Office Applications and Entertainment, Magic Cubes

5.0   Quinary Composition (5 x 5 x 5)

5.1   Introduction

Any number n = 0 ... 124 can be written as n = b₁ + 5 * b₂ + 25 * b₃ with b_i = 0, 1, 2, 3, 4 for i = 1, 2, 3 (Quinary representation).

Consequently any Magic Cube C of order 5 with the numbers 1 ... 125 can be written as C = B1 + 5 * B2 + 25 * B3 + [1] where the matrices B1, B2 and B3 - further referred to as Quinary Cubes - contain only the integers 0, 1, 2, 3 and 4.

5.2   Sudoku Comparable Cubes

Quinary Cubes for which the rows, columns and pillars contain each the five integers 0, 1, 2, 3 and 4 are also referred to as Sudoku Comparable Cubes. The rows, columns and pillars of Sudoku Comparable Cubes sum to 10.

Solutions based on Sudoku Comparable Cubes can be found for:

• Semi Pan Magic Cubes, as described in Section 4.7;
  
  
• Pantriagonal Magic Cubes, as described in Section 4.8;
  
  
• Moriyama Magic Cubes, as described in Section 4.9;
  
  
• Almost Perfect Center Symmetric Magic Cubes, as described in Section 4.6;

5.3   Semi Pan Magic Cubes (1)

Semi Pan Magic Sudoku Comparable Cubes can be obtained by applying the equations as deducted in Section 4.7, however for a Magic Sum 10.

An optimized guessing routine (SudCube5a) produced 32 Sudoku Comparable Cubes within 1,2 seconds, which are shown in Attachment 5.3.1.

Semi Pan Magic Cubes can be generated by selecting combinations of Quinary Cubes (B1, B2, B3) while ensuring that the resulting Cube C contains all integers 1 thru 125.

This was achieved with routine CnstrCbs5a, which checked the 29760 (= 32*31*30) possibilities and produced 12288 Semi Pan Magic Cubes of the 5^th order within 11 minutes.

It should be noted that, as the Sudoku Comparable Cubes were generated based on the equations deducted in Section 4.7, the resulting Semi Pan Magic Cubes have symmetric space diagonals.

An alternative method of obtaining Sudoku Comparable Cubes, allowing also the generation of Semi Pan Magic Cubes with non symmetrical space diagonals, is described in Exhibit III.

5.4   Semi Pan Magic Cubes (2)

Another suitable method for Semi Pan Magic Cubes with symmetric space diagonals is to generate Pan Magic Top Squares based on 3 Quinary Pan Magic Squares as available in Attachment 3.7.1 and to calculate the remaining 75 elements with the equations deducted in Section 4.7.

The required routine (CnstrCbs5b) generated, with the first Sudoku Comparable Square B1 (Nr. 240) fixed, 384 Semi Pan Magic Cubes (ref. Attachment 5.4.1) within 133 seconds.

The same routine counted a total of 12288 Semi Pan Magic Cubes in ca. 8 hours. After defining the occurring numbers of the first square B1 in subject routine, the 12288 solutions were found within an hour.

5.5   Pantriagonal Magic Cubes, General

Pantriagonal Magic Sudoku Comparable Cubes can be obtained by applying the equations as deducted in Section 4.8, however for a Magic Sum 10.

An optimized guessing routine (SudCube5c) produced 480 Sudoku Comparable Cubes within 1038 seconds, which are shown in Attachment 5.5.1.

Pantriagonal Magic Cubes can be generated by selecting combinations of Quinary Cubes (B1, B2, B3) while ensuring that the resulting Cube C contains all integers 1 thru 125.

This was achieved with routine CnstrCbs5a, which checked, with B1 = constant, 228962 (= 479*478) possibilities and produced 86400 Pantriagonal Magic Cubes of the 5^th order within 40 minutes.

The total number of Pantriagonal Magic Cubes of the 5^th order which can be found, based on the 480 Sudoku Comparable Cubes mentioned above, is 480 * 86400 = 41472000 (= 125 * 331776).

5.6a  Pantriagonal Magic Cubes, Moriyama (1)

Sudoku Comparable Moriyama Cubes (Associated and Non-Associated) can be obtained by applying the equations as deducted in Section 4.9, however for a Magic Sum 10.

An optimized guessing routine (SudCube5d1) produced 96 Sudoku Comparable Cubes within 15 seconds, which are shown in Attachment 5.6.1.

Subject Pantriagonal Magic Cubes, for which the center diagonals sum to the Magic Sum, can be generated by selecting combinations of Quinary Cubes (B1, B2, B3) while ensuring that the resulting Cube C contains all integers 1 thru 125.

This was achieved with routine CnstrCbs5a, which checked the 857280 (= 96*95*94) possibilities and produced 331776 Pantriagonal Magic Cubes of the 5^th order within 2,6 hours.

5.6b  Pantriagonal Magic Cubes, Moriyama (2)

Sudoku Comparable Moriyama Cubes (Associated) can be obtained by applying the equations as deducted in Section 4.9, however for a Magic Sum 10.

An optimized guessing routine (SudCube5d2) produced 32 Sudoku Comparable Cubes within 3,8 seconds, which are shown in Attachment 5.6.2.

The Associated Pantriagonal Magic Cubes can be generated by selecting combinations of Quinary Cubes (B1, B2, B3) while ensuring that the resulting Cube C contains all integers 1 thru 125.

This was achieved with routine CnstrCbs5a, which checked the 29760 (= 32*31*30) possibilities and produced 12288 Associated Pantriagonal Magic Cubes of the 5^th order within 11,5 minutes.

5.7   Pan Diagonal/Triagonal Magic Cubes

Quinary Pan Diagonal/Triagonal Magic Cubes can be obtained by applying the equations as deducted in Section 4.10, however for a Magic Sum 10 under the restriction that the elements of each Pan Diagonal should be different.

Because of the, even for the integers 0, 1, 2, 3 and 4, large amount of independent variables the process was broken down into two steps:

• An optimized guessing routine (SudCube5e1) produced 161280 Top Squares with Sudoku Comparable (pan) diagonals within ca. 8 hours.
  
  
• Another guessing routine (SudCube5e2) produced, based on these generated Top Squares, 720 Quinary Magic Cubes within 27 minutes, which are shown in Attachment 5.7.1.

Pan Diagonal/Triagonal Magic Cubes can be generated by selecting combinations of Quinary Cubes (B1, B2, B3) while ensuring that the resulting Cube C contains all integers 1 thru 125.

This was achieved with routine CnstrCbs5a, which checked, with B1 = constant, 516252 (= 719*718) possibilities and produced 288000 Pan Diagonal/Triagonal Magic Cubes of the 5^th order within ca. 1,5 hours.

5.8   Almost Perfect Center Symmetric Magic Cubes, Andrews

Almost Perfect Center Symmetric Sudoku Comparable Cubes can be obtained by applying the equations as deducted in Section 4.6, however for a Magic Sum 10.

An optimized guessing routine (SudCube5f) produced 128 Sudoku Comparable Cubes within 15 seconds, which are shown in Attachment 5.8.1.

Almost Perfect Center Symmetric Magic Cubes can be generated by selecting combinations of Quinary Cubes (B1, B2, B3) while ensuring that the resulting Cube C contains all integers 1 thru 125.

This was achieved with routine CnstrCbs5a, which checked the 1685040 (= 120*119*118) possibilities and produced 24576 Almost Perfect Center Symmetric Magic Cubes of the 5^th order within 5,8 hours.

5.9    Summary

The obtained results regarding the miscellaneous types of order 5 Magic Cubes as deducted and discussed in previous sections are summarized in following table:

Note 1: Produced with B1 = constant.