Office Applications and Entertainment, Magic Cubes

3.0    Magic Cubes (4 x 4 x 4)

3.16   Quaternary Composition (2)

       Introduction

Any number m = 0 ... 63 can be written as m = b₁ + 4 * b₂ + 16 * b₃ with b_i = 0, 1, 2, 3 for i = 1, 2, 3 (Quaternary representation).

Consequently any Magic Cube C of order 4 with the numbers 1 ... 64 can be written as C = B1 + 4 * B2 + 16 * B3 + [1] where the matrices B1, B2 and B3 - further referred to as Quaternary Cubes - contain only the integers 0, 1, 2 and 3.

Quaternary Cubes for which the rows, columns and pillars contain each the four integers 0, 1, 2 and 3 are also referred to as Sudoku Comparable Cubes. The rows, columns and pillars of Sudoku Comparable Cubes sum to 6.

3.16.1 Simple Magic Cubes, Horizontal Associated Magic Planes

Sudoku Comparable Magic Cubes, as defined above, can be obtained by applying the equations as deducted in Section 3.13.1, however for a Magic Sum 6.

An optimized guessing routine (SudCube41) produced 176 Sudoku Comparable Magic Cubes within 16.8 seconds, which are shown in Attachment 3.16.1.

The defined Magic Cubes can be generated by selecting combinations of Sudoku Comparable Cubes (B1, B2, B3) while ensuring that the resulting Cube C contains all integers 1 thru 64.

This was achieved with routine CnstrCbs4b, which checked the 5359200 (= 176*175*174) possibilities and produced 36864 Simple Magic Cubes of the 4^th order within ca. 7,6 hours.

3.16.2 Simple Magic Cubes, Horizontal Pan Magic Planes (3D-Compact)

Sudoku Comparable Magic Cubes, as defined above, can be obtained by applying the equations as deducted in Section 3.13.2, however for a Magic Sum 6.

An optimized guessing routine (SudCube42) produced 112 Sudoku Comparable Magic Cubes within 8.2 seconds, which are shown in Attachment 3.16.2.

The defined Magic Cubes can be generated by selecting combinations of Sudoku Comparable Cubes (B1, B2, B3) while ensuring that the resulting Cube C contains all integers 1 thru 64.

This was achieved with routine CnstrCbs4b, which checked the 1367520 (= 112*111*110) possibilities and produced 36864 Simple Magic Cubes of the 4^th order within ca. 3,5 hours.

3.16.3 Simple Magic Cubes, Associated with Horizontal Magic Planes

Sudoku Comparable Magic Cubes, as defined above, can be obtained by applying the equations as deducted in Section 3.13.3, however for a Magic Sum 6.

An optimized guessing routine (SudCube43) produced 192 Sudoku Comparable Magic Cubes within 20 seconds, which are shown in Attachment 3.16.3.

The defined Magic Cubes can be generated by selecting combinations of Sudoku Comparable Cubes (B1, B2, B3) while ensuring that the resulting Cube C contains all integers 1 thru 64.

This was achieved with routine CnstrCbs4b, which checked the 6967680 (= 192*191*190) possibilities and produced 49152 Simple Magic Cubes of the 4^th order within 13 hours.

3.16.4 Simple Magic Cubes, Associated and 3D-Compact

Sudoku Comparable Magic Cubes, as defined above, can be obtained by applying the equations as deducted in Section 3.13.4, however for a Magic Sum 6.

An optimized guessing routine (SudCube44) produced 144 Sudoku Comparable Magic Cubes within 15 seconds, which are shown in Attachment 3.16.4.

The defined Magic Cubes can be generated by selecting combinations of Sudoku Comparable Cubes (B1, B2, B3) while ensuring that the resulting Cube C contains all integers 1 thru 64.

This was achieved with routine CnstrCbs4b, which checked the 2924064 (= 144*143*142) possibilities and produced 104448 Simple Magic Cubes of the 4^th order within 7,2 hours.

3.16.5 Simple Magic Cubes, Plane Symmetrical

Sudoku Comparable Magic Cubes, as defined above, can be obtained by applying the equations as deducted in Section 3.13.5, however for a Magic Sum 6.

An optimized guessing routine (SudCube45) - with the applicable option activated - could produce:

Sudoku Comparable Cubes with Sudoku Comparable Space Diagonals (1216 ea);
Sudoku Comparable Cubes with Sudoku Comparable Horizontal Main Diagonals (48 ea).

The second collection is shown in Attachment 3.16.5.

The corresponding Magic Cubes can be generated by selecting combinations of Sudoku Comparable Cubes (B1, B2, B3) while ensuring that the resulting Cube C contains all integers 1 thru 64.

This was achieved with routine CnstrCbs4b, which checked the 103776 (= 48*47*46) possibilities and produced 18432 Simple Magic Cubes of the 4^th order within 0,7 hours.

3.16.6 Pantriagonal Magic Cubes, Complete

Quaternary Pantriagonal Complete Magic Cubes can be obtained by applying the equations deducted in Section 3.14.1, however for a Magic Sum 6.

The following - each partly overlapping - sets could be generated with an optimized guessing routine (SudCube46):

Sudoku Comparable Cubes (1536 ea);
Quaternary Cubes with Sudoku Comparable Pan Triagonals (4504 ea).

Pantriagonal Complete Magic Cubes can be generated by selecting combinations of Quaternary Cubes (B1, B2, B3) while ensuring that the resulting Cube C contains all integers 1 thru 64.

This was achieved with routine CnstrCbs4b, which checked - for the Sudoku Comparable Cubes with B1 = constant - 2354690 (= 1535*1534) possibilities and produced 34048 Pantriagonal Complete Magic Cubes within about 4,3 hours.

3.16.7 Pantriagonal Magic Cubes, Complete, Horizontal Magic Planes

Quaternary Pantriagonal Magic Cubes, as defined above, can be obtained by applying the equations deducted in Section 3.14.2, however for a Magic Sum 6.

The following - each partly overlapping - sets could be generated with an optimized guessing routine (SudCube47):

Sudoku Comparable Cubes (96 ea);
Quaternary Cubes with Sudoku Comparable Pan Triagonals (80 ea).

The resulting unique Quaternary Cubes (144 ea) are shown in Attachment 3.16.7.

The defined Pantriagonal Magic Cubes can be generated by selecting combinations of Quaternary Cubes (B1, B2, B3) while ensuring that the resulting Cube C contains all integers 1 thru 64.

This was achieved with routine CnstrCbs4b, which checked the 2924064 (= 144*143*142) possibilities and produced 135168 Pantriagonal Magic Cubes within ca. 7 hours.

3.16.8 Summary

The obtained results regarding the miscellaneous types of order 4 Magic Cubes as deducted and discussed in previous sections are summarized in following table:

Note 1: Produced with B1 = constant.