Office Applications and Entertainment, Magic Cubes

8.0   Magic Cubes (6 x 6 x 6)

8.1   Historical Background

The historical development, from the first order 6 Simple Magic Cube to the order 6 Perfect Magic Cube and later the Pantriagonal Associated Magic Cube, can be summarised as follows:

Type	Author	Year
Simple Magic Cube	W. Firth	1889
Magic Center Planes	John Worthington	1910
Simple Magic Cube, Associated	Ir. Weidemann	1922
Pantriagonal Complete	Gahuko Abe	1948
Magic Border Planes (s-Magic)	Walter Trump	2003, Sept
Perfect Magic Cube	Walter Trump	2003, Sept.
Perfect Magic Cube	Mitsutoshi Nakamura	2004, July
Pantriagonal Associated	Mitsutoshi Nakamura	2008

The Magic Cubes listed above are shown in Attachment 8.1.1 and Attachment 8.1.2.

Following sections will describe and illustrate how comparable cubes can be constructed or generated.

8.2   Simple Magic Cubes

An efficient method to generate Simple Magic Cubes of order 6 is described in Section 6.5.2.

Miscellaneous examples are shown in Attachment 6.5.2.

8.3   Associated Magic Cubes

An efficient method to generate Associated Magic Cubes of order 6 is described in Section 6.5.3.

Miscellaneous examples are shown in Attachment 6.5.4.

8.4   Magic Cubes with Magic Border Planes (s-Magic)

The first Magic Cube with Magic Border Planes, as constructed by Walter Trump (2003), was based on John Worthington’s Magic Cube with Magic Center Planes (1910).

Section 6.5.4 describes an efficient method to generate Associated Magic Cubes with Magic Center Planes.

Attachment 6.5.41 contains miscellaneous examples of Associated Magic Cubes with Magic Center Planes.

Attachment 6.5.42 contains the corresponding Associated Magic Cubes with Magic Border Planes (s-Magic).

8.5   Bordered Magic Cubes

8.5.1 Border Construction

Comparable with the method discussed in Section 4.3b, order 6 Bordered Magic Cubes can be constructed based on Complementary Anti Symmetric Magic Squares of order 6.

Examples of such squares, which can be used as top squares for Bordered Magic Cubes, are shown in Attachment 8.5.1.

The relation between opposite surface squares can be represented as follows:

with Pr3 = s6 / 3 the pair sum for the corresponding Magic Sum s6.

With c(i) the cube variables and the substitution:

the defining equations of the Left Magic Square can be written as:

a(8)  = s6 - a(15) - a(22) - a(29) - a(1)  - a(36)
a(11) = s6 - a(17) - a(23) - a(29) - a(5)  - a(35)
a(16) = s6 - a(21) - a(26) - a(11) - a(6)  - a(31)
a(14) = s6 - a(20) - a(26) - a(8)  - a(2)  - a(32)
a(10) = s6 - a(28) - a(16) - a(22) - a(4)  - a(34)
a(9)  = s6 - a(27) - a(21) - a(15) - a(3)  - a(33)
a(25) = s6 - a(30) - a(27) - a(28) - a(26) - a(29)
a(19) = s6 - a(24) - a(20) - a(21) - a(23) - a(22)
a(13) = s6 - a(18) - a(14) - a(16) - a(17) - a(15)
a(12) = s6 - a(30) - a(18) - a(24) - a(6)  - a(36)
a(7)  = s6 - a(12) - a(9)  - a(10) - a(11) - a(8)

with a(i) independent for i = 26 ... 30,  20 ... 24 and i = 15, 17, 18
and  a(i) defined     for i = 1 ... 6 and i = 31 ... 36

Based on a comparable substitution:

the defining equations of the Magic Back Square can be written as:

a(8)  = s6 -  a(1)  - a(15) - a(22) - a(29) - a(36)
a(16) = s6 -  a(21) - a(15) - a(22) +
           - (a(1)  + a(3)  + a(4)  + a(6)  - a(7) - a(12) - a(25) - a(30) + a(31) + a(33) + a(34) + a(36))/2
a(11) = s6 -  a(6)  - a(16) - a(21) - a(26) - a(31)
a(27) = s6 -  a(25) - a(26) - a(28) - a(29) - a(30)
a(10) = s6 -  a(16) - a(22) - a(28) - a(4)  - a(34)
a(9)  = s6 -  a(15) - a(21) - a(27) - a(3)  - a(33)
a(20) = s6 -  a(21) - a(22) - a(23) - a(19) - a(24)
a(17) = s6 -  a(11) - a(23) - a(29) - a(5)  - a(35)
a(14) = s6 -  a(15) - a(16) - a(17) - a(13) - a(18)

with a(i) independent for i = 26, 28, 29,  21 ... 23 and i = 15
and  a(i) defined     for i = 1 ... 6, i = 31 ... 36 and i = 7, 12, 13, 18, 19, 24, 25, 30

Based on the equations listed above, a guessing routine can be written to generate Bordered Magic Cubes of order 6 within a reasonable time (MgcCube6i).

Attachment 8.5.2 shows, for the Anti Symmetric Magic Squares enclosed in Attachment 8.5.1, the first occurring border.

For each suitable top square numerous borders can be generated (n6, dependent from the integers applied in subject top square).

Moreover, each border corresponds with:

• 48 borders which can be obtained by means of rotation and reflection;
•  4 borders which can be obtained by interchanging plane n with plane (7 - n) for n = 2, 3;
•  2 borders which can be obtained by permutating plane 2, 3 and simultaneously plane 5, 4.

Consequently each border corresponds with 2 * 4 * 48 * n6 = 384 * n6 suitable borders.

Attachment 8.5.3 shows for Border C002 the eight borders which can be obtained from each other by plane exchange.

8.5.2 Center Cubes

Any of the following order 4 Magic Cubes, based on the integers 77, 78 ... 140 = (1, 2 ... 64) + 76, can be used as Center Cube for the borders deducted in previous section:

• Simple, Associated
• Simple, Associated and 3D-Compact
• Simple, Associated with Horizontal Magic Planes
• Simple, Horizontal Associated Magic Planes
• Simple, Horizontal Pan Magic Planes (3D-Compact)
• Simple, Plane Symmetrical
• Simple, Plane Symmetrical with Horizontal Magic Planes
  
  
• Pantriagonal, Complete
• Pantriagonal, Complete with Horizontal Magic Planes
• Pantriagonal, 2D-Compact
• Pantriagonal, 2D-Compact and Plane Symmetrical
• Pantriagonal, 2D Compact and Complete
• Pantriagonal, Associated

In general the resulting Bordered Magic Cube will be s-Magic. For Center Cubes with Horzontal Magic Planes, the six horizontal planes will be magic.

It should be noted that order 4 Almost Perfect Magic Cubes are not suitable, as the Space Diagonals don't sum to the Magic Sum s4 (ref. Section 3.3).

However order 4 Plane Symmetrical Cubes with Horizontal Magic Planes can be used for the construction of Concentric Perfect Magic Cubes, which will be discussed in Section 8.6.

8.6   Perfect Magic Cubes

8.6.1 Center Cubes

As mentioned above order 4 Almost Perfect Magic Cubes are not suitable for the construction of Concentric Magic Cubes.

This would require a border with corner pairs which are non symmetric over the diagonals, which is not possible (ref. Exhibit VIIIa).

The Perfect Concentric Magic Cubes as constructed by Walter Trump (2003) and Mitsutoshi Nakamura (2004) are based on order 4 Plane Symmetrical Cubes with Horizontal Magic Planes (ref. Attachment 8.1.1).

These center cubes have the additional property that the vertical plane diagonals sum to 2 * s4 per plane, which facilitates the border construction.

8.6.2 Border Construction

Mitsutoshi Nakamura applied a border for which the top and bottom square are concentric, with exception of the corner points which are symmetric over the space diagonals.

This border type allows for a construction comparable with the border described in Section 8.5.1 above (ref. Exhibit VIIIb).

Attachment 8.6.2 shows some additional order 6 Perfect Concentric Magic Cubes, based on a selection of order 4 Plane Symmetrical Cubes.

Attachment 8.6.3 shows some additional order 6 Perfect Concentric Magic Cubes, based on miscellaneous top squares.

8.6.3 Transformations

Comparable with order 6 Magic Squares (ref. 'Magic Squares' Section 6.3), Perfect Magic Cubes of order 6 might be subject to following transformations:

• Any plane n can be interchanged with plane (7 - n), as well as the combination of these permutations.
  The possible number of unique transformations is 2³ / 2 = 4.
  
  
• Any permutation can be applied to the planes 1, 2, 3 provided that the same permutation is applied to the planes 6, 5, 4. The possible number of transformations is 3! = 6.
  
  
• Combination of abovementioned transformations will result in 24 unique solutions, which are shown in Attachment 8.6.4.

Note: Secondary properties, like the applied symmetry, are not invariant to the transformations described above.

Based on these 24 transformations and the 48 cubes which can be found by means of rotation and/or reflection any 6^th order (Perfect) Magic Cube corresponds with a Class of 24 * 48 = 1152 (Perfect) Magic Cubes.

8.6.4 Enumeration (Partial)

Although a complete enumeration of order 6 Perfect Concentric Magic Cubes is beyond the scope of this section, a partial enumeration can be made based on the results of previous sections.

The number of surface planes which can be generated with the variables of the edge constant are:

• 256 Top  Squares based on the 32 remaining integers of the top and bottom squares;
•   2 Left Squares based on the 32 remaining integers of the left and right squares;
•   1 Back Square  based on the 32 remaining integers of the back and front squares.

The number of suitable Plane Symmetrical Center Cubes which can be generated with the edge constant is 128.

This results in 2 * 256 * 128 = 65536 Perfect Concentric Magic Cubes, not counting rotation, reflection or transformation as discussed in Section 8.6.3 above.

8.6.5 Higher Order Perfect Concentric Magic Cubes

Mitsutoshi Nakamura has proven that Perfect Concentric Magic Cubes can be constructed for any even order higher than 4, and provides on his website examples of such cubes for order 6 to 40.

8.7   Pantriagonal Magic Cubes

Mitsutoshi Nakamura provides on his website, amongst others, algorithms to construct Pantriagonal Magic Cubes of order m = 4x + 2 for m >= 6.

8.7.1 Pantriagonal and Complete

The algorithm to construct a Pantriagonal Complete Magic Cube of order m = 6 has been incorporated in procedure CnstrPntr6. The resulting cube is shown below:

A Pantriagonal Magic Cube can be transformed into another Pantriagonal Magic Cube by moving an orthogonal plane from one side of the cube to the other.

Consequently a Pantriagonal Magic Cube belongs to a collection of 6³ * 48 = 10368 elements which can be found by means of rotation, reflection or planar shifts.

The Class of 48 elements which can be obtained by rotation/reflection of a Pantriagonal Magic Cube is shown in Attachment 8.7.2.

The Class of 216 elements which can be obtained by planar shifts of a Pantriagonal Magic Cube is shown in Attachment 8.7.1. Each cube of Attachment 8.7.2 can be used as a Base for Attachment 8.7.1.

It should be noted that the planar shifts are from right to left (L1 ... L5), from front to back (B1 ... B5) and from bottom to top (T1 ... T5).

The cube shown above is essential different from the cube of Gahuko Abe as it can’t be obtained by any of the operations described above (rotation, reflection, planar shifts).

8.7.2 Pantriagonal and Associated

The algorithm to construct a Pantriagonal Associated Magic Cube of order m = 6 has been incorporated in procedure AssPntr6. The resulting cube is shown below:

Although other Pantriagonal Magic Cubes can be constructed by means of rotation, reflection and/or planar shifts, the associated property is not invariant to planar shifts.

Attachment 8.7.3 shows some additional Pantriagonal Associated Magic Cubes which could be found with the method described in Exhibit VIIIc.

8.8   Summary

The obtained results regarding the miscellaneous types of order 6 Magic Cubes as deducted and discussed in previous sections are summarized in following table:

Next section will provide some methods for the construction and generation of order 7 Magic Cubes.