Magic Cubes and Tesseracts

• What are magic cubes and magic tesseracts?
• Terms (Updated on July 5, 2008)    Updated!
• 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
• Links and bibliography
• Site map
• History of updating this site

What's new

• Added an algorithm to construct an pan and strictly magic tesseract (Algorithms to make magic cubes and magic tesseracts), and updated the page Terms. [July 5, 2008]    New!
• Added an algorithm to construct an associated pantriagonal magic cube of singly-even order, including order 6 (Algorithms to make magic cubes and magic tesseracts). [June 9, 2008]
• Added some magic hypercubes of dimensions 5 to 7 (Works on magic tesseracts and hypercubes). [May 25, 2008]
• Added an associated panmagic hypercube of order 6 and dimension 5 (Works on magic tesseracts and hypercubes). [March 16, 2008]
• I constructed an order-6 associated pantriagonal magic cube on February 6, 2008.
  Listed this cube to "Magic cubes of each order (orders 6 and 7)" and "Works on magic cubes", and modified "Classes of magic cubes and magic tesseracts" and some other pages related to the cube. [March 2, 2008]
• Added some comments on order-4 magic cubes. (Magic cubes of each order) [March 2, 2008]

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. This site is concerned only with normal magic cubes in principle.
   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

an order-4 magic cube (the magic constant is 130) [Yoshihiro Kurushima (?-1757)] : the first magic cube in the world

an order-5 diagonal magic cube (the magic constant is 315) [Walter Trump & Christian Boyer, 2003] : minimum diagonal magic cube

an order-3 magic tesseract (the magic constant is 123) : minimum magic tesseract

   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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