Home Site Map

**What are magic cubes and magic tesseracts?****Terms**(Updated on August 10, 2008)**Classes of magic cubes and magic tesseracts****Magic cubes of each order**,**Magic tesseracts of each order****Works on magic cubes**,**Works on magic tesseracts and hypercubes****An order-16 bordered diagonal magic cube****Algorithms to make magic cubes and magic tesseracts**Updated!**Theorems on magic cubes and magic tesseracts****Magic rectangles****Non-normal magic cube**New!**Links and bibliography****Site map****History of updating this site**

If you have an RSS (RDF Site Summary) reader, you can receive update information of this site.

To receive it, click the right icon and import the URL of the XML file into your RSS reader.

**Moved this website. The new URL is http://magcube.la.coocan.jp/magcube/en/ .**[August 8, 2016]- Added an algorithm to construct a
**magic knight jump hypercube**of dimension n and order 4. (**Algorithms to make magic cubes and magic tesseracts**) [August 26, 2016] New! - Released the page
**Non-normal magic cube**. [August 15, 2016] New!

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 `m`^{3}, 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 `m`^{4}. 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.

**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.

Home Site Map

Mitsutoshi Nakamura (Feedback) To send an email to me, please enable JavaScript on your browser.

[Legal indication] Prohibit the send of any spam or advertising.

"Magic Cubes and Tesseracts" http://magcube.la.coocan.jp/magcube/en/

Copyright © 2004-2016, Mitsutoshi Nakamura. All rights reserved.