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# Classes of magic cubes and magic tesseracts

[Magic cubes] [Magic tesseracts]

## 1. Classes of magic cubes

Magic cubes are classified according to the following six classes:

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.

Does there exist a magic cube for each order and each class?
ordersimplepantriagonaldiagonalpantriagonal
diagonal
pandiagonalNasik
not assoc.assoc.not assoc.assoc.not assoc.assoc.not assoc.assoc.not assoc.assoc.not assoc.assoc.
3 NoYesNoNoNoNoNoNoNoNoNoNo
4 YesYesYesYesNoNoNoNoNoNoNoNo
5 YesYesYesYesYesNoNoNoNoNoNoNo
6 YesYesYesYesYesNoNoNoNoNoNoNo
7 YesYesYesYesYesYes??YesYesNoNo
8 YesYesYesYesYesYesYesYesYesYesYesNo
> 8, odd YesYesYesYesYesYes??YesYesYesYes
> 8, 4x+2YesYesYesYesYesNoNoNoNoNoNoNo
> 8, 8x+4YesYesYesYesYesYesYes?NoNoNoNo
> 8, 8x YesYesYesYesYesYesYesYesYesYesYesYes
assoc. = associated, Yes = exist, No = not exist, ? = unknown

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

Note See the summary page by Harvey D. Heinz to know who made the first magic cube for each order and class.
Note Every order-3 magic cube is associated. There are four order-3 magic cubes. Look at this.
Note There cannot exist an order-(4x+2) associated magic hypercube of even dimension (magic square, tesseract, etc.), but this does not hold for odd dimension (magic cube, etc.).
Note There cannot exist an order-(4x+2) pandiagonal or Nasik magic cube. If such a cube were possible, its surfaces would be order-(4x+2) (non-normal) pandiagonal magic squares with the odd magic constant, but it is known that such a magic square cannot exist.
Note There cannot exist an order-(8x+4) pandiagonal or Nasik magic cube. See Aale de Winkel's site.
Note Hans-Bernhard Meyer proved there cannot exist an order-5 associated diagonal magic cube. See his site.
Note There cannot exist an order-(4x+2) associated diagonal magic cube. If such a cube were possible, its oblique squares would be order-(4x+2) (non-normal) associated magic squares with the odd constant, but it is known that such a magic square cannot exist.
Note There cannot exist an order-(4x+2) pantriagonal diagonal magic cube. If such a cube were possible, its oblique squares would be order-(4x+2) (non-normal) pandiagonal magic squares with the odd constant, but it is known that such a magic square cannot exist.
Note There cannot exist an order-8 associated Nasik magic cube. Look at this page.
Note It is unknown whether an odd order pantriagonal diagonal magic cube that is not Nasik can exist or not.
I constructed an order-25 pantriagonal diagonal magic cube that is not Nasik in October 2006. See this file (CSV file).
Note Here are examples of the following cubes for orders within 40. These cubes can also construct for any order higher than 40.
Here is another series of diagonal magic cubes. These cubes are bordered diagonal magic cubes.
An order-16 bordered diagonal magic cube (colored) is here.

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## 2. Classes of magic tesseracts

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
Recall that the definition of magic tesseracts requires magicness of their quadragonals.

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.

Here are algorithms to make magic tesseracts for each of these classes.

### Note

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? Clickpossibly not minimum
Diag. + Triag. (2,3,4-agonal) 8? Clickpossibly not minimum
Diag. + Triag. + Pan4 8? Clickpossibly 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? Clickpossibly not minimum
Pan2 + Pan3 15 Click
Pan2 + Pan3 + Pan4 (Nasik) 16 Click

(*1)(*2) Nakamura, August 1, 2010.

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