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). |
Order-12 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
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 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 |
1-144 |
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(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). |
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 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 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.
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 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 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. 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.
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.
|
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
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; |
Order-8 Bimagic square
|
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. |
Order-16 Bimagic
square
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 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 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
|
|
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 |
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. |
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