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Definitions of terms are here. Works on magic cubes are here.

**order 4**- an order-4
**2,4-agonal**magic tesseract (**associated**) (CSV file) [Nakamura, December 2004]

This tesseract is associated and 2,4-agonal. An order-4 strictly magic tesseract (that is, a 2,3,4-agonal magic tesseract)**cannot**exist. - an order-4
**pan-3-agonal**magic tesseract (**associated**) (CSV file) [Nakamura, September 2007]

This tesseract is associated and every pantriagonal is magic. Of course, every quadragonal is also magic.

- an order-4
**order 8**- an order-8
**pan-3,4-agonal**magic tesseract (**associated**) (CSV file) [Nakamura, August 2004]

This tesseract is associated and pan-3,4-agonal. - an order-8
**pan-3,4-agonal**magic tesseract (**associated**and**2D-compact**) (CSV file) [Nakamura, September 2007]

This tesseract is associated, 2D-compact, and pan-3,4-agonal. - an order-8
**2,3,4-agonal**magic tesseract (**associated**) (CSV file) [Nakamura, December 2004]

This tesseract is associated and strictly magic (that is, 2,3,4-agonal). - an order-8
**pan-4-agonal**&**2,3,4-agonal**magic tesseract (non-associated) (CSV file) [Nakamura, September 2006]

This tesseract is both panmagic (that is, pan-4-agonal) and strictly magic (that is, 2,3,4-agonal). - an order-8
**pan-3,4-agonal**&**2,3,4-agonal**magic tesseract (non-associated) (CSV file) [Nakamura, September 2006]

This tesseract is pan-3,4-agonal and 2,3,4-agonal. In other words, it is panmagic, strictly magic, and pantriagonal.

An order-8 Nasik (that is, pan-2,3,4-agonal) magic tesseract**cannot**exist.

- an order-8
**order 10**- an order-10
**pan-3-agonal**magic tesseract (non-associated) (CSV file) [Nakamura, August 2007]

Every pantriagonal of this tesseract is magic. Of course, every quadragonal is also magic. - an order-10
**2,3,4-agonal**magic tesseract (non-associated) (CSV file) [Nakamura, January 2013]

This magic tesseract strictly magic, that is, every diagonal, triagonal, and quadragonal of this tesseract is magic.

- an order-10
**order 12**- an order-12
**pan-4-agonal**magic tesseract (**associated**) (CSV file, 116KB) [Nakamura, August 2004]

This tesseract is associated and panmagic (that is, pan-4-agonal). - an order-12
**pan-3,4-agonal**magic tesseract (**associated**and**2D-compact**) (CSV file, 116KB) [Nakamura, September 2007]

This tesseract is associated, 2D-compact, and pan-3,4-agonal. - an order-12
**2,3,4-agonal**magic tesseract (**associated**) (CSV file, 116KB) [Nakamura,September 2006]

This tesseract is associated and strictly magic (that is, 2,3,4-agonal). - an order-12
**pan-4-agonal**&**2,3,4-agonal**magic tesseract (non-associated) (CSV file, 116KB) [Nakamura, September 2006]

This tesseract is both panmagic (that is, pan-4-agonal) and strictly magic (that is, 2,3,4-agonal).

An order-12 Nasik (that is, pan-2,3,4-agonal) magic tesseract**cannot**exist.

- an order-12
**order 14**- an order-14
**pan-3-agonal**magic tesseract (non-associated) (CSV file, 222KB) [Nakamura, August 2007]

Every pantriagonal of this tesseract is magic. Of course, every quadragonal is also magic.

**No**magic tesseract of singly even order can be pandiagonal or panquadragonal. - an order-14
**2,3,4-agonal**magic tesseract (non-associated) (CSV file, 222KB) [Nakamura, January 2013]

This magic tesseract strictly magic, that is, every diagonal, triagonal, and quadragonal of this tesseract is magic.

- an order-14
**order 15**- an order-15
**2,3,4-agonal**magic tesseract (**associated**) (CSV file, 295KB) [Nakamura, January 2013]

This magic tesseract strictly magic, that is, every diagonal, triagonal, and quadragonal of this tesseract is magic.

- an order-15
**order 18**- an order-18
**2,3,4-agonal**magic tesseract (non-associated) (zipped CSV file, 286KB) [Nakamura, January 2013]

This magic tesseract strictly magic, that is, every diagonal, triagonal, and quadragonal of this tesseract is magic.

- an order-18

**order 4**- an order-4
**pan-3,5-agonal**5-D magic hypercube (**associated**) (CSV file) [Nakamura, August 2004]

This magic hypercube is associated and panmagic (that is, pan-5-agonal). It is also pan-3-agonal. - an order-4
**pan-5-agonal**5-D magic hypercube (**associated**and**4D-compact**) (CSV file) [Nakamura, January 2012]

This magic hypercube is associated, 4D-compact (that is, 2x2x2x2-compact), and panmagic (that is, pan-5-agonal).

There does**not**exist a hypercube of dimension 5 and order 4 which is "associated, 3D-compact (that is, 2x2x2-compact), and panmagic (that is, pan-5-agonal)". - an order-4
**2,5-agonal**5-D magic hypercube (non-associated) (CSV file) [Nakamura, May 2008] - an order-4
**4,5-agonal**5-D magic hypercube (**associated**) (CSV file) [Nakamura, May 2008] - an order-4
**3,5-agonal**5-D magic hypercube (non-associated,**knight tour**) (CSV file) [Nakamura, April 2009]

The 1,024 consecutive integers of this magic hypercube trace out a**magic knight tour**. It is also 3-agonal and 5-agonal.

- an order-4
**order 6**- an order-6
**pan-5-agonal**5-D magic hypercube (non-associated, complete) (CSV file) [Nakamura, November 2004]

This magic hypercube is panmagic (that is, pan-5-agonal) and complete. - an order-6
**pan-5-agonal**5-D magic hypercube (**associated**) (CSV file) [Nakamura, March 2008]

This magic hypercube is associated and panmagic (that is, pan-5-agonal).

- an order-6
**order 8**- an order-8
**pan-3,5-agonal**5-D magic hypercube (**associated**) (CSV file, 197KB) [Nakamura, August 2004]

This magic hypercube is associated and panmagic (that is, pan-5-agonal). It is also pan-3-agonal. - an order-8
**pan-3,5-agonal**5-D magic hypercube (**associated**,**3D-compact**) (CSV file, 197KB) [Nakamura, June 2009]

This magic hypercube is associated, 3D-compact (that is, 2x2x2-compact), and panmagic (that is, pan-5-agonal). It is also pan-3-agonal.

There is**no**normal magic hypercube of**odd**dimension which is both associated and 2D-compact (that is, 2x2-compact). - an order-8
**2,3,4,5-agonal**5-D magic hypercube (**associated**) (CSV file, 197KB) [Nakamura, December 2004]

This magic hypercube is associated and strictly magic (that is, 2,3,4,5-agonal). - an order-8
**3,5-agonal**5-D magic hypercube (non-associated,**knight tour**) (CSV file) [Nakamura, May 2009]

The 32,768 consecutive integers of this magic hypercube trace out a**magic knight tour**. It is also 3-agonal and 5-agonal.

- an order-8
**order 10**- an order-10
**pan-3,5-agonal**5-D magic hypercube (**associated**) (zipped CSV file, 245KB) [Nakamura, May 2008]

This magic hypercube is associated and panmagic (that is, pan-5-agonal). It is also pan-3-agonal.

- an order-10

**order 4**- an order-4
**pan-3,5-agonal**6-D magic hypercube (**associated**) (CSV file) [Nakamura, September 2007]

This magic hypercube is associated and pan-3,5-agonal. - an order-4
**2,5,6-agonal**6-D magic hypercube (**associated**) (CSV file) [Nakamura, May 2008] - an order-4
**4,6-agonal**6-D magic hypercube (**associated**) (CSV file) [Nakamura, May 2008]

- an order-4
**order 6**- an order-6
**5,6-agonal**6-D magic hypercube (non-associated) (zipped CSV file, 127KB) [Nakamura, January 2012]

- an order-6
**order 8**- an order-8
**pan-3,5,6-agonal**6-D magic hypercube (**associated**) (zipped CSV file, 717KB) [Nakamura, August 2004]

This magic hypercube is associated and panmagic (that is, pan-6-agonal). It is also pan-3,5-agonal. - an order-8
**pan-3,5,6-agonal**6-D magic hypercube (**associated**and**2D-compact**) (zipped CSV file, 659KB) [Nakamura, September 2007]

This magic hypercube is associated, 2D-compact, and panmagic (that is, pan-6-agonal). It is also pan-3,5-agonal. - an order-8
**2,3,4,5,6-agonal**6-D magic hypercube (**associated**) (zipped CSV file, 773KB) [Nakamura, December 2004]

This magic hypercube is associated and strictly magic (that is, 2,3,4,5,6-agonal).

- an order-8

**order 4**- an order-4
**pan-3,5,7-agonal**7-D magic hypercube (**associated**) (CSV file, 112KB) [Nakamura, August 2004]

This magic hypercube is associated and panmagic (that is, pan-7-agonal). It is also pan-3,5-agonal. - an order-4
**pan-7-agonal**7-D magic hypercube (**associated**and**4D-compact**) (CSV file, 113KB) [Nakamura, January 2012]

This magic hypercube is associated, 4D-compact (that is, 2x2x2x2-compact), and panmagic (that is, pan-7-agonal).

There does**not**exist a hypercube of dimension 7 and order 4 which is "associated, 3D-compact (that is, 2x2x2-compact), and panmagic (that is, pan-7-agonal)". - an order-4
**3,5,7-agonal**7-D magic hypercube (non-associated,**knight tour**) (CSV file, 112KB) [Nakamura, April 2009]

The 16,384 consecutive integers of this magic hypercube trace out a**magic knight tour**. It is also 3-agonal, 5-agonal, and 7-agonal.

- an order-4
**order 6**- an order-6
**pan-7-agonal**7-D magic hypercube (**associated**) (zipped CSV file, 863KB) [Nakamura, May 2008]

This magic hypercube is associated and panmagic (that is, pan-7-agonal).

- an order-6

**1. Associated panmagic hypercubes**

**(1) Order m = 4**

I proved in August 2004 that an associated panmagic hypercube exists for order m = 4 and any **odd** dimension n >= 3.

(Examples for dimension 5 and dimension 7 are shown above, and an example for dimension 3 is here.)

On the other hand, every order-4 panmagic hypercube of **even** dimension is complete, so an order-4 (normal) panmagic hypercube of **even** dimension **cannot** be associated.

In particular, **no** order-4 associated pandiagonal magic square exists, and **no** order-4 associated panquadragonal magic tesseract can exist.

**(2) Order m = 4x (m >= 8)**

I proved in August 2004 that an associated panmagic hypercube exists for any order m = 4x >= 8 and any dimension n >= 2.

(Examples for order 8 and dimension 4, dimension 5, and dimension 6 are shown above, and an example for order 8 and dimension 3 is here.)

In addition, if the dimension n is **even**, an associated 2D-compact panmagic hypercube can exist for any order m = 4x >= 8.

(Examples for order 8 and dimension 4 and dimension 6 are shown above. A magic hypercube of **odd** dimension **cannot** be both associated and 2D-compact.)

**(3) Order 4x+2 (m >= 6)**

I proved in May 2008 that an associated panmagic hypercube exists for any order m = 4x+2 >= 6 and any **odd** dimension n >= 3.

(Examples for order 6 and dimension 5 and dimension 7 are shown above, and an example for order 6 and dimension 3 is here.)

On the other hand, if the dimension is **even** and the order is **singly-even**, a magic hypercube can be **neither** panmagic **nor** associated.

**2. Strictly magic hypercubes**

(1) I proved in December 2004 that a strictly magic hypercube of order m and dimension n exists if the order m is divisible by 2^{e}, where e is the least integer such that 2^{e} > n+1.

For example, a strictly magic hypercube of order 8 exists for any dimension lower than 7, and a strictly magic hypercube of order 16 exists for any dimension lower than 15.

(Examples for order 8 and dimension 4, dimension 5, and dimension 6 are shown above, and an example for order 8 and dimension 3 is here.)

I proved in May 2008 that a strictly magic hypercube of order 8 and dimension 7 **cannot** exist.

(2) I proved in January 2013 that a strictly magic tesseract (that is, 2,3,4-agonal magic tesseract) exists for any **even** order greater than 7, and for any **odd** order greater than 14. Examples for orders 8, 10, 12, 14, 15, and 18 are shown above. (For order 16 and 17, there exists a Nasik (stronger than strictly magic) tesseract.) It is an open problem whether a strictly magic tesseract of order 6, 7, 9, 11, or 13 exists or not (it is impossible for order 3, 4, or 5).

**3. Order-4 diagonal magic hypercubes**

I proved in May 2008 the following theorem on the existence of a diagonal (namely, 2-agonal) magic hypercube of order 4:

A diagonal magic hypercube of order 4 and dimension n >= 2 **cannot** exist if __n = 3 (mod. 4)__, and exists if not.

For example, such a magic hypercube exists for dimension 2, 4, 5, 6, or 8, and **cannot** exist for dimension 3, 7, or 11.

(Examples for dimension 4, dimension 5, and dimension 6 are shown above. It is well known that an order-4 diagonal magic hypercube **cannot** exist for dimension 3.)

On the other hand, a triagonal (namely, 3-agonal) magic hypercube of order 4 and dimension n exists for any dimension n >= 3.

**4. Magic hypercubes with knight tours**

I proved in May 2009 that a magic hypercube with a **knight tour** exists for any order m = 4x >= 4 and any **odd** dimension n >= 3.

(Examples for dimension 5 and order 4, dimension 5 and order 8, and dimension 7 and order 4 are shown above, and an example for dimension 3 and order 4 is here.)

On the other hand, a magic hypercube with a **knight tour** is not found yet for **even** dimension n >= 4. **Awani Kumar** constructed an order-4 semimagic tesseract with a **knight tour**, whose quadragonals are **not** magic, in September 2008. He constructed various (semi)magic hypercubes with **knight tours**, for example, a magic square of order 12 with a knight tour, magic cubes of orders 8, 12, and 16 with knight tours, a 5-dimensional semimagic hypercube of order 4 with a knight tour (whose quintagonals are not magic), 2-dimensional magic tours of order 8 on the surfaces of a cube and a tesseract (whose diagonals are not magic), etc.

See the following sites to study magic knight tours:

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This page was last updated on August 8, 2016.

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