Office Applications and Entertainment, Magic Cubes

9.0   Construction Methods (Single Even Order)

Many construction methods can't be applied for Magic Cubes of Single Even Order.

Following sections describe a few methods which can be used.

The algorithms applied for Pantriagonal Magic Cubes and Perfect Magic Cubes are based on the algoritms as previously published by Misutoshi Nakamura.

9.1   Magic Cubes of order 6

The construction of Magic Cubes of order 6 has been described in following sections:

• Section 6.5 Composition by means of Trenkler Cubes;
• Section 8.5 Bordered Magic Cubes;
• Section 8.6 Perfect Magic Cubes;
• Section 8.7 Pantriagonal Magic Cubes.

Comparable methods can be used for the construction of Magic Cubes of order 10, 14 and 18.

9.2   Magic Cubes of order 10

9.2.1 Composition by means of Trenkler Cubes

For Simple Magic Cubes of order 10 the Trenkler Method can be summarised as follows:

• Construct a Trenkler Cube T composed of 125 order 2 Cubelets, such that all rows, columns, pillars and the four space diagonals sum to 35;
• Select a 5 x 5 x 5 Magic Cube B (e.g. from Attachment 4.3.2 or Attachment 4.3.4);
• Construct the 10 x 10 x 10 Simple Magic Cube C by adding 125 * t(i,j) to b(i) for i = 1 ... 125 and j = 1 ... 8.

Comparable with the Medjig Method for order 10 Concentric Magic Squares, as discussed in Section 6.6.1 of 'Magic Squares', it is quite easy to construct Concentric Trenkler Cubes of order 10.

A possible construction method for the 10 x 10 x 10 border, based on a 6 x 6 x 6 Center Cube and Plane Symmetrical Cubelets, is described below:

• Select the Corner Cubelets of the order 10 Concentric Cube such that the space diagonals sum to 35;
• Copy Plane 6-1 to the center of Plane 10-1 and complete Plane 10-1,
  Plane 10-2 is now defined (complementary);
• Copy the resulting border of Plane 10-1 to Plane 10-3, 5 and 7,
  Plane 10-4, 6 and 8 are now defined (complementary to resp. 3, 5 and 7);
• Copy Plane 6-6 to the center of Plane 10-10 and Complete Plane 10-10,
  Plane 10-9 is now defined (complementary).

Note: Plane  6-1 thru  6- 6 are the horizontal planes of the order  6 Center Cube.
      Plane 10-1 thru 10-10 are the horizontal planes of the order 10 Concentric Cube.

Numerous borders can be obtained based on the 10 x 10 x 10 border described above, by means of:

• Rotation and reflection;
• Exchange of row-, column- and pillar cubelets (with the corner cubelets constant).

Any of these borders, combined with any of the 6 x 6 x 6 Trenkler Cubes described in Section 6.5.2, will result in a Concentric Trenkler Cube T.

A 10 x 10 x 10 Concentric Trenkler Cube T can be combined with:

• a 5 x 5 x 5 Bordered Magic Cube B, resulting in a (Simple) Concentric Magic Cube C;
• any other 5 x 5 x 5 Magic Cube B (ref. Section 5.9), resulting in a Simple Magic Cube C.

An example of the construction of a Concentric Magic Cube C is shown in Attachment 9.2.1.

9.2.2 Pantriagonal Magic Cubes

The algorithm to construct a Pantriagonal Associated Magic Cube of order m = 10 is available in procedure AssPntr6.
The resulting cube is shown in Attachment 9.2.2.

9.2.3 Perfect Diagonal Magic Cubes

The algorithm to construct a Perfect Magic Cube of order m = 10 is available in procedure Perfect10.
The resulting cube is shown in Attachment 9.2.3.

9.2.4 Transformations

Comparable with order 6 Perfect Magic Cubes (ref. Section 8.6.3), Perfect Magic Cubes of order 10 might be subject to following transformations:

• Any plane n can be interchanged with plane (11 - n), as well as the combination of these permutations.
  The possible number of unique transformations is 2⁵ / 2 = 16.
  
  
• Any permutation can be applied to the planes 1, 2, 3, 4, 5 provided that the same permutation is applied to the planes 10, 9, 8, 7, 6. The possible number of transformations is 5! = 120.
  
  
• Combination of abovementioned transformations will result in 1920 unique solutions.

Note: Secondary properties, like the applied symmetry in Concentric Perfect Magic Cubes, are not invariant to the transformations described above.

Based on these 1920 transformations and the 48 cubes which can be found by means of rotation and/or reflection any 10^th order Perfect Magic Cube corresponds with a Class of 48 * 1920 = 92160 Perfect Magic Cubes.

9.3   Magic Cubes of order 14

9.3.1 Composition by means of Trenkler Cubes

For Simple Magic Cubes of order 14 the Trenkler Method can be summarised as follows:

• Construct a Trenkler Cube T composed of 343 order 2 Cubelets, such that all rows, columns, pillars and the four space diagonals sum to 49;
• Construct a 7 x 7 x 7 Magic Cube B;
• Construct the 14 x 14 x 14 Simple Magic Cube C by adding 343 * t(i,j) to b(i) for i = 1 ... 343 and j = 1 ... 8.

The construction of a Concentric Trenkler Cube of order 14 can be executed as described in Section 9.2.1 above.

The resulting 14 x 14 x 14 Trenkler Cube is concentric and can be used for the construction of 14 x 14 x 14 (Concentric) Magic Cubes.

An example is shown in Attachment 9.3.1 (C is Simple but not Concentric as B is Associated).

9.3.2 Pantriagonal Magic Cubes

The algorithm to construct a Pantriagonal Associated Magic Cube of order m = 14 is available in procedure AssPntr6.
The resulting cube is shown in Attachment 9.3.2.

9.3.3 Perfect Diagonal Magic Cubes

The algorithm to construct a Perfect Magic Cube of order m = 14 is available in procedure Perfect10.
The resulting cube is shown in Attachment 9.3.3.

9.3.4 Transformations

Perfect Magic Cubes of order 14 might be subject to following transformations:

• Any plane n can be interchanged with plane (15 - n), as well as the combination of these permutations.
  The possible number of unique transformations is 2⁷ / 2 = 64.
  
  
• Any permutation can be applied to the planes 1, 2 ... 7 provided that the same permutation is applied to the planes 14, 13 ... 8. The possible number of transformations is 7! = 5040.
  
  
• Combination of abovementioned transformations will result in 322560 unique solutions.

Note: Secondary properties are not invariant to the transformations described above.

Based on these 322560 transformations and the 48 cubes which can be found by means of rotation and/or reflection any 14^th order Perfect Magic Cube corresponds with a Class of 48 * 322560 = 15482880 Perfect Magic Cubes.

9.4   Magic Cubes of order 18

9.4.1 Composition by means of Trenkler Cubes

The construction of Associated Magic Cubes of order 18 has been described in Section 6.5.7.
An example is shown in Attachment 6.5.6.

Based on the order 9 Composed Magic Cubes as deducted in Section 6.2, order 18 Composed Magic Cubes can be constructed.
An example is shown in Attachment 9.4.1.

9.4.2 Pantriagonal Magic Cubes

The algorithm to construct a Pantriagonal Associated Magic Cube of order m = 18 is available in procedure AssPntr6.
The resulting cube is shown in Attachment 9.4.2.

9.4.3 Perfect Diagonal Magic Cubes

The algorithm to construct a Perfect Magic Cube of order m = 18 is available in procedure Perfect18.
The resulting cube is shown in Attachment 9.4.3.

9.4.4 Transformations

Perfect Magic Cubes of order 18 might be subject to following transformations:

• Any plane n can be interchanged with plane (19 - n), as well as the combination of these permutations.
  The possible number of unique transformations is 2⁹ / 2 = 256.
  
  
• Any permutation can be applied to the planes 1, 2 ... 9 provided that the same permutation is applied to the planes 18, 17 ... 10. The possible number of transformations is 9! = 362880.
  
  
• Combination of abovementioned transformations will result in 92897280 unique solutions.

Note: Secondary properties are not invariant to the transformations described above.

Based on these 92897280 transformations and the 48 cubes which can be found by means of rotation and/or reflection any 14^th order Perfect Magic Cube corresponds with a Class of 48 * 92897280 = 4459069440 Perfect Magic Cubes.

9.4.5 Composition by means of Sub Cubes

Composed Magic Cubes of order 18 can be constructed with the method described in Section 6.2.
Following compositions are possible:

• Composed of 216 Sub Cubes 3 x 3 x 3
• Composed of  27 Sub Cubes 6 x 6 x 6

The number of possible combinations for both cases, n6 * 192²¹⁶ and 192 * n6²⁷, is beyond imagination.

9.5   Summary

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