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Since May 9, 2004 (Last updated on August 26, 2016)
Magic Cubes and Tesseracts
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A magic cube is defined as a cubical array such that all rows, columns, pillars, and four triagonals of the array sum to the same value (called the (magic) constant or the magic sum). Magic cubes are, as it were, three-dimensional magic squares. An order-m magic cube is called a normal magic cube if the cube consists of consecutive integers from 1 to m3, and called a non-normal magic cube if not. A magic cube in this site is normal if its normality is unspecified.
It is not required that (2-dimensional) diagonals of a magic cube sum to the constant. A magic cube with the feature that every diagonal sums to the constant is called a diagonal magic cube. A diagonal magic cube can exist only for orders higher than 4.
Similarly, a magic tesseract is a four-dimensional hypercube whose rows, columns, pillars, files, and eight quadragonals sum to the constant. An order-m normal magic tesseract consists of consecutive integers from 1 to m4. A magic tesseract is called a strictly magic tesseract if all diagonals and all triagonals of the tesseract are magic. The smallest known strictly magic tesseract is an order-8 strictly magic tesseract constructed by the author in 2004.
Generally, n-dimensional magic hypercubes are defined for every integer n > 1. Marián Trenkler proved that a normal magic hypercube of dimension n and order m can exist for every integer n > 1 and every integer m > 2.
Examples
an order-3 magic cube (the magic constant is 42) : minimum magic cube
Plane No.1
10 | 24 | 8 |
23 | 7 | 12 |
9 | 11 | 22 |
Plane No.2
26 | 1 | 15 |
3 | 14 | 25 |
13 | 27 | 2 |
Plane No.3
6 | 17 | 19 |
16 | 21 | 5 |
20 | 4 | 18 |
an order-4 magic cube (the magic constant is 130) [Yoshihiro Kurushima (?-1757)] : the first magic cube in the world
Plane No.1
1 | 62 | 63 | 4 |
44 | 23 | 22 | 41 |
24 | 43 | 42 | 21 |
61 | 2 | 3 | 64 |
Plane No.2
60 | 7 | 6 | 57 |
17 | 46 | 47 | 20 |
45 | 18 | 19 | 48 |
8 | 59 | 58 | 5 |
Plane No.3
56 | 11 | 10 | 53 |
29 | 34 | 35 | 32 |
33 | 30 | 31 | 36 |
12 | 55 | 54 | 9 |
Plane No.4
13 | 50 | 51 | 16 |
40 | 27 | 26 | 37 |
28 | 39 | 38 | 25 |
49 | 14 | 15 | 52 |
an order-5 diagonal magic cube (the magic constant is 315) [Walter Trump & Christian Boyer, 2003] : minimum diagonal magic cube
Plane No.1
25 | 16 | 80 | 104 | 90 |
115 | 98 | 4 | 1 | 97 |
42 | 111 | 85 | 2 | 75 |
66 | 72 | 27 | 102 | 48 |
67 | 18 | 119 | 106 | 5 |
Plane No.2
91 | 77 | 71 | 6 | 70 |
52 | 64 | 117 | 69 | 13 |
30 | 118 | 21 | 123 | 23 |
26 | 39 | 92 | 44 | 114 |
116 | 17 | 14 | 73 | 95 |
Plane No.3
47 | 61 | 45 | 76 | 86 |
107 | 43 | 38 | 33 | 94 |
89 | 68 | 63 | 58 | 37 |
32 | 93 | 88 | 83 | 19 |
40 | 50 | 81 | 65 | 79 |
Plane No.4
31 | 53 | 112 | 109 | 10 |
12 | 82 | 34 | 87 | 100 |
103 | 3 | 105 | 8 | 96 |
113 | 57 | 9 | 62 | 74 |
56 | 120 | 55 | 49 | 35 |
Plane No.5
121 | 108 | 7 | 20 | 59 |
29 | 28 | 122 | 125 | 11 |
51 | 15 | 41 | 124 | 84 |
78 | 54 | 99 | 24 | 60 |
36 | 110 | 46 | 22 | 101 |
an order-3 magic tesseract (the magic constant is 123) : minimum magic tesseract
Plane (1,1)
65 | 24 | 34 |
22 | 35 | 66 |
36 | 64 | 23 |
Plane (1,2)
31 | 71 | 21 |
72 | 19 | 32 |
20 | 33 | 70 |
Plane (1,3)
27 | 28 | 68 |
29 | 69 | 25 |
67 | 26 | 30 |
Plane (2,1)
6 | 43 | 74 |
44 | 75 | 4 |
73 | 5 | 45 |
Plane (2,2)
80 | 3 | 40 |
1 | 41 | 81 |
42 | 79 | 2 |
Plane (2,3)
37 | 77 | 9 |
78 | 7 | 38 |
8 | 39 | 76 |
Plane (3,1)
52 | 56 | 15 |
57 | 13 | 53 |
14 | 54 | 55 |
Plane (3,2)
12 | 49 | 62 |
50 | 63 | 10 |
61 | 11 | 51 |
Plane (3,3)
59 | 18 | 46 |
16 | 47 | 60 |
48 | 58 | 17 |
There exist four order-3 magic cubes and 58 order-3 magic tesseracts. For order 4 or higher, the number of magic cubes or magic tesseracts is still unknown. According to Water Trump, the number of order-4 associated magic cubes is exactly 44,447,308,800.
Here are other examples of magic cubes and magic tesseracts.
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"Magic Cubes and Tesseracts" http://magcube.la.coocan.jp/magcube/en/
Copyright © 2004-2016, Mitsutoshi Nakamura. All rights reserved.
Magic Cubes and Tesseracts