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

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

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

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.


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).


An order 9
associated bimagic square. Also an order16 trimagic square, but not with
consecutive numbers. 

An order 8
and an order 16 bimagic square designed using a spreadsheet. 

George
Chen discusses multigrades, using the Melancholia square as an
example 

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 
1144 
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(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). 
Order12 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. (Trimagic12.xls)
While doing my everyday work during the week, I started to improve the
trimagic program on Saturday
morning. At 9.00 the first semitrimagic 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 Order12 Trimagic Square1
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 Order12 Trimagic Square2
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 Order12 Trimagic Square3
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 order12 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 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 Hendricks Bimagics
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.
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 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 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, 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.
[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.)
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.
[1] J.
R. Hendricks, A Bimagic Cube Order 25, selfpublished, 0968470076,
2000
[2] Holger Danielsson, Printout of a Bimagic Cube Order 25,
selfpublished, 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 (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
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 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;
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,
selfpublished, 1991. 
Gil
Lamb's Bimagics
Order8 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 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.
The square with the squared numbers (degree 2) is not pandiagonal. S2
= 11,180. 
Order16 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 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 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 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
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
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 
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 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.
June 16, 2002 Aale provided a spreadsheet to produce new order12
trimagic squares based on transformations of Walter's original. 
