Office Applications and Entertainment, Magic Cubes

3.12   Quaternary Composition (1)

3.12.1 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.

3.12.2 Sudoku Comparable Cubes

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.

Solutions based on Sudoku Comparable Cubes can be found for:

• Almost Perfect Magic Cubes, as described in Section 3.2.2;
  
  
• Associated Magic Cubes, as described in Section 3.9.2;
  
  
• Pantriagonal/Associated Magic Cubes, as described in Section 3.10.2;

3.12.3 Almost Perfect Magic Cubes

Sudoku Comparable Cubes, with also the plane diagonals summing to 6, can be obtained by applying the equations as deducted in Section 3.2.2, however for a Magic Sum 6.

An optimized guessing routine (SudCube4a) produced 56 Sudoku Comparable Cubes within 2,25 seconds, which are shown in Attachment 3.6.1.

Almost Perfect 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 CnstrCbs4a, which checked the 160272 (= 56*55*54) possibilities and produced 27648 Almost Perfect Magic Cubes of the 4^th order within 1,5 hours.

The collection of 27648 Almost Perfect Magic Cubes includes 3072 Plane Symmetric Cubes (horizontal).

3.12.4 Pantriagonal Magic Cubes

Quaternary Pantriagonal Magic Cubes can be obtained by applying the equations as deducted in Section 3.7.2, however for a Magic Sum 6.

An optimized guessing routine (SudCube4b) produced 304 Quaternary Pantriagonal Magic Cubes within 13 seconds, which are shown in Attachment 3.6.4.

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, with B1 = constant, 91506 (= 303*302) possibilities and produced 384 of the 46080 Pantriagonal Magic Cubes of the 4^th order within 520 seconds.

3.12.5 Pandiagonal/Triagonal Magic Cubes

Quaternary Pan Diagonal/Triagonal Magic Cubes can be obtained by applying the equations as deducted in Section 3.8.2, however for a Magic Sum 6 under the restriction that the elements of each Pan Triagonal should be different.

An optimized guessing routine (SudCube4c) produced 384 Quaternary Pan Diagonal/Triagonal Magic Cubes within 36 seconds, which are shown in Attachment 3.6.5.

Pan Diagonal/Triagonal 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, with B1 = constant, 146306 (= 383*382) possibilities and produced 6144 Pan Diagonal/Triagonal Magic Cubes of the 4^th order within ca. 15 minutes.

3.12.6 Associated Magic Cubes

Associated Sudoku Comparable Magic Cubes can be obtained by applying the equations as deducted in Section 3.9.2, however for a Magic Sum 6.

An optimized guessing routine (SudCube4d) produced 1536 Associated Sudoku Comparable Magic Cubes within 75 seconds, which are shown in Attachment 3.6.6.

Associated 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, with B1 = constant, 2357760 (= 1536 * 1535) possibilities and produced 15360 Associated Magic Cubes of the 4^th order within 4 hours.

3.12.7 Pantriagonal/Associated Magic Cubes

Pantriagonal/Associated Sudoku Comparable Magic Cubes can be obtained by applying the equations as deducted in Section 3.10.2, however for a Magic Sum 6.

An optimized guessing routine (SudCube4e) produced 112 Pantriagonal/Associated Sudoku Comparable Magic Cubes within 2 seconds, which are shown in Attachment 3.6.7.

Pantriagonal/Associated 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 55296 Pantriagonal/Associated Magic Cubes of the 4^th order within 1,8 hours.

3.12.8 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.