Office Applications and Entertainment, Magic Cubes

6.0   Construction Methods (Higher Order)

6.5   Composition by means of Trenkler Cubes

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.

6.5.7 Simple Magic Cubes of order 8

For Simple Magic Cubes of order 8 the Trenkler Method can be summarised as follows:

• Construct a Trenkler Cube T composed of 64 order 2 Cubelets, such that all rows, columns, pillars and the four space diagonals sum to 28;
• Select a suitable 4 x 4 x 4 Magic Cube B e.g. from the cubes constructed in Section 3.16.6;
• Construct the 8 x 8 x 8 Simple Magic Cube C by adding 64 * t(i,j) to b(i) for i = 1 ... 64 and j = 1 ... 8.

A numerical example is shown below:

The resulting Magic Cube C constructed above is Pantriagonal and Complete.

Attachment 6.5.7 shows a few more examples of order 8 Pantriagonal Magic Cubes based on the Trenkler Cube applied above and following order 4 Base Cubes:

• Pantriagonal Magic Cube, Complete, Horizontal Magic Planes
• Pantriagonal Magic Cube, Plane Symmetrical

Attachment 6.5.8 contains an order 8 Simple Magic Cube with Horizontal Pan Magic Planes based on the Trenkler Cube shown above and an order 4 Simple Magic Base Cube with Horizontal Pan Magic Planes (ref. Section 3.13.2).

Notes:

1. It should be noted that Almost Perfect Magic Cubes are not suitable as Base Cubes, as the space diagonals don't sum to the Magic Sum s4 (ref. Section 3.3).
2. John Hendricks used the Trenkler Cube applied above for the construction of Pantriagonal, Perfect Pandiagonal, Compact and Complete Magic Cubes based on order 8 Most Perfect Base Squares.

6.5.8 Associated Magic Cubes of order 8

The Trenkler Method can also be used for the construction of Associated Magic Cubes, as described below:

• Construct an Associated Trenkler Cube T composed of 64 order 2 Cubelets, such that all rows, columns and pillars sum to 28 and the associated pairs sum to 7;
• Select an Associated 4 x 4 x 4 Magic Cube B e.g. from the cubes constructed in Section 3.16.3;
• Construct the 8 x 8 x 8 Associated Magic Cube C by adding 64 * t(i,j) to b(i) for i = 1 ... 64 and j = 1 ... 8.

A numerical example is shown below:

It can be noticed that the Horizontal Magic Planes of the resulting order 8 Associated Magic Cube C are Magic.

6.5.9 Associated Magic Cubes of order 16

Based on the results of Section 6.5.8 above, Associated Magic Cubes of order 16 can be constructed as described below:

• Construct an Associated Trenkler Cube T composed of 512 order 2 Cubelets, such that all rows, columns and pillars sum to 56 and the associated pairs sum to 7;
• Select an 8 x 8 x 8 Associated Magic Cube B e.g. from the cubes constructed in Section 6.5.9 (ref. Attachment 6.5.8);
• Construct the 16 x 16 x 16 Associated Magic Cube C by adding 512 * t(i,j) to b(i) for i = 1 ... 512 and j = 1 ... 8.

A numerical example is shown in Attachment 6.5.9.

Note: The applied Associated Trenkler Cube (MC = 56) is composed out of 8 identical Associated Trenkler Cubes (MC = 28).

6.5.10 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 of single even order (10, 14 and 18).