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:

simple magic cubes (3-agonal)
pantriagonal magic cubes (pan-3-agonal)
diagonal magic cubes (2,3-agonal)
pantriagonal diagonal 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, 2-compact, 3-compact, 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?**
order	simple		pantriagonal		diagonal		pantriagonal diagonal		pandiagonal		Nasik
order	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
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	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

assoc. = associated, Yes = exist, No = not exist, ? = unknown

Here are examples of magic cubes for orders from 3 to 8, 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.
For more information, see Aale de Winkel's site.

Note There cannot exist an order-(8x+4) pandiagonal or Nasik magic cube. See Aale de Winkel's 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.

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)

Here is another series of diagonal magic cubes. These cubes are bordered diagonal magic cubes.

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

An order-16 bordered diagonal magic cube (colored) is here.

Top

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
every quadragonal is magic, every panquadragonal 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 2-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 order divisible by 4 and higher than 7, or for any odd order higher than 14, and cannot exist for order 3, 4, or 5. It is unknown what condition is required and sufficient for the existence of strictly magic tesseracts.

(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 the classes of magic tesseracts

No.	Condition
(1)	Quadrag.
(2)	Pan4
	Triag. + Quadrag.
	Triag. + Pan4
(3)	Pan3 + Quadrag.
(4)	Pan3 + Pan4
	Diag. + Quadrag.
	Diag. + Pan4
(5)	Diag. + Triag.
(6)	Diag. + Triag. + Pan4
	Diag. + Pan3 + Quadrag.
	Diag. + Pan3 + Pan4
	Pan2 + Quadrag.
	Pan2 + Pan4
	Pan2 + Triag. + Quadrag.
	Pan2 + Triag. + Pan4
	Pan2 + Pan3 + Quadrag.
(7)	Pan2 + Pan3 + Pan4

are listed below:
Lowest Order Example 3 Click 4 Click 4 Click 4 Click 4 Click 4 Click 4 Click 8? Click + Quadrag. 8? Click 8? Click ? unknown 8 Click 9? Click 13? Click 16? Click 16? Click 15? Click 16 Click

Top

Home

Site Map

Mitsutoshi Nakamura (Feedback) To send an email to me, please enable JavaScript on your browser.
This page was last updated on July 5, 2008.
"Magic Cubes and Tesseracts" http://homepage2.nifty.com/googol/magcube/en/
Copyright © 2004-2008, Mitsutoshi Nakamura. All rights reserved.

	blog SEO tool