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 order-16 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 |
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 |

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. |
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