Multimagic Squares

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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 hypercube work in higher dimensions, 'n' is reserved to indicate dimension.

Contents

Multimagic Squares - some history & comparisons

 Table showing a chronological history of multimagic squares (and 1 cube).

Order-12 Trimagic Square

Walter Trump announced the successful completion of this square on June 9, 2002!

Tetramagic & 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 order-16 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 Squares - some history & comparisons

Multimagic Degree

Order
n 

Creator

Date

Number range

Magic constant

Equations
(for series 1 to n)

2. - Bimagic

8

G. Pfeffermann

1890

1 - 64

S1 = 260
S2 = 11,180

S1 = m(m˛+1)/2
S2 = S1*(2m˛+1)/3

2. - Bimagic

9

G. Pfeffermann

1891

1 - 81

S1 = 369
S2 = 20,049

 

3. - Trimagic

128

Gaston Tarry

1905

1 - 16384

S1 = 1,048,640
S2 = 11,454,294,720
S3 = 140,754,668,748,800

 

S3 = m * S1˛

3. Trimagic

64

E. Cazalas

1933

1 - 4096

S1 = 131,104
S2 = 11,201,200
S3 = 1,100,048,564,224

 

3. Trimagic

32

W. H. Benson

1976

1 - 1024

S1 = 16,400
S2 = 11,201,200
S3 =8,606,720,000

 

3. Trimagic

12

Walter Trump

June, 2002

1-144

S1 = 870
S2 = 83,800
S3 = 9,082,800

 

4. Tetramagic

512

Christian Boyer & André Viricel
 

August 2001

1 -262144
(0 – 262,143)

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

 


S4 = S2 * (6m * S1 - 1) / 5

5. Pentamagic

1024

Christian Boyer & André Viricel
 

August 2001

1 – 1,048,576
(Actual series used  was  (0 – 1,048,575)

S1 = 536871424
S2 = 375299432077824
S3 = 295147342229667841024
S4 = 247587417561640996243121664
S5 = 216345083469423421673932062721024

 




S5 = (3m * S2˛ - S3) / 2
2. Bimagic Cube 25 John Hendricks June 2000 1 - 15,625 S1 = 195,325
S2 = 2,034,700,525		 S1 = m(m3 +1)/2
S2 = m(m3+ 1)(2m3 + 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).

Order-12 Trimagic Square

In an email to friends

Subject: First trimagic square of order 12
Date: Sun, 09 Jun 2002 05:46:27 -0400 (EDT)
From: TrumpNbg@aol.com
Dear friends,
I proudly present the first trimagic square of order 12. See attachment. (Trimagic-12.xls)
While doing my everyday work during the week, I started to improve the trimagic program on Saturday
morning. At 9.00 the first semi-trimagic squares were found. All those squares also were bimagic and
the 88th square happened to be trimagic. First I didn't trust the solution and searched for an error in
the square. But it seems to be absolutely correct.
Walter

In an earlier email from Walter on May 12, 2002 he said

in the attachment you find a word document.
It contains analytical proofs for the nonexistence of trimagic squares of orders 9, 10, 11 and 14. On
the other hand it shows, how many odd integers must be in the trimagic series of order 12, 13, 15
and which combinations of those series are possible.

I strongly assume that there are trimagic squares of order 12 and I guess that at least one such square will be found in the next months.

It took him less the one month!

The Order-12 Trimagic Square-1

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

The Order-12 Trimagic Square-2

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

The Order-12 Trimagic Square-3

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

A different order-12 trimagic square

  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).

Tetramagic and Pentamagic

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 m2 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 m2 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 self-similar. 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 self-similar because they are symmetric across both the vertical and horizontal axis.

John Hendricks Bimagics

John used his 'digital equation' method to form 'H' 9x9 magic (and semi-bimagic) 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.
1 33 62 23 52 75 18 38 67
65 13 45 60 8 28 79 21 50
48 77 25 40 72 11 35 55 6
43 66 14 29 58 9 51 80 19
26 46 78 12 41 70 4 36 56
63 2 31 73 24 53 68 16 39
76 27 47 71 10 42 57 5 34
32 61 3 54 74 22 37 69 17
15 44 64 7 30 59 20 49 81

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.
43 51 29 66 80 58 14 19 9
26 4 12 46 36 41 78 56 70
63 68 73 2 16 24 31 39 53
76 57 71 27 5 10 47 34 42
32 37 54 61 69 74 3 17 22
15 20 7 44 49 30 64 81 59
1 18 23 33 38 52 62 67 75
65 79 60 13 21 8 45 50 28
48 35 40 77 55 72 25 6 11

As a bonus, each individual 3x3 sub-square 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 square was formed from the one to the left by column change. Then the top 3 x 9 rectangle was moved to the bottom.
This method and the first square (above) is from J. R. Hendricks, Bimagic Squares: Order 9, self-publ., 0-9684700-6-8, Dec.1999.

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. [1][2] 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.)
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 same feature as the order 9 square above). 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 15,625 (252) 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  (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 this cube.

[1] J. R. Hendricks, A Bimagic Cube Order 25, self-published, 0-9684700-7-6, 2000
[2] Holger Danielsson, Printout of a Bimagic Cube Order 25, self-published, 2001.

Collison's Orders 9 and 16 multimagic

28 13 9 59 66 79 51 44 20
50 8 19 81 58 65 43 30 15
11 77 70 42 46 35 4 27 57
75 33 53 22 2 18 68 61 37
6 72 56 34 41 48 26 10 76
45 21 14 64 80 60 29 49 7
25 55 78 47 36 40 12 5 71
67 52 39 17 24 1 63 74 32
62 38 31 3 16 23 73 69 54
David M. Collison (1937-1991) (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 order-9 bimagic square in 1976.
Order-16 Trimagic
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

Collison constructed this order-16 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!  S1 = 9520,     S2 = 8,228,000,     S3 = 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;
S4 = 1,815,549,271,049,335,956; and S5 = 33,830,849,951,944,563,638,700.
This square is listed in appendix B of J. R. Hendricks, Magic Square Course, self-published, 1991.

Gil Lamb's Bimagics

Order-8 Bimagic square
3 55 44 32 6 50 45 25
29 41 54 2 28 48 51 7
56 4 31 43 49 5 26 46
42 30 1 53 47 27 8 52
59 15 20 40 62 10 21 33
37 17 14 58 36 24 11 63
16 60 39 19 9 61 34 22
18 38 57 13 23 35 64 12
This is one of a whole series of bimagic order-8 (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.
The square with the squared numbers (degree 2) is not pandiagonal. S2 = 11,180.
Order-16 Bimagic square
227 52 5 114 153 170 209 70 238 61 12 127 152 167 224 75
76 223 174 157 128 11 56 231 69 210 163 148 113 6 57 234
55 232 123 16 173 158 79 220 58 233 118 1 164 147 66 213
214 65 154 169 2 117 228 51 219 80 151 168 15 124 237 62
176 155 72 215 60 239 126 13 161 150 73 218 53 226 115 4
9 122 225 54 211 68 149 162 8 119 240 59 222 77 156 175
146 165 212 67 230 49 10 121 159 172 221 78 235 64 7 120
125 14 63 236 71 216 171 160 116 3 50 229 74 217 166 145
19 196 245 130 105 90 33 182 30 205 252 143 104 87 48 187
188 47 94 109 144 251 200 23 181 34 83 100 129 246 201 26
199 24 139 256 93 110 191 44 202 25 134 241 84 99 178 37
38 177 106 89 242 133 20 195 43 192 103 88 255 140 29 206
96 107 184 39 204 31 142 253 81 102 185 42 197 18 131 244
249 138 17 198 35 180 101 82 248 135 32 203 46 189 108 95
98 85 36 179 22 193 250 137 111 92 45 190 27 208 247 136
141 254 207 28 183 40 91 112 132 243 194 21 186 41 86 97

After finding the large group of order-8 bimagic's, Gil thought he would do something different. He was able to use the same method to construct order-16 bimagic, i.e. he did the unusual and went from smaller to bigger. Here too, the first (degree1) square is pandiagonal. S1 = 2056. The degree 2 square is not pandiagonal with S2 = 351,576.

George Chen's Trigrades

I received this material from George in a spreadsheet on Feb. 13, 2002

16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1
2 13 16 3
11 8 5 10
7 12 9 6
14 1 4 15
The second square is transposed from the first one.
The melancholia square is semi-pandiagonal and is Frenicle's # 175 (after normalizing). The transposed square is # 360. Both are associated.
22 + 8 2 + 92 + 152 = 32 + 52 + 122 + 142 = 374
23 + 8 3 + 93 + 153 = 33 + 53 + 123 + 143 = 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 order-5

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

Both of these magic squares are pandiagonal associated. S = 65

102 + 2 2 + 132 + 242 + 162 =  222 + 62 + 132 + 202 + 42 = 1105
103 + 2 3 + 133 + 243 + 163 = 223 + 63 + 133 + 203 + 43 = 21125

And an order-7. 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
342 + 362 + 472 + 252 + 32 + 142 + 162 =  102 + 442 + 152 + 252 + 352 + 62 + 402 = 1105
343 + 363 + 473 + 253 + 33 + 143 + 163 =  103 + 443 + 153 + 253 + 353 + 63 + 403 = 21125

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Credits and Links

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 order-5 and order-7 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 self-similar pandiagonal magic order-7 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, 0-9684700-6-8, Dec.1999.
J. R. Hendricks, A Bimagic Cube Order 25, self-published, 0-9684700-7-6, 2000
Holger Danielsson, Printout of a Bimagic Cube Order 25, self-published, 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 (1937-1991, 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 penta-semimagic 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 3-D 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 3-D magic stars. His ever expanding Encyclopedia is  http://www.magichypercubes.com/Encyclopedia/index.html.
June 16, 2002 Aale provided a spreadsheet to produce new order-12 trimagic squares based on transformations of Walter's original. 

 

This page was originally posted June 2002
It was last updated October 15, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz