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- simple magic cubes (3-agonal)
- pantriagonal magic cubes (pan-3-agonal)
- diagonal magic cubes (2,3-agonal)
- pantriagonal diagonal (PantriagDiag) magic cubes (2-agonal and pan-3-agonal)
- pandiagonal magic cubes (pan-2-agonal and 3-agonal)
- Nasik magic cubes (pan-2,3-agonal)

Some magic cubes have additional properties, for example, complete, associated, plane symmetrical, 2D-compact, 3D-compact, inlaid, *etc.*.

(Definitions of terms are here.)

Whether a magic cube can exist or not depends on its order and class. The following table shows the existence or non-existence of magic cubes for each order and class.

order | simple | pantriagonal | diagonal | pantriagonal diagonal | pandiagonal | Nasik | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

not assoc. | assoc. | not assoc. | assoc. | not assoc. | assoc. | not assoc. | assoc. | not assoc. | assoc. | not assoc. | assoc. | |

3 | No | Yes | No | No | No | No | No | No | No | No | No | No |

4 | Yes | Yes | Yes | Yes | No | No | No | No | No | No | No | No |

5 | Yes | Yes | Yes | Yes | Yes | No | No | No | No | No | No | No |

6 | Yes | Yes | Yes | Yes | Yes | No | No | No | No | No | No | No |

7 | Yes | Yes | Yes | Yes | Yes | Yes | ? | ? | Yes | Yes | No | No |

8 | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | No |

> 8, odd | Yes | Yes | Yes | Yes | Yes | Yes | ? | ? | Yes | Yes | Yes | Yes |

> 8, 4x+2 | Yes | Yes | Yes | Yes | Yes | No | No | No | No | No | No | No |

> 8, 8x+4 | Yes | Yes | Yes | Yes | Yes | Yes | Yes | ? | No | No | No | No |

> 8, 8x | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes |

Here are examples of magic cubes for orders from 3 to 10, and algorithms to make magic cubes for each of the classes.

For more information, see Aale de Winkel's site.

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

- order-(4x) associated
**pantriagonal**magic cubes (for orders from 4 to 40) - order-(4x+2) associated
**pantriagonal**magic cubes (for orders from 10 to 38) - order-(4x) associated
**diagonal**magic cubes (for orders from 8 to 40) - order-(4x+2)
**diagonal**magic cubes (for orders from 10 to 38) - order-(4x)
**pantriagonal diagonal**magic cubes (for orders from 8 to 40) - order-(8x) associated
**pandiagonal**magic cubes (for orders from 16 to 40, not Nasik) - order-(8x) associated
**Nasik**magic cubes (for orders 16 to 40)

- order-(2x)
**bordered diagonal magic cubes**(for orders from 6 to 40)

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Magic tesseracts have the following properties on their (pan)diagonals, (pan)triagonal, and (pan)quadragonals:

- no condition on (pan)diagonals, every diagonal is magic, every pandiagonal is magic
- no condition on (pan)triagonals, every triagonal is magic, every pantriagonal is magic
- every quadragonal is magic, every panquadragonal is magic

Magic tesseracts are classified into the 18 classes given by combinations of the above properties. The following are major classes of magic tesseracts.

**(1) simple magic tesseracts** (4-agonal)

This class is given by the combination of conditions "no condition on (pan)diagonals", "no condition on (pan)triagonals", and "every quadragonal is magic".

**(2) panmagic tesseracts** (pan-4-agonal)

This class is given by the combination of conditions "no condition on (pan)diagonals", "no condition on (pan)triagonals", and "every panquadragonal is magic".

A panmagic tesseract can exist only if its order is divisible by 4, or **odd** and higher than 6.

**(3) pantriagonal magic tesseracts** (pan-3-agonal and 4-agonal)

This class is given by the combination of conditions "no condition on (pan)diagonals", "every pantriagonal is magic", and "every quadragonal is magic".

A pantriagonal magic tesseract can exist 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.

**(4) pan-3,4-agonal magic tesseracts**

This class is given by the combination of conditions "no condition on (pan)diagonals", "evry pantriagonal is magic", and "every panquadragonal is magic".

Every 2D-compact magic tesseract exactly belongs to this class.

A pan-3,4-agonal magic tesseract can exist only if its order is divisible by 4, or **odd** and higher than 8.

**(5) strictly magic tesseracts** (2,3,4-agonal)

This class is given by the combination of conditions "every diagonal is magic", "every triagonal is magic", and "every quadragonal is magic".

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. It is unknown whether a strictly magic tesseracts of order 6, 7, 9, 11, or 13 exists or not. Updated!

**(6) pan and strictly magic tesseracts** (2,3,4-agonal and pan-4-agonal)

This class is given by the combination of conditions "every diagonal is magic", "every triagonal is magic", and "every panquadragonal is magic".

A pan and strictly magic tesseract exists for any order divisible by 4 and higher than 7, and **cannot** exist for order 3, 4, 5, or singly-even integer. It is unknown what condition is required and sufficient for the existence of pan and strictly magic tesseracts.

**(7) Nasik magic tesseracts** (pan-2,3,4-agonal)

This class is given by the combination of conditions "every pandiagonal is magic", "every pantriagonal is magic", and "every panquadragonal is magic". Note that this condition is essentially stronger than the condition of pan and strictly magic.

A Nasik magic tesseract can exist only if its order is divisible by 16, or **odd** and higher than 16.

All of the 18 classes of magic tesseracts are listed below. None of these classes is empty.

Class | Min. Order | Example | Note |
---|---|---|---|

Simple (4-agonal) | 3 | Click | |

Pan4 (panmagic) | 4 | Click | |

Triag. | 4 | Click | |

Triag. + Pan4 | 4 | Click | |

Pan3 | 4 | Click | |

Pan3 + Pan4 | 4 | Click | |

Diag. | 4 | Click | |

Diag. + Pan4 | 8? | Click | possibly not minimum |

Diag. + Triag. (2,3,4-agonal) | 8? | Click | possibly not minimum |

Diag. + Triag. + Pan4 | 8? | Click | possibly not minimum |

Diag. + Pan3 | 8? | Click(*1) | possibly not minimum |

Diag. + Pan3 + Pan4 | 8 | Click | |

Pan2 | 9 | Click | |

Pan2 + Pan4 | 13 | Click | |

Pan2 + Triag. | 16? | Click(*2) | possibly not minimum |

Pan2 + Triag. + Pan4 | 16? | Click | possibly not minimum |

Pan2 + Pan3 | 15 | Click | |

Pan2 + Pan3 + Pan4 (Nasik) | 16 | Click |

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This page was last updated on August 8, 2016.

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

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