Office Applications and Entertainment, Magic Cubes

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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.

It is possible to construct Sudoku Comparable Cubes by combining 5 Quinary Squares, as deducted in Section 3.7.2 of 'Magic Squares', applying following selection criteria (ref. SudCube5b):

  • Select a top square S1i (i = 1 ... n3) with elements s1(p,q) and p,q = 1 ... 5 and s1(3,3) ≠ 2;

  • Select a second square S2j (j > i) with elements s2(p,q) and p,q = 1 ... 5 and s2(3,3) ≠ 2,
    such that s2(p,q) ≠ s1(p,q) for p,q = 1 ... 5;

  • Select a third square S3k (k > j) with elements s3(p,q) and p,q = 1 ... 5 and s3(3,3) = 2,
    such that s3(p,q) ≠ s2(p,q) and s3(p,q) ≠ s1(p,q) for p,q = 1 ... 5;

  • Select a forth square S4l (l > k) with elements s4(p,q) and p,q = 1 ... 5 and s4(3,3) ≠ 2,
    such that s4(p,q) ≠ s3(p,q), s4(p,q) ≠ s2(p,q) and s4(p,q) ≠ s1(p,q) for p,q = 1 ... 5;

  • The fifth square S5m can be calculated by s5(p,q)= 10 Σ si(p,q) for i = 1 ... 4 and p,q = 1 ... 5
    . i

The resulting cube is a Sudoku Comparable Cube with a center element 2, for which the rows, columns, pillars and at least the (pan) diagonals of the horizontal planes sum to 10.

Based on Attachment 3.7.2 (Sudoku Comparable Pan Magic Squares), a collection of 2016 Sudoku Comparable Cubes with horizontal Pan Magic Planes could be generated within 12 minutes (ref. SudCube5b).

The collection could be completed by means of permutation of the planes around the center planes.

While generating the permutations, the cubes with the space diagonals summing to 10 were selected (ref. SudPerm5) and a collection of 864 Sudoku Comparable Cubes remained.

From this collection 48 unique Sudoku Comparable Semi Pan Magic Cubes could be filtered:

  • 24 with the vertical planes from left to right Pan Magic (ref. Attachment 5.3.2) and another
  • 24 with the vertical planes from back to front Pan Magic (ref. Attachment 5.3.3).

which collections could be completed with cubes obtained by 180o rotation around the vertical and/or horizontal back to front axes (ref. Rotate5), leaving two collections of 96 Sudoku Comparable Semi Pan Magic Cubes each.

The second collection contains indeed the 32 Sudoku Comparable Semi Pan Magic Cubes with symmetrical space diagonals as deducted in Section 5.3 (ref. Attachment 5.3.1).

The routine CnstrCbs5a generated :

  • 331776 Semi Pan Magic Cubes with Pan Magic vertical planes from left to right and
  • 331776 Semi Pan Magic Cubes with Pan Magic vertical planes from back to front

both series within 2,6 hours. Both collections contain 12288 cubes with symmetrical space diagonals.


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