Home Site Map Next 3-D ( 1 2 3 4 5 ) 4-D ( 1 2 3 4 5 6 )

**Magic cubes****pantriagonal magic cubes (pan-3-agonal)****diagonal magic cubes (2,3-agonal)****pantriagonal diagonal magic cubes (pan-3-agonal and 2,3-agonal)****pandiagonal magic cubes (pan-2-agonal and 2,3-agonal)****Nasik magic cubes (pan-2,3-agonal)**

**Magic tesseracts****panmagic tesseracts (pan-4-agonal**)**pantriagonal magic tesseracts (pan-3-agonal and 3,4-agonal)****pan-3,4-agonal magic tesseracts****strictly magic tesseracts (2,3,4-agonal)****pan and strictly magic tesseracts (pan-4-agonal and 2,3,4-agonal)****Nasik magic tesseracts (pan-2,3,4-agonal)**

**Other n-dimensional magic hypercubes**

This section explains general algorithms to construct magic cubes of classes **pantriagonal (pan-3-agonal)**, **diagonal (2,3-agonal)**, **pantriagonal diagonal (pan-3-agonal and 2,3-agonal)**, **pandiagonal (pan-2-agonal and 2,3-agonal)**, and **Nasik (pan-2,3-agonal)**. The algorithms for the two classes **pantriagonal** and **diagonal** support magic cubes of **singly-even** order (that is, order (4x+2)), which used to be said to be difficult to construct.

Let m be the order of the cube and (a_{ijk}), where i,j,k = 0,...,m-1, be the cube (note that the indexes start from zero). The function **gcd**(x, y) means the greatest common divisor of x and y.

Definitions of terms are here. Explanation of classes of magic cubes is here.

A pantriagonal magic cube (or, panmagic cube) always exists if the order is higher than 3. In this case, an associated pantriagonal magic cube exists, too. See also algorithms for higher classes, **pantriagonal diagonal** and **Nasik**.

**Case m = 4x****Case m = 2x+1 and m >= 5****Case m = 4x+2 and m >= 6**

A diagonal magic cube (or, strictly magic cube) can exist only if the order is higher than 4. The following algorithms work for orders higher than 6. See also algorithms for higher classes, **pantriagonal diagonal**, **pandiagonal**, and **Nasik**.

**Case m = 4x and m >= 8 (associated)****Case m = 2x+1 and m >= 7 (associated)****Case m = 4x+2 and m >= 10 (non-associated)**

A pantriagonal diagonal magic cube (or, a **PantriagDiag** magic cube) of **even** order can exist only if the order is higher than 7 and divisible by 4. See also algorithms for a higher class, **Nasik**.

I constructed an order-25 pantriagonal diagonal magic cube which is

It is still unknown whether a

A pandiagonal magic cube can exist only if the order is divisible by 8, or **odd** and higher than 6. In this case, an associated pandiagonal magic cube can exist. See also algorithms for a higher class, **Nasik**.

**Case m = 8x****Case m = 2x+1 and m >= 7**

A Nasik magic cube (or, a pan-2,3-agonal magic cube) can exist only if the order is divisible by 8, or **odd** and higher than 8. In this case, if the order is higher than 8, an associated Nasik magic cube exists. Furthermore, if the order is higher than 8 and divisible by 8, a Nasik magic cube exists which is both associated and 3D-compact. There **cannot** exist a 2D-compact Nasik magic cube.

**Case m = 8x****Non-associated (complete, 3D-compact)****Associated (m >= 16) (not 3D-compact)****Associated and 3D-compact (m >= 16)****Nasik and bimagic (m = 64)**

**Case m = 2x+1 and m >= 9**

This section explains general algorithms to construct magic tesseracts of classes **panmagic (pan-4-agonal)**, **pantriagonal (pan-3-agonal and 3,4-agonal)**, **pan-3,4-agonal**, **strictly magic (2,3,4-agonal)**, **pan and strictly magic (pan-4-agonal and 2,3,4-agonal)**, and **Nasik (pan-2,3,4-agonal)**. The algorithm for the class **pantriagonal** supports magic tesseracts of **singly-even** order (that is, order (4x+2)), which used to be said to be difficult to construct.

Let m be the order of the tesseract and (a_{ijkh}), where i,j,k,h = 0,...,m-1, be the tesseract (note that the indexes start from zero). The function **gcd**(x, y) means the greatest common divisor of x and y.

Definitions of terms are here. Explanation of classes of magic tesseracts is here.

A panmagic tesseract (namely, a pan-4-agonal or panquadragonal magic tesseract) can exist only if the order is divisible by 4, or **odd** and higher than 6. In this case, if the order is higher than 6, an associated panmagic tesseract can exist. See also algorithms for higher classes, **pan-3,4-agonal** and **Nasik**.

**Case m = 4x****Case m = 2x+1 and m >= 7**

A pantriagonal magic tesseract exists for any order except 3 and 6. (An order-3 pantriagonal magic tesseract **cannot** exist, and it is unknown whether an order-6 pantriagonal magic tesseract can exist or not.) If the order is **odd** or divisible by 4, an associated pantriagonal magic tesseract exists. See also algorithms for higher classes, **pan-3,4-agonal** and **Nasik**.

**Case m = 4x****Case m = 2x+1 and m >= 5****Case m = 4x+2 and m >= 10**

A pan-3,4-agonal magic tesseract can exist only if the order is divisible by 4, or **odd** and higher than 8. In this case, if the order is higher than 7, an associated pan-3,4-agonal tesseract can exist. See also algorithms for a higher class, **Nasik**.

**Case m = 4x****Case m = 2x+1 and m >= 9**

It is unknown what condition is required and sufficient for the existence of a strictly magic tesseract, namely, a 2,3,4-agonal magic tesseract. A strictly magic tesseract exists for any **even** order higher than 7 and for any **odd** order higher than 14, and **cannot** exist for order 3, 4, or 5. The following algorithms work for **even** orders divisible by 4 and higher than 7, and **odd** orders higher than 14. See also algorithms for higher classes, **pan and strictly magic** and **Nasik**.

**Case m = 4x and m >= 8(associated)****Case m = 2x+1 and m >= 15**

A pan and strictly magic tesseract, namely, a pan-4-agonal and 2,3,4-agonal magic tesseract, of **even** order can exist only if the order is divisible by 4 and higher than 7.

A Nasik magic tesseract, namely, a pan-2,3,4-agonal magic tesseract, can exist only if the order is divisible by 16, or **odd** and higher than 16. In this case, if the order is higher than 16, an associated Nasik magic tesseract can exist. Furthermore, if the order is higher than 16 and divisible by 16, a Nasik magic tesseract exists which is both associated and 4D-compact.

There **cannot** exist a Nasik magic tesseract which is 2D-compact or 3D-compact.

**Case m = 16x****Non-associated (complete, 4D-compact)****Associated (m >= 32) (not 4D-compact)****Associated and 4D-compact (m >= 32)**

**Case m = 2x+1 and m >= 17**

Top

This section refers to algorithms to construct n-dimensional magic hypercubes with particular properties.

A **magic knight tour hypercube** is a magic hypercube in which the path from the cell (a) to the cell (a+1) of the hypercube, where a = 1,2,..., is always (2-dimensional) knight jump. This section refers to the case of dimension n and order 4.

A **magic knight tour hypercube** of dimension n and order 4 exists if n is **odd** and n > 2. In this case, the hypercube can construct by a general construction algorithm using binary digits. Such a hypercube cannot exist for n = 2 (to begin with, a mere knight tour is impossible), and it is unknown whether it exists for **even** n > 3.

Examples of order-4 magic knight tour hypercube of dimension 5 and dimension 7 are here.

Kiyomi Ohmori,

Akira Hirayama & Gakuho Abe,

William H. Benson & Oswald Jacoby,

Marián Trenkler,

Marián Trenkler,

Marián Trenkler,

Marián Trenkler,

Thomas R. Hagedorn,

Thomas R. Hagedorn,

Guenter Stertenbrink & Jean Charles Meyrignac,

Home Site Map Next 3-D ( 1 2 3 4 5 ) 4-D ( 1 2 3 4 5 6 )

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.

This page was last updated on August 26, 2016.

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

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