8 corners of many sub-cubes sum to 8/16 of S₁. In fact, the corners of all the order 9 sub-cubes (including wraparound) within this cube, sum to S/2.

Notice that all of Christian’s bimagic cubes start off with the number of the year in which he discovered them!

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

Boyer order 16-2

The cube shown here 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.
i.e. numbers from 0 to 4095, S 1 = 32760, S 2 = 89445720. The number 2003 is once again in the top, left, back cell of the cube. Christian has retained this feature in all his subsequent multimagic cubes to date.

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 S₁ so there are 48 (plus 6 oblique) order 16 magic squares.
However, these planar diagonals are not magic in the degree 2 cube so it contains no 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 bi-magic.

The planar broken diagonals are incorrect except for the S₁ 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 S₂ cube but triagonals will be incorrect.

Another feature is that the eight corners of many of the sub-cubes within the main cube sum to 8/16 of S₁.
It is interesting to note that this feature is much more evident in Christian’s first cube (Jan. 20). In fact, corners of ALL order 9 sub-cubes (including wrap-around) sum correctly in that cube!

There are no correct 5x5x5 or 9x9x9 sub-cubes starting on the top 2 horizontal planes (so none in the cube?).

The complete cube listing is here.

Boyer Trimagic

Some email announcements from Christian Boyer!

This email from Christian Boyer
27 Jan 2003 19:38:01 +0100 (CET)
Dear friends,
A new progression in multimagic cubes.
I have constructed today a bimagic cube with this supplemental fact: its diagonals are bimagic (only magic in my previous cube).
Order 32, so with numbers from 0 to 32767. See the top layer here.
And some trimagic capabilities are already there: the four triagonals are trimagic.
Thanks to Walter, he has already checked the cube.
I can send you also the Excel file. Let me know if you want it. The size of the zipped file is 108Kb.
I think to be able to announce the first trimagic cube this week, or next week.
Kind regards.
Christian.

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

02 Feb 2003 13:01:13 +0100 (CET)
Subject: The first trimagic cube
Dear friends,
I am happy to announce that I have found, yesterday Feb 1st, the first trimagic cube.
So a magic cube remaining magic when all its numbers are squared, and again remaining magic when all its numbers are cubed.
It has been confirmed trimagic by our friend Walter.
This cube is order 64 (means 64x64x64), and so is using numbers from 0 to 262143, each number only once...
The 4096 rows, 4096 columns and 4096 pillars are trimagic.
The 4 triagonals (means 4 main diagonals of the cube) are trimagic.
The 384 diagonals -not asked in the definition of a standard magic cube- are bimagic "only", sorry guys!
The magic sums are:
            S 1 = 8388576
            S 2 = 1466007115104
            S 3 = 288228177132650496
If you want the cube, just send me a message. But be aware that the zipped Excel file is a 1Mb file!
Kind regards.
Christian.

Another email from Christian on Feb. 5/03

Dear friends,
For your information, the story continues.
A trimagic cube with all its trimagic diagonals (and of course always trimagic rows, columns, pillars and triagonals) have been constructed.
The object is monstrous, order 256. Numbers from 0 to 16,777,215.

The cube is saved in a file having a size of 170Mb. Really too big to send by email...
Walter has kindly accept that I send him the "monster" on a CD-R.
CD-R sent Monday by mail from France, received and tested today by Walter in Germany.
Kind regards.
Christian.

... 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 difference between the order-64 and order-256 trimagic cubes are diagonals.
For both of them, their rows + lines + pillars + triagonals (=the 4 main diagonals) are of course trimagic.
But the 384 diagonals of the order-64 cube are BI-magic "only", compared to the 1536 diagonals of the order-256 cube which are TRI-magic.
So the order 256 trimagic cube has 3 x 256 or 768 order 256 orthogonal trimagic squares.

Christian Boyer has an excellent site on multimagic squares and cubes.