Office Applications and Entertainment, Magic Cubes

6.0   Construction Methods (Higher Order)

6.5   Composition by means of Trenkler Cubes

6.5.1 Introduction

Comparable with the Medjig Method as discussed in Section 6.6.1 of 'Magic Squares', it is possible to construct Magic Cubes C of order 2n based on Magic Cubes B of order n.

This method, as published by Professor M. Trenkler (ref. Math. Gaz. 82, March 1998), combines a Magic Cube B of order n with an Auxiliary Cube T composed of n³ order 2 Cubelets, each containing the numbers 0, 1 ... 7.

6.5.2 Simple Magic Cubes of order 6

For Simple Magic Cubes of order 6 the Trenkler Method can be summarised as follows:

• Construct a Trenkler Cube T composed of 27 order 2 Cubelets, such that all rows, columns, pillars and the four space diagonals sum to 21;
• Construct a 3 x 3 x 3 Magic Cube B (ref. Attachment 1);
• Construct the 6 x 6 x 6 Simple Magic Cube C by adding 27 * t(i,j) to b(i) for i = 1 ... 27 and j = 1 ... 8.

A numerical example is shown below:

Notes: The order 2 Cubelets applied above are plane symmetrical.
       An example of this method was previously published by A. Sayles (ref. The Monist 20, 1910 pp 299-303).

An optimized guessing routine (MgcCube6a) counted, based on the properties defined above, the following number of Trenkler Cubes per Unique Center Cubelet:

resulting in a total of 8 * 37229184 = 2,98 10⁸ Trenkler Cubes, of which a few are shown in Attachment 6.5.1.

Attachment 6.5.2 contains the resulting Simple Magic Cubes, based on the order 3 Magic Cube B shown above.

6.5.3 Associated Magic Cubes of order 6

With some minor modifications the Trenkler Method can be used for the construction of Associated Magic Cubes:

• Construct an Associated Trenkler Cube T composed of 27 order 2 Cubelets, such that:
  - all rows, columns and pillars sum to 21 and
  - the associated pairs sum to 7;
• Construct a 3 x 3 x 3 Magic Cube B (ref. Attachment 1);
• Construct the 6 x 6 x 6 Associated Magic Cube C by adding 27 * t(i,j) to b(i) for i = 1 ... 27 and j = 1 ... 8.

A numerical example is shown below:

Attachment 6.5.3 shows the first occurring 48 Associated Trenkler Cubes (ref. MgcCube6b).

Attachment 6.5.4 shows the resulting Associated Magic Cubes, based on the order 3 Magic Cube B shown above.

6.5.4 Associated Magic Cubes of order 6
      Magic Center Planes

The Trenkler Method can also be used for the construction of Associated Magic Cubes with Magic Center Planes:

• Construct an Associated Trenkler Cube T composed of 27 order 2 Cubelets, such that:
  - all rows, columns, pillars and the main diagonals of the six center planes sum to 21 and
  - the associated pairs sum to 7;
• Construct a 3 x 3 x 3 Magic Cube B (ref. Attachment 1);
• Construct the 6 x 6 x 6 Associated Magic Cube C by adding 27 * t(i,j) to b(i) for i = 1 ... 27 and j = 1 ... 8.
  The diagonals of the 6 center planes will sum to the magic sum.

A numerical example is shown below:

Note: The deduction of the applied guessing routine MgcCube6c is described in Exhibit V.

Attachment 6.5.31 shows the first occurring 48 Associated Trenkler Cubes with Magic Center Planes (generated within 40 sec).

Attachment 6.5.41 shows the resulting Associated Magic Cubes, based on the order 3 Magic Cube B shown above.

6.5.5 Associated Magic Cubes of order 6
      Magic Border Planes (s-Magic)

Associated Magic Cubes with Magic Center Planes as deducted in Section 6.5.4 above can be transformed into Associated Magic Cubes with Magic Border Planes (s-Magic).

Attachment 6.5.42 shows the s-Magic Associated Magic Cubes, based on the Associated Magic Cubes shown in Attachment 6.5.41.

6.5.6 Associated Magic Cubes of order 12

Based on the results of Section 6.5.3 above, Associated Magic Cubes of order 12 can be constructed as described below:

• Construct an Associated Trenkler Cube T composed of 216 order 2 Cubelets, such that all rows, columns and pillars sum to 42 and the associated pairs sum to 7;
• Select a 6 x 6 x 6 Associated Magic Cube B e.g. from the cubes constructed in Section 6.5.3 (ref. Attachment 6.5.4);
• Construct the 12 x 12 x 12 Associated Magic Cube C by adding 216 * t(i,j) to b(i) for i = 1 ... 216 and j = 1 ... 8.

A numerical example is shown in Attachment 6.5.5.

Note: The applied Associated Trenkler Cube (MC = 42) is composed out of 8 identical Associated Trenkler Cubes (MC = 21).

6.5.7 Associated Magic Cubes of order 18

Based on the results of Section 6.5.3 above, Associated Magic Cubes of order 18 can be constructed as described below:

• Construct an Associated Trenkler Cube T composed of 729 order 2 Cubelets, such that all rows, columns and pillars sum to 63 and the associated pairs sum to 7;
• Select a 9 x 9 x 9 Associated Magic Cube B e.g. from the cubes constructed in Section Section 6.3.2;
• Construct the 18 x 18 x 18 Associated Magic Cube C by adding 729 * t(i,j) to b(i) for i = 1 ... 216 and j = 1 ... 8.

A numerical example is shown in Attachment 6.5.6.

Note: The applied Associated Trenkler Cube (MC = 63) is composed out of 27 identical Associated Trenkler Cubes (MC = 21).

6.5.8 Summary

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

Next section will provide some examples of the construction of Magic Cubes based on Trenkler Cubes of order 8 and 16.