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Since May 9, 2004 (Last updated on August 26, 2016)

# Magic Cubes and Tesseracts

## What are magic cubes and magic tesseracts?

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

 10 24 8 23 7 12 9 11 22
 26 1 15 3 14 25 13 27 2
 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

 1 62 63 4 44 23 22 41 24 43 42 21 61 2 3 64
 60 7 6 57 17 46 47 20 45 18 19 48 8 59 58 5
 56 11 10 53 29 34 35 32 33 30 31 36 12 55 54 9
 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

 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
 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
 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
 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
 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

 65 24 34 22 35 66 36 64 23
 31 71 21 72 19 32 20 33 70
 27 28 68 29 69 25 67 26 30

 6 43 74 44 75 4 73 5 45
 80 3 40 1 41 81 42 79 2
 37 77 9 78 7 38 8 39 76

 52 56 15 57 13 53 14 54 55
 12 49 62 50 63 10 61 11 51
 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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