Multimagic squares are regular magic squares i.e. they have the property that all rows, all columns, and the two main diagonals sum to the same value. However, a bimagic square has the additional property that if each number in the square is multiplied by itself (squared, or raised to the second power) the resulting row, column, and diagonal sums are also magic. In addition, a trimagic square has the additional property that if each number in the square is multiplied by itself twice (cubed, or raised to the third power) the square is still magic. And so on for tetra and penta magic squares.
This page represents multimagic object facts as I know them. Please let me know if you disagree or are aware of other material that perhaps should be on this page. Notice that I have adopted the new convention of using 'm' to denote order of the magic object. With the rapid increase in work on higher dimensions, 'n' is reserved to indicate dimension.
Multimagic Squares  some history & comparisons 
Table showing a chronological history of multimagic squares (and 1 cube). 
Order12 Trimagic Square 
Walter Trump announced the successful
completion of this square on June 9, 2002! 
Tetramagic and Pentamagic 
Christian Boyer in collaboration with his
88 years old friend André Viricel,
announced the completion of the first Tetramagic and Pentamagic squares in
August, 2001. 
John Hendricks Bimagics 
In a booklet published in June, 2000,
John Hendricks announced the completion of the first bimagic cube. Also presented is one of a family of order 9 bimagic squares with special properties (which also appear in the cube). 
Collison's Orders 9 and 16 multimagic 
An order 9 associated bimagic square. Also an
order16 trimagic square, but not with consecutive numbers. 
Gil Lamb's Bimagics 
An order 8 and an order 16 bimagic square designed
using a spreadsheet. 
George Chen's Trigrades 
George Chen discusses multigrades, using the Melancholia square
as an example 
Credits and Links 
Contributors to this branch of magic squares and links to their sites. 
Multimagic Degree 
Order 
Creator 
Date 
Number range 
Magic constant 
Equations 
2.  Bimagic 
8 
G. Pfeffermann 
1890 
1  64 
S1 = 260 
S1 = m(m²+1)/2 
2.  Bimagic 
9 
G. Pfeffermann 
1891 
1  81 
S1 = 369 

3.  Trimagic 
128 
Gaston Tarry 
1905 
1  16384 
S1 = 1,048,640 

3. Trimagic 
64 
E. Cazalas 
1933 
1  4096 
S1 = 131,104 

3. Trimagic 
32 
W. H. Benson 
1976 
1  1024 
S1 = 16,400 

3. Trimagic 
12 
Walter Trump 
June, 2002 
1144 
S1 = 870 

4. Tetramagic 
512 
Christian Boyer & André Viricel 
August 2001 
1 262144 
S1 = 67,109,120 

5. Pentamagic 
1024 
Christian Boyer & André Viricel 
August 2001 
1 – 1,048,576 
S1 = 536871424 

2. Bimagic Cube  25  John Hendricks  June 2000  1  15,625 
S1 = 195,325 S2 = 2,034,700,525 
S1 = m(m^{3} +1)/2 S2 = m(m^{3}+ 1)(2m^{3} + 1)/6 
Note: This table lists only multimagic objects using consecutive numbers. Earlier multimagic squares have been built but not with consecutive numbers (notably by Collison). 
In an email to friends
Subject:
First trimagic square of order 12 
In an earlier email from Walter on May 12, 2002 he said
in the
attachment you find a word document. 
It took him less the one month!
The normal square (each number to the 1st power). 
870  
1  22  33  41  62  66  79  83  104  112  123  144  870 
9  119  45  115  107  93  52  38  30  100  26  136  870 
75  141  35  48  57  14  131  88  97  110  4  70  870 
74  8  106  49  12  43  102  133  96  39  137  71  870 
140  101  124  42  60  37  108  85  103  21  44  5  870 
122  76  142  86  67  126  19  78  59  3  69  23  870 
55  27  95  135  130  89  56  15  10  50  118  90  870 
132  117  68  91  11  99  46  134  54  77  28  13  870 
73  64  2  121  109  32  113  36  24  143  81  72  870 
58  98  84  116  138  16  129  7  29  61  47  87  870 
80  34  105  6  92  127  18  53  139  40  111  65  870 
51  63  31  20  25  128  17  120  125  114  82  94  870 
870  870  870  870  870  870  870  870  870  870  870  870  870 
Each number of the first square raised to the 2nd power. This is called degree 2. 
83810  
1  484  1089  1681  3844  4356  6241  6889  10816  12544  15129  20736  83810 
81  14161  2025  13225  11449  8649  2704  1444  600  10000  676  18496  83810 
5625  19881  1225  2304  3249  196  17161  7744  9409  12100  16  4900  83810 
5476  64  11236  2401  144  1849  10404  17689  9216  1521  18769  5041  83810 
19600  10201  15376  1764  3600  1369  11664  7225  10609  441  1936  25  83810 
14884  5776  20164  7396  4489  15876  361  6084  3481  9  4761  529  83810 
3025  729  9025  18225  16900  7921  3136  225  100  2500  13924  8100  83810 
17424  13689  4624  8281  121  9801  2116  17956  2916  5929  784  169  83810 
5329  4096  4  14641  11881  1024  12769  1296  576  20449  6561  5184  83810 
3364  9604  7056  13456  19044  256  16641  49  841  3721  2209  7569  83810 
6400  1156  11025  36  8464  16219  324  2809  19321  1600  12321  4225  83810 
2601  3969  961  400  625  16384  289  14400  15625  12996  6724  8836  83810 
83810  83810  83810  83810  83810  83810  83810  83810  83810  83810  83810  83810  83810 
Each number of the first square raised to the 3rd power. This is called degree 3. 
9082800  
1  10648  35937  68921  238328  287496  493039  571787  1124864  1404928  1860867  2985984  9082800 
729  1685159  91125  1520875  1225043  804357  140608  54872  27000  1000000  17576  2515456  9082800 
421875  2803221  42875  110592  185193  2744  2248091  681472  912673  1331000  64  343000  9082800 
405224  512  1191016  117649  1728  79507  1061208  2352637  884736  59319  2571353  357911  9082800 
2744000  1030301  1906624  74088  216000  50653  1259712  614125  1092727  9261  85184  125  9082800 
1815848  438976  2863288  636056  300763  2000376  6859  474552  205379  27  328509  12167  9082800 
166375  19683  857375  2460375  2197000  704969  175616  3375  1000  125000  1643032  729000  9082800 
2299968  1601613  314432  753571  1331  970299  97336  2406104  157464  456533  21952  2197  9082800 
389017  262144  8  1771561  1295029  32768  1442897  46656  13824  2924207  531441  373248  9082800 
195112  941192  592704  1560896  2628072  4096  2146689  343  24389  226981  103823  658503  9082800 
512000  39304  1157625  216  778688  2048383  5832  148877  2685619  64000  1367631  274625  9082800 
132651  250047  29791  8000  15625  2097152  4913  1728000  1953125  1481544  551368  830584  9082800 
9082800  9082800  9082800  9082800  9082800  9082800  9082800  9082800  9082800  9082800  9082800  9082800  9082800 
870  
1  41  112  66  83  22  123  62  79  33  104  144  870 
74  49  39  43  133  8  137  12  102  106  96  71  870 
58  116  61  16  7  98  47  138  129  84  29  87  870 
122  86  3  126  78  76  69  67  19  142  59  23  870 
132  91  77  99  134  117  28  11  46  68  54  13  870 
9  115  100  93  38  119  26  107  52  45  30  136  870 
80  6  40  127  53  34  111  92  18  105  139  65  870 
140  42  21  37  85  101  44  60  108  124  103  5  870 
55  135  50  89  15  27  118  130  56  95  10  90  870 
75  48  110  14  88  141  4  57  131  35  97  70  870 
73  121  143  32  36  64  81  109  113  2  24  72  870 
51  20  114  128  120  63  82  25  17  31  125  94  870 
870  870  870  870  870  870  870  870  870  870  870  870  870 
This trimagic square is derived from Walter's original
using a spreadsheet designed by Aale de Winkel.
From any magic square, there are always a family of additional squares that may
be obtained by various transformations. Notice that the leading diagonal almost
consists of a series of increasing and then decreasing values (spoilt only
by the 94 in the lower right cell). The series in the right diagonal decreases
and then increases (again spoiled by just one number). Also, any two numbers in
the same row and an equal distance on either side of the center vertical line
sum to 145 (as do the numbers in Walter's square).
Christian Boyer (France), in collaboration with his 88 years old friend André Viricel, constructed the first known tetramagic square in May 2001. Then in June 2001 they completed the first pentamagic square. These were both announced to the public in August 2001 in the French edition of Scientific American.
Because of copyright restrictions, not too many details are available. I present a few here. More are available on his Web site (called Multimagic Squares.), including files of the two squares that may be downloaded.
The Tetramagic Square (corners only)
0  139,938  18,244  ٠ ٠ ٠  243,899  122,205  262,143 
140,551  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  121,592 
18,959  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  243,184 
٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠ 
242,703  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  19,440 
121,607  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  140,536 
261,632  122,018  244,036  ٠ ٠ ٠  18,107  140,125  511 
This square is 512 x 512 (order 512) and uses
the numbers 0 to 262,143. The initial design of magic squares is always
simplified if such a consecutive series starting from zero is used.
To convert to the more conventional 1 to m^{2} simply add one to the
number in each cell. The magic sums must then be increased by 512.
S1 = 67,109,120
S2 = 11,728,056,921,344
S3 =
2,305,825,417,061,204,480
S4 = 483,565,716,171,561,366,524,672
The Pentamagic Square (corners only)
0  733,632  419,712  ٠ ٠ ٠  628,863  314,943  1,048,575 
866,545  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  182,030 
685,538  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  363,037 
٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠ 
685,597  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  362,978 
867,086  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  ٠ ٠ ٠  181,489 
1,013  733,759  418,943  ٠ ٠ ٠  629,632  314,816  1,047,552 
This square is 1024 x 1024 (order 1024) and
uses the numbers 0 to 1,048,575. To convert to the more conventional 1 to m^{2}
simply add one to the number in each cell. The magic sums must then be increased
by 1024.
S1 = 536,870,400
S2 = 375,299,432,076,800
S3 = 295,147,342,229,667,840,000
S4 = 247,587,417,561,640,996,243,120,640
S5 = 216,345,083,469,423,421,673,932,062,720,000
Christian has shown it can be done. Now, can you find smaller orders of tetra and penta magic squares?
NOTES:
The 2 squares by Boyer and
the one by Trump are all selfsimilar. They are symmetric across the
vertical center line. If each number in the square is changed to it’s complement
(n + 1 – number) the resulting square will be the same square but reflected
horizontally. Because they are symmetric across only one axis,
they are not associated magic squares (sometimes called symmetric). Associated
magic squares, of course, are also selfsimilar because they are symmetric
across both the vertical and horizontal axis.
John used his 'digital equation' method to
form 'H' 9x9 magic (and semibimagic) squares as the one below. Then by row or
column interchange he formed a bimagic square.
He then moved on to a higher dimension, and constructed a 25x25x25 bimagic cube.
An (almost) Bimagic. The diagonals of the
degree 2 square do not sum correctly on this associated magic square. John's initial (and mine also) sums to 389 as well.
The order of columns (or rows) of the square may be changed so that the square of the numbers in the diagonals sum correctly to 20,049 and the diagonals are still correct for the ordinary magic square. In the case of this square the columns are rearranged so that the new top row is 1, 18, 23, 33, 38, 52, 62,67,75. The result is a bimagic square. 
This is a true bimagic square. All
rows, columns and main diagonals sum correctly for this and when each number
is squared.
As a bonus, each individual 3x3 subsquare also sums
to 369. Any 3 x 9 rectangle may be moved from one side of the
square to the other to create a new bimagic square. 

This method and the first square (above) is from J. R. Hendricks, Bimagic Squares: Order 9, selfpubl., 0968470068, Dec.1999. 
Order25 Bimagic Cube
The 25 x 25 square shown here is the top
horizontal layer of John Hendricks 25 x 25 x 25 bimagic cube. Each of the 25
horizontal planes is bimagic. The 25 vertical planes parallel to the front, and
the 25 vertical planes parallel to the side are simple magic. (One or both
diagonals of the degree 2 squares are incorrect.)
S_{1}
= 195,325. S_{2} =
2,034,700,525.
On all 75 of the degree 1 magic squares,
the 25 cells of each 5x5 subsquares
also sum to 195,325 (the same feature as the order 9 square
above). The cube, of course, has the same
S_{1} and
S_{2} 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 15,625 (25^{2}) but the construction used the quinary number system with numbers from 000,000 to 444,444. The coordinate equations also used the the quinary system with numbers from 00 to 44 instead of decimal numbers 1 to 25.
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  15054  1034  5139 
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).
Note of Interest. David M. Collison (19371991) 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 this cube.
J. R. Hendricks, A Bimagic Cube Order 25, selfpublished,
0968470076, 2000
Holger Danielsson, Printout of a Bimagic Cube Order 25, selfpublished, 2001.

David M. Collison (19371991) (U. S. A.) sent this
magic square from his home in California, with no explanation, to John R.
Hendricks (Canada) just before he died. The magic sum, as shown is 369. If each number is squared, the sum is then 20,049. This square (degree1) is associated. Odd order multimagic squares are relatively rare. Benson & Jacoby published an associated order9 bimagic square in 1976. 
Order16 Trimagic
Collison constructed this order16 trimagic square about the same time (late 1980'?). Note however, that it does not use consecutive numbers but 256 numbers ranging from 1 to 1189. But still a significant accomplishment as it was probably the smallest trimagic prior to Trump's! S_{1} = 9520, S_{2} = 8,228,000, S_{3} = 7,946,344,000
David Collison also
constructed an order 36 multimagic square. It was fully trimagic, but
the diagonals are incorrect for tetramagic and pentamagic (he called them
quadrimagic and quintamagic) although all rows and columns gave the correct
sums. The square did not use consecutive numbers. Therefore the magic sums
may seem strange. S1 = 374,940; S2 = 5,811,077,364; S3 =
100,225,960,155,180; 
Order8 Bimagic square

This is one of a whole series of bimagic
order8 (and 16) squares sent to me by Gil Lamb (Thailand) in Feb., 2002.
They are composed by the use of a spreadsheet to first produce
'generating squares'. In each case, the first
square (degree 1) is pandiagonal with S1 = 260. 
Order16 Bimagic
square
After finding the large group of order8 bimagic's, Gil thought he would do something different. He was able to use the same method to construct order16 bimagic, i.e. he did the unusual and went from smaller to bigger. Here too, the first (degree1) square is pandiagonal. S1 = 2056. The degree2 square is not pandiagonal with S2 = 351,576. 
I received this material from George in a spreadsheet on Feb. 13, 2002


The second square is transposed from the
first one. The melancholia square is semipandiagonal and is Frenicle's # 175 (after normalizing). The transposed square is # 360. Both are associated. 
2^{2} + 8^{ 2} + 9^{2} + 15^{2}  =  3^{2} + 5^{2} + 12^{2} + 14^{2}  =  374 
2^{3} + 8^{ 3} + 9^{3} + 15^{3}  =  3^{3} + 5^{3} + 12^{3} + 14^{3}  =  4624 
George goes on to say;
With this amazing feature, people
tried to find bimagic and trimagic squares. As you know, there are no bimagic or trimagic squares of prime orders. However, we can find similar features (to the above) in any prime orders. Following are examples for orders 5 and 7. 
Two order5


Both of these magic squares are pandiagonal associated. S = 65
10^{2} + 2^{ 2} + 13^{2} + 24^{2} + 16^{2}  =  22^{2} + 6^{2} + 13^{2} + 20^{2} + 4^{2}  =  1105 
10^{3} + 2^{ 3} + 13^{3} + 24^{3} + 16^{3}  =  22^{3} + 6^{3} + 13^{3} + 20^{3} + 4^{3}  =  21125 
And an order7. Not pandiagonal but associated.
43  26  4  10  21  30  41 
18  31  42  44  27  1  12 
28  2  13  15  33  39  45 
34  36  47  25  3  14  16 
5  11  17  35  37  48  22 
38  49  23  6  8  19  32 
9  20  29  40  46  24  7 
34^{2} + 36^{2} + 47^{2} + 25^{2} + 3^{2} + 14^{2} + 16^{2}  =  10^{2} + 44^{2} + 15^{2} + 25^{2} + 35^{2} + 6^{2} + 40^{2}  =  1105 
34^{3} + 36^{3} + 47^{3} + 25^{3} + 3^{3} + 14^{3} + 16^{3}  =  10^{3} + 44^{3} + 15^{3} + 25^{3} + 35^{3} + 6^{3} + 40^{3}  =  21125 
Christian Boyer 
Christian Boyer (France), in collaboration with his 88 years old friend André Viricel, constructed the first known tetramagic square in May 2001. Then in June 2001 they completed the first pentamagic square. These were both announced to the public in August 2001 in the French edition of Scientific American. Christian has an excellent Web site called
Multimagic Squares. 
Walter Trump 
Walter Trump (Germany), has done intensive work enumerating order5 and order7 magic squares. His extensive knowledge of the basic programming language, and his willingness to always help, has been of great benefit to me.
His Web page on
selfsimilar pandiagonal magic order7 squares is
Here . 
John Hendricks  John Hendricks (Canada) (1929 
2007) was the
most prolific producer of modern day magic object material. He has
extensively investigated the relationship between magic hypercubes of
different dimension with the end result of a new definition for Nasik (Perfect)
magic cubes. He also has been very prolific in developing Inlaid
magic squares, cube and Tesseracts. His most recent contributions have been
the Bimagic Cube and the Nasik (Perfect) Magic Tesseract! Some of his work is displayed on Holger's site (see below) and on various pages on my site, especially here. Material I used for this page is from: J. R. Hendricks, Bimagic Squares: Order 9, 0968470068, Dec.1999. J. R. Hendricks, A Bimagic Cube Order 25, selfpublished, 0968470076, 2000 Holger Danielsson, Printout of a Bimagic Cube Order 25, selfpublished, 2001. John Hendricks books are now all
out of print but some are available in PDF format. See his memorial site at
http://members.shaw.ca/johnhendricksmath/ 
David Collison  David M. Collison (19371991,
California, U.S.A.), was unknown to me. John Hendricks exchanged correspondence with him and has presented some of his (Collison's) work in his numerous books and articles. Highlights seem to be his work with multimagic squares (which included an order 16 trimagic square and an order 36 pentasemimagic square) and with perfect cubes (new definition) which included an order11. 
Gil Lamb 
Gil Lamb (Thailand) has much to offer in any
magic square discussion. His expertise with spreadsheets give him the ability to
investigate a subject quickly and present his ideas simply and elegantly. The
squares presented here are via private correspondence and printouts from his
spreadsheets in January and February of 2002. 
George Chen 
George Chen's (Taiwan) vast knowledge of magic
squares and willingness to add to any discussion is an inspiration to all magic
square enthusiasts. He is an active participant in almost any aspect of magic
squares. His material presented on this page was from an email of Feb. 13, 2002. 
Aale de Winkel 
Aale de Winkel (The Netherlands) has a wide interest in
magic squares. He gets involved in discussion groups and is a great source of
inspiring and original ideas. A few years ago, he collaborated with me on an
investigation of Quadrant Magic Squares. When I mentioned casually that I would
like to construct a 3D magic star but could not even visualize it, he suggested
how it could be done. Later, after I pursued a few false leads, he came up with
the correct solution. Here are my pages on Quadrant magic
squares and 3D magic
stars. His ever expanding Encyclopedia is
http://www.magichypercubes.com/Encyclopedia/index.html. 
Please send me Feedback about my Web site!
Harvey Heinz harveyheinz@shaw.ca
This page was originally posted on June 16, 2002
This page was last updated
October 31, 2009
Copyright © 2002 by Harvey D. Heinz