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# Magic cubes of each order

Definitions of terms are here. Explanation of classes of magic cubes is here. Examples of magic tesseracts are here.
[Order 3] [Order 4] [Order 5] [Order 6] [Order 7] [Order 8] [Order 9] [Order 10]

## Order-3 magic cubes (the constant is 42)

There are exactly four order-3 magic cubes. John R. Hendricks (1929-2007) proved that in 1972 ([1])(*). All order-3 magic cubes are associated and simple.
(*) Before Hendricks, the following persons claimed independent of one another that the number of order-3 magic cubes is 4 ([2]):
Eisuke Ishikawa(1933), D. N. Lehmer(1934), Gensho Abe(1939).

[1] John R. Hendricks, The Third-Order Magic Cube Complete, Journal of Recreational Mathematics 5:1(1972), 43-50.
[2] Akira Hirayama & Gakuho Abe, Researches in Magic Squares (Japanese), Osaka Kyoikutosho, 1983, 151.

 18 22 2 20 9 13 4 11 27
 23 3 16 7 14 21 12 25 5
 1 17 24 15 19 8 26 6 10
 16 23 3 20 9 13 6 10 26
 24 1 17 7 14 21 11 27 4
 2 18 22 15 19 8 25 5 12
 12 23 7 22 9 11 8 10 24
 26 1 15 3 14 25 13 27 2
 4 18 20 17 19 6 21 5 16
 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
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

T. Hugel made the first order-3 magic cube in 1876. His cube is equivalent to the first of the above cubes.

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## Order-4 magic cubes (the constant is 130)

There exist order-4 magic cubes of classes simple and pantriagonal. Associated magic cubes of order 4 can exist for both classes. The number of order-4 magic cubes is unknown. Water Trump discovered in 2003 that the number of order-4 associated magic cubes is exactly 44,447,308,800.

### (1) Order-4 simple magic cubes

an order-4 simple magic cube (axis symmetrical, horizontal-diagonal) [Yoshihiro Kurushima (?-1757)]
 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
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.154

an order-4 simple magic cube (associated) [Yoshihiro Kurushima]
 49 32 48 1 12 37 21 60 8 41 25 56 61 20 36 13
 15 34 18 63 54 27 43 6 58 23 39 10 3 46 30 51
 14 35 19 62 55 26 42 7 59 22 38 11 2 47 31 50
 52 29 45 4 9 40 24 57 5 44 28 53 64 17 33 16
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.154

An order-4 magic cube was first constructed by a Japanese mathematician, Yoshihiro Kurushima (?-1757). These two magic cubes were found from his manuscript.
Either of these cubes is the first magic cube in the world.

an order-4 simple magic cube (associated, horizontal-diagonal) [Gakuho Abe]
 8 61 1 60 11 50 14 55 62 7 59 2 49 12 56 13
 45 24 44 17 34 27 39 30 23 46 18 43 28 33 29 40
 25 36 32 37 22 47 19 42 35 26 38 31 48 21 41 20
 52 9 53 16 63 6 58 3 10 51 15 54 5 64 4 57
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.107

This magic cube is associated and each of the four horizontal planes of the cube is an order-4 magic square.
The four horizontal planes compose the following order-8 associated magic square.
 8 61 1 60 45 24 44 17 11 50 14 55 34 27 39 30 62 7 59 2 23 46 18 43 49 12 56 13 28 33 29 40 25 36 32 37 52 9 53 16 22 47 19 42 63 6 58 3 35 26 38 31 10 51 15 54 48 21 41 20 5 64 4 57

an order-4 simple magic cube (non-associated, horizontal-pandiagonal, 3D-compact) [Gensho Abe, 1938]
 64 6 43 17 33 27 54 16 22 48 1 59 11 49 32 38
 2 60 21 47 31 37 12 50 44 18 63 5 53 15 34 28
 61 7 42 20 36 26 55 13 23 45 4 58 10 52 29 39
 3 57 24 46 30 40 9 51 41 19 62 8 56 14 35 25
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.157

Every horizontal plane of this cube is an order-4 pandiagonal magic square which is also complete and compact.

an order-4 simple magic cube (associated, 3D-compact) [Nakamura, November 2007]
 1 48 57 24 63 18 7 42 36 13 28 53 30 51 38 11
 60 21 4 45 6 43 62 19 25 56 33 16 39 10 31 50
 15 34 55 26 49 32 9 40 46 3 22 59 20 61 44 5
 54 27 14 35 12 37 52 29 23 58 47 2 41 8 17 64
the source: original.

Every order-4 associated and 3D-compact magic cube belongs to the class 'simple'.

### (2) Order-4 pantriagonal magic cubes

an order-4 pantriagonal magic cube (non-associated, complete) [A. H. Frost, 1878]
 33 31 30 36 28 38 39 25 14 52 49 15 55 9 12 54
 24 42 43 21 45 19 18 48 59 5 8 58 2 64 61 3
 16 50 51 13 53 11 10 56 35 29 32 34 26 40 37 27
 57 7 6 60 4 62 63 1 22 44 41 23 47 17 20 46
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.157

This cube seems to be the first order-4 pantriagonal magic cube.

an order-4 pantriagonal magic cube (non-associated, complete, 2D-compact) [Arata Sakai (1908-1964), 1938]
 43 21 42 24 18 48 19 45 39 25 38 28 30 36 31 33
 6 60 7 57 63 1 62 4 10 56 11 53 51 13 50 16
 27 37 26 40 34 32 35 29 23 41 22 44 46 20 47 17
 54 12 55 9 15 49 14 52 58 8 59 5 3 61 2 64
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.157

This magic cube is pantraigonal, complete, and 2D-compact.
Every 2D-compact magic cube is pantraigonal.

an order-4 pantriagonal magic cube (non-associated, non-complete, 2D-compact) [Kanji Setsuda]
 1 56 25 48 64 9 40 17 5 52 29 44 60 13 36 21
 63 10 39 18 2 55 26 47 59 14 35 22 6 51 30 43
 4 53 28 45 61 12 37 20 8 49 32 41 57 16 33 24
 62 11 38 19 3 54 27 46 58 15 34 23 7 50 31 42
the source: Kanji Setsuda's site (http://kanjisetsuda.la.coocan.jp/pages/EnglishP1.html)

This magic cube is pantraigonal and 2D-compact but not complete. (A 2D-compact magic cube cannot be associated.)
Kanji Setsuda has beed made a lot of studies of order-4 2D-compact pantriagonal magic cubes.

an order-4 pantriagonal magic cube (non-associated, complete, horizontal-diagonal) [Shigematsu Urata (1889-1958), 1941]
 1 40 62 27 59 30 8 33 24 49 43 14 46 11 17 56
 60 29 7 34 2 39 61 28 45 12 18 55 23 50 44 13
 22 51 41 16 48 9 19 54 3 38 64 25 57 32 6 35
 47 10 20 5 21 52 42 15 58 31 5 36 4 37 63 26
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, p.157

This magic cube is pantriagonal and complete, and each of the four horizontal planes of the cube is an order-4 magic square.
An order-4 pantriagonal magic cube cannot be 2D-compact or 3D-compact if all of its horizontal planes are magic.

an order-4 pantriagonal magic cube (non-associated, non-complete, 'knight-tour') [Guenter Stertenbrink, 2003]
 20 41 14 55 39 30 57 4 10 51 24 45 61 8 35 26
 15 54 17 44 60 1 38 31 21 48 11 50 34 27 64 5
 42 19 56 13 29 40 3 58 52 9 46 23 7 62 25 36
 53 16 43 18 2 59 32 37 47 22 49 12 28 33 6 63
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

This magic cube is pantriagonal and the 64 consecutive integers of the cube trace out a magic knight tour.

an order-4 pantriagonal magic cube (plane-symmetrical, 2D-compact) [Nakamura, May 2004]
 1 32 33 64 56 41 24 9 13 20 45 52 60 37 28 5
 62 35 30 3 11 22 43 54 50 47 18 15 7 26 39 58
 4 29 36 61 53 44 21 12 16 17 48 49 57 40 25 8
 63 34 31 2 10 23 42 55 51 46 19 14 6 27 38 59
 1 32 33 64 56 41 24 9 13 20 45 52 60 37 28 5
 62 35 30 3 11 22 43 54 50 47 18 15 7 26 39 58
 19 14 51 46 38 59 6 27 31 2 63 34 42 55 10 23
 48 49 16 17 25 8 57 40 36 61 4 29 21 12 53 44
the source: original

Both of these magic cubes are pantriagonal, plane symmetrical, and 2D-compact.
These pantriagonal magic cubes are essentially diffrent from each other though they share the 1st and the 2nd plane.

an order-4 pantriagonal magic cube (associated) [Nakamura, May 2004]
 1 55 14 60 40 29 43 18 30 44 17 39 59 2 56 13
 31 38 20 41 53 3 58 16 4 57 15 54 42 32 37 19
 46 28 33 23 11 50 8 61 49 7 62 12 24 45 27 34
 52 9 63 6 26 48 21 35 47 22 36 25 5 51 10 64
the source: original. Here is this cube of CSV format.

This is an order-4 magic cube which is both pantriagonal and associated. It was once believed that such a magic cube could not exist. According to Water Trump, there are exactly 37,824 order-4 associated pantriagonal magic cubes.
An order-4 associated pantriagonal magic cube cannot be 2D-compact or 3D-compact.
There is a study of order-4 associated pantriagonal magic cubes by Kanji Setsuda.

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## Order-5 magic cubes (the constant is 315)

There exist order-5 magic cubes of classes simple, pantriagonal, and diagonal. An associated magic cube of order 5 can exist for the classes simple and pantriagonal. It is unknown whether an associated diagonal magic cube of order 5 can exist or not.

an order-5 simple magic cube (associated) [Hermann Schubert, 1898]
 121 27 83 14 70 10 61 117 48 79 44 100 1 57 113 53 109 40 91 22 87 18 74 105 31
 2 58 114 45 96 36 92 23 54 110 75 101 32 88 19 84 15 66 122 28 118 49 80 6 62
 33 89 20 71 102 67 123 29 85 11 76 7 63 119 50 115 41 97 3 59 24 55 106 37 93
 64 120 46 77 8 98 4 60 111 42 107 38 94 25 51 16 72 103 34 90 30 81 12 68 124
 95 21 52 108 39 104 35 86 17 73 13 69 125 26 82 47 78 9 65 116 56 112 43 99 5
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

an order-5 simple magic cube (associated, horizontal-diagonal) [Theodor Hugel, 1876]
 93 121 62 4 35 52 9 45 98 111 50 88 101 57 19 106 67 24 40 78 14 30 83 116 72
 12 29 85 118 71 95 123 61 2 34 51 7 44 100 113 49 90 103 56 17 108 66 22 39 80
 110 68 21 37 79 11 27 84 120 73 94 125 63 1 32 53 6 42 99 115 47 89 105 58 16
 46 87 104 60 18 109 70 23 36 77 13 26 82 119 75 92 124 65 3 31 55 8 41 97 114
 54 10 43 96 112 48 86 102 59 20 107 69 25 38 76 15 28 81 117 74 91 122 64 5 33
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

This cube is associated and every horizontal plane of the cube is an order-5 pandiagonal magic square.
This cube seems to be the first order-5 magic cube in the world.

an order-5 simple magic cube (non-associated, s-magic, bordered) [Walter Trump, 2003]
 33 22 113 42 105 2 89 106 19 99 85 82 4 119 25 98 90 6 112 9 97 32 86 23 77
 95 80 111 11 18 91 55 65 69 35 16 66 70 53 110 5 68 54 67 121 108 46 15 115 31
 100 17 48 34 116 114 75 52 62 12 83 50 63 76 43 8 64 74 51 118 10 109 78 92 26
 38 102 3 125 47 81 59 72 58 45 30 73 56 60 96 87 57 61 71 39 79 24 123 1 88
 49 94 40 103 29 27 37 20 107 124 101 44 122 7 41 117 36 120 14 28 21 104 13 84 93
the source: Walter Trump's site (http://www.trump.de/magic-squares/magic-cubes/cubes-1.html)

This cube is an s-magic cube. Furthermore, the cube contains within it an order-3 magic cube, which consists of consecutive integers from 50 to 76, so it is also a bordered (namely, consecutively concentric) magic cube.
Trump says that an order-5 bordered diagonal magic cube cannot exist.

an order-5 pantriagonal magic cube (associated) [Yoshio Moriyama, 1967]
 21 40 54 93 107 112 1 45 59 98 78 117 6 50 64 69 83 122 11 30 35 74 88 102 16
 37 51 95 109 23 3 42 56 100 114 119 8 47 61 80 85 124 13 27 66 71 90 104 18 32
 53 92 106 25 39 44 58 97 111 5 10 49 63 77 116 121 15 29 68 82 87 101 20 34 73
 94 108 22 36 55 60 99 113 2 41 46 65 79 118 7 12 26 70 84 123 103 17 31 75 89
 110 24 38 52 91 96 115 4 43 57 62 76 120 9 48 28 67 81 125 14 19 33 72 86 105
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, pp.163-164

Is this cube the first order-5 pantriagonal magic cube?

an order-5 diagonal magic cube (non-associated) [Walter Trump & Christian Boyer, 2003]
 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
the source: Walter Trump's site (http://www.trump.de/magic-squares/magic-cubes/cubes-1.html)

This cube by Walter Trump & Christian Boyer is the first order-5 diagonal magic cube in the world. It was once believed that such a magic cube could not exist. They found this cube by computer. They also found other order-5 diagonal magic cubes, which are listed to Aale de Winkel's site.

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