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Definitions of terms are here. Explanation of classes of magic cubes is here. Works on magic tesseracts and hypercubes are here.

**order-(2**magic cubes`x`) bordered diagonal

These magic cubes are probably the first even order bordered (namely,**consecutively concentric**) magic cubes in the world, and probably the first bordered diagonal magic cubes in the world.

**Here is a colored page for an order-16 bordered diagonal magic cube.****order-(4**magic cubes`x`+2) diagonal**order-(4**magic cubes (also`x`+2) pantriagonal**associated**)

These magic cubes are probably the first order-(4`x`+2) associated pantriagonal magic cubes in the world.

**Added an order-6 associated pantriagonal magic cube constructed in February 2008.****order-(8**magic cubes (also`x`) Nasik**associated**)

These magic cubes are probably the first order-(8`x`) associated Nasik magic cubes in the world.**order-(8**magic cubes (also`x`) Nasik**associated**and**3D-compact**)

These magic cubes are probably the first order-(8`x`) Nasik magic cubes in the world which are both associated and 3D-compact.**order-(8**magic cubes (also`x`) pandiagonal**associated**)

These magic cubes are pandiagonal but**not**Nasik.**order-(4**magic cubes`x`) pantriagonal diagonal

These magic cubes are both pantriagonal and diagonal but**not**Nasik.**order-(4**magic cubes (also`x`) diagonal**associated**)**order-(4**magic cubes (also`x`) pantriagonal**associated**)**order-(2**magic cubes (also`x`+1)*pseudo-pantriagonal*diagonal**associated**)

These magic cubes have a special feature, that is, though they are**not**pantriagonal magic cubes, all pantriagonals on their oblique squares are magic. I call the feature*pseudo-pantriagonal*.

The cubes are both*pseudo-pantriagonal*and diagonal.**bimagic cubes**

These magic cubes are bimagic, that is, they are still magic when each cell of them is squared.

**order-(2**`x`) bordered diagonal magic cubes

The following magic cubes are probably the first bordered (namely,**consecutively concentric**) magic cubes of even order in the world, and probably the first bordered diagonal magic cubes in the world.

Each of these cubes is a diagonal magic cube and contains (non-normal) subcubes of all even orders lower than the order of the cube.

Every subcube consists of consecutive integers, and every subcube (except the order-4 one) is a diagonal magic cube.

(An order-4 magic cube cannot be diagonal unless it contains duplicate numbers.)

- an order-6 bordered diagonal magic cube (CSV file) [Nakamura, July 2004] Web version (colored) is here.
- an order-8 bordered diagonal magic cube (CSV file) [Nakamura, July 2004] Web version (colored) is here.
- an order-10 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]
- an order-12 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]
- an order-14 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]
- an order-16 bordered diagonal magic cube (CSV file) [Nakamura, July 2004] Web version (colored) is here. Excel version (colored) is here.
- an order-18 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]
- an order-20 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]
- an order-22 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]
- an order-24 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]
- an order-26 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]
- an order-28 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]
- an order-30 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]
- an order-32 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]
- an order-34 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]
- an order-36 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]
- an order-38 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]
- an order-40 bordered diagonal magic cube (CSV file) [Nakamura, July 2004]

Top**Note**Walter Trump made an order-5 bordered magic cube (s-magic but**not**diagonal) in June 2003. He says that an order-5**bordered****diagonal**magic cube cannot exist.

**order-(4**`x`+2) diagonal magic cubes- an order-10 diagonal magic cube (CSV file) [Nakamura, March 2004]
- an order-14 diagonal magic cube (CSV file) [Nakamura, March 2004]
- an order-18 diagonal magic cube (CSV file) [Nakamura, March 2004]

This cube is probably the first order-18 diagonal magic cube in the world. - an order-22 diagonal magic cube (CSV file) [Nakamura, March 2004]
- an order-26 diagonal magic cube (CSV file) [Nakamura, March 2004]
- an order-30 diagonal magic cube (CSV file) [Nakamura, March 2004]
- an order-34 diagonal magic cube (CSV file) [Nakamura, March 2004]
- an order-38 diagonal magic cube (CSV file) [Nakamura, March 2004]

**Note**Also refer to bordered diagonal magic cubes.**Note**Li Wen made the first order-10 diagonal magic cube in 1988.**Note**William H. Benson & Oswald Jacobi published the first order-14 diagonal magic cube in 1981. The cube is shown in the following book:

Benson & Jacoby,*Magic Cubes New Recreations*, Dover,1981, ISBN 0-486-24140-8, pp.116-126.Top**Note**An order-(4`x`+2) associated diagonal magic cube cannot exist. If such a cube were possible, its oblique squares would be order-(4`x`+2) (non-normal) associated magic squares with the odd constant. However, such a magic square cannot exist.

**order-(4**`x`+2) pantriagonal magic cubes (also associated)

The following magic cubes are probably the first order-(4`x`+2) associated pantriagonal magic cubes in the world.- an order-6 pantriagonal magic cube (associated) (CSV file) [Nakamura, February 2008]
- an order-10 pantriagonal magic cube (associated) (CSV file) [Nakamura, April 2004]
- an order-14 pantriagonal magic cube (associated) (CSV file) [Nakamura, April 2004]
- an order-18 pantriagonal magic cube (associated) (CSV file) [Nakamura, April 2004]
- an order-22 pantriagonal magic cube (associated) (CSV file) [Nakamura, April 2004]
- an order-26 pantriagonal magic cube (associated) (CSV file) [Nakamura, April 2004]
- an order-30 pantriagonal magic cube (associated) (CSV file) [Nakamura, April 2004]
- an order-34 pantriagonal magic cube (associated) (CSV file) [Nakamura, April 2004]
- an order-38 pantriagonal magic cube (associated) (CSV file) [Nakamura, April 2004]

**Note**Gakuho Abe made the first order-6**pantriagonal**magic cube (**not**associated) in 1948. He also made an order-10**3D-compact****pantriagonal**magic cube (**not**associated) in 1959.**Note**An order-(4`x`+2) associated magic hypercube of**even**dimension**cannot**exist, but this does not hold for**odd**dimensions.Top**Note**An order-(4`x`+2) panmagic hypercube of**even**dimension**cannot**exist, but this does not hold for**odd**dimensions.

In fact, Abhinav Soni made some order-(4`x`+2) panmagic hypercubes of odd dimension higher than 3 in July 2004.

**order-(8**`x`) Nasik magic cubes (also associated)

The following magic cubes are probably the first order-(8`x`) associated Nasik magic cubes (that is, pan-2,3-agonal magic cubes) in the world.- an order-16 Nasik magic cube (associated) (CSV file) [Nakamura, March 2004]
- an order-24 Nasik magic cube (associated) (CSV file) [Nakamura, March 2004]
- an order-32 Nasik magic cube (associated) (CSV file) [Nakamura, March 2004]
- an order-40 Nasik magic cube (associated) (CSV file) [Nakamura, March 2004]

**Note**A Nasik magic cube of even order can exist only if the order is divisible by 8. On the other hand, a Nasik magic cube of odd order can exist only if the order is higher than 8.**Note**An order-8 associated Nasik magic cube cannot exist. Look at this.F. A. P. Barnard published the first order-8 Nasik magic cube (Top**not**associated) in 1888.

**order-(8**`x`) Nasik magic cubes (also associated and 3D-compact)

The following magic cubes are probably the first order-(8`x`) Nasik magic cubes (that is, pan-2,3-agonal magic cubes) in the world which are both associated and 3D-compact.- an order-16 Nasik magic cube (associated and 3D-compact) (CSV file) [Nakamura, October 2007]
- an order-24 Nasik magic cube (associated and 3D-compact) (CSV file) [Nakamura, October 2007]
- an order-32 Nasik magic cube (associated and 3D-compact) (CSV file) [Nakamura, October 2007]
- an order-40 Nasik magic cube (associated and 3D-compact) (CSV file) [Nakamura, October 2007]

**order-(8**`x`) pandiagonal magic cubes (also associated)

The following magic cubes are pandiagonal but**not**pan-2,3-agonal.- an order-16 pandiagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-24 pandiagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-32 pandiagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-40 pandiagonal magic cube (associated) (CSV file) [Nakamura, May 2004]

**Note**A pandiagonal magic cube of even order can exist only if the order is divisible by 8.Top**Open problem**Can there exist an order-8 pandiagonal magic cube also associated?

**order-(4**`x`) pantriagonal diagonal magic cubes

The following magic cubes are both pantriagonal and diagonal but NOT pan-2,3-agonal.- an order-8 pantriagonal diagonal magic cube (CSV file) [Nakamura, April 2004]
- an order-12 pantriagonal diagonal magic cube (CSV file) [Nakamura, March 2004]
- an order-16 pantriagonal diagonal magic cube (CSV file) [Nakamura, March 2004]
- an order-20 pantriagonal diagonal magic cube (CSV file) [Nakamura, March 2004]
- an order-24 pantriagonal diagonal magic cube (CSV file) [Nakamura, March 2004]
- an order-28 pantriagonal diagonal magic cube (CSV file) [Nakamura, March 2004]
- an order-32 pantriagonal diagonal magic cube (CSV file) [Nakamura, March 2004]
- an order-36 pantriagonal diagonal magic cube (CSV file) [Nakamura, March 2004]
- an order-40 pantriagonal diagonal magic cube (CSV file) [Nakamura, March 2004]

**Open problem**Can there exist an order-8 pantriagonal diagonal magic cube also associated?Top**Open problem**Can there exist an order-(8`x`+4) pantriagonal diagonal magic cube also associated?

**order-(4**`x`) diagonal magic cubes (also associated)

- an order-8 diagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-12 diagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-16 diagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-20 diagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-24 diagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-28 diagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-32 diagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-36 diagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-40 diagonal magic cube (associated) (CSV file) [Nakamura, May 2004]

Top**Note**Order-4 diagonal magic cubes cannot exist. Richard Schroeppel proved this in 1972.

Walter Trump says that an order-4 s-magic cube cannot exist, either.

**order-(4**`x`) pantriagonal magic cubes (also associated)

- an order-4 pantriagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-8 pantriagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-12 pantriagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-16 pantriagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-20 pantriagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-24 pantriagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-28 pantriagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-32 pantriagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-36 pantriagonal magic cube (associated) (CSV file) [Nakamura, May 2004]
- an order-40 pantriagonal magic cube (associated) (CSV file) [Nakamura, May 2004]

**order-(2**`x`+1)*pseudo-pantriagonal*diagonal magic cubes (also associated)

The folowing magic cubes have a special feature, that is, though they are**not**pantriagonal magic cubes, all pantriagonals on their oblique squares are magic. I call the feature*pseudo-pantriagonal*.

These cubes are both*pseudo-pantriagonal*and diagonal. I made these cube by exchanging some cells of pan-2,3-agonal magic cubes of the same order.- an order-9
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, June 2004]

- an order-11
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, March 2004]

- an order-13
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, March 2004]

- an order-15
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura & Abhinav Soni, April 2004]

- an order-17
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, March 2004]

- an order-19
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, March 2004]

- an order-21
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, June 2004]

- an order-23
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, March 2004]

- an order-25
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, June 2004]

- an order-27
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, June 2004]

- an order-29
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, June 2004]

- an order-31
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, June 2004]

- an order-33
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, June 2004]

- an order-35
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, June 2004]

- an order-37
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, June 2004]

- an order-39
*pseudo-pantriagonal*diagonal magic cube (associated) (CSV file) [Nakamura, June 2004]

Top**Open problem**Can there exist an order-7*pseudo-pantriagonal*diagonal magic cube?

(An order-7 pan-2,3-agonal magic cube cannot exist, so we cannot make order-7*pseudo-pantriagonal*diagonal magic cubes by using pan-2,3-agonal magic cubes.)

- an order-9
**bimagic cubes**

- an order-16 bimagic cube (also diagonal) (CSV file) [Nakamura, April 2004]

This cube is both bimagic and diagonal, and all hirozontal planes of the cube are bimagic squares. - an order-27 bimagic cube (also diagonal and associated) (CSV file) [Nakamura, June 2004]

This cube has the same feature as the order-16 bimagic cube. In addition, this cube is associated.

*finite fields*,**F**and_{16}**F**._{27}

- an order-64 Nasik & bimagic cube (bi-1,2,3-agonal) (zipped CSV file, 781KB) [Nakamura, December 2009]

This cube is both complete Nasik (**mono-pan-1,2,3-agonal**) and bidiagonal bimagic (**bi-1,2,3-agonal**). The four triagonals are also trimagic (**tri-3-agonal**).

Here is a construction method of this cube.

Top**Note**Christian Boyer has made a lot of studies on multimagic squares and cubes (bimagic, trimagic, ...). See his site.

- an order-16 bimagic cube (also diagonal) (CSV file) [Nakamura, April 2004]

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