Office Applications and Entertainment, Magic Cubes

7.0   Special Cubes, Prime Numbers

7.8   Perfect Concentric Magic Cubes (6 x 6 x 6)

7.8.1 Introduction

The Perfect Concentric Magic Cubes as constructed by Walter Trump (2003) and Mitsutoshi Nakamura (2004) for consecutive integers, are based on order 4 Plane Symmetrical Center Cubes with Horizontal Magic Planes, as
deducted in Section 7.2.8.

These center cubes have the additional property that the vertical plane diagonals sum to 2 * s4 per plane, which facilitates the border construction.

Mitsutoshi Nakamura applied a border for which the top and bottom square are concentric, with exception of the corner points which are symmetric along the space diagonals.

This border type allows for a construction comparable with the construction method described in Section 7.4.4.

7.8.2 Border Construction (General)

This section describes the construction of order 6 Prime Number Perfect Concentric Magic Cubes based on the application of:

• Order 4 Plane Symmetric Center Cubes with horizontal magic planes
• Concentric top and bottom planes
  (with exception of the corner points which are symmetric along the space diagonals)

The construction method, based on this principle, can be summarized as follows:

• Generate, for the applicable Magic Sums, pair collections reduced with the pairs required for one of the possible order 4 Plane Symmetric Center Cubes (ref. Attachment 7.2.8);
• Construct, based on these reduced collections, the magic top - and resulting bottom plane;
• Determine, based on the selected top - and resulting bottom planes, the back - and front plane;
• Determine, based on the top -, bottom -, back - and front planes, the left - and resulting right plane.

The relation between the top and bottom plane can be represented as follows:

with Pr3 = s6 / 3 the pair sum for the corresponding Magic Sum s6,
and d(i) the sum of the variables of diagonal i (i = 1 ... 16) of the vertical planes of the center cube.

The relation between the left and right plane can be represented as follows:

A comparable relation exists between the variables of the back and front plane.

7.8.3 Horizontal Magic Border Planes

With c(i) the cube variables and the substitution a(i) = c(i) for i = 1 ... 36,
the equations describing the top plane can be written as:

a(5) = s6 / 3 - a(35)
a(4) = s6 / 3 - a(34)
a(3) = s6 / 3 - a(33)
a(2) = s6 / 3 - a(32)

a(25) = s6 / 3 - a(30)
a(19) = s6 / 3 - a(24)
a(13) = s6 / 3 - a(18)
a(7)  = s6 / 3 - a(12)

a(6)  = s4 - a(31) - a(1)  - a(36)
a(32) = s6 - a(33) - a(34) - a(35) - a(31) - a(36)
a(12) = s6 - a(18) - a(24) - a(30) - a(6)  - a(36)
a(8)  = s6 - a(15) - a(22) - a(29) - a(1)  - a(36)
a(16) = s4 - a(21) - a(15) - a(22)
a(11) = s6 - a(16) - a(21) - a(26) - a(6)  - a(31)
a(27) = s4 - a(28) - a(26) - a(29)
a(10) = s6 - a(28) - a(16) - a(22) - a(4)  - a(34)
a(9)  = s4 - a(10) - a(11) - a(8)
a(20) = s4 - a(23) - a(21) - a(22)
a(17) = s4 - a(23) - a(11) - a(29)
a(14) = s4 - a(20) - a(26) - a(8)

with the independent variables (16 ea) highlighted in red.

Solutions for these equations can be generated quite fast by calculating sequentially the bottom row, the right column, the main diagonals and the remaining rows and columns.

7.8.4 Vertical Magic Border Planes (Back/Front)

With c(i) the cube variables and the substitution:

the defining equations of the Back Magic Square can be written as:

a(8)  = s6 - a(15) - a(22) - a(29) - a(1)  - a(36)
a(11) = s6 - a(17) - a(23) - a(29) - a(5)  - a(35)
a(16) = s6 - a(21) - a(26) - a(11) - a(6)  - a(31)
a(14) = s6 - a(20) - a(26) - a(8)  - a(2)  - a(32)
a(10) = s6 - a(28) - a(16) - a(22) - a(4)  - a(34)
a(9)  = s6 - a(27) - a(21) - a(15) - a(3)  - a(33)
a(25) = s6 - a(30) - a(27) - a(28) - a(26) - a(29)
a(19) = s6 - a(24) - a(20) - a(21) - a(23) - a(22)
a(13) = s6 - a(18) - a(14) - a(16) - a(17) - a(15)
a(12) = s6 - a(30) - a(18) - a(24) - a(6)  - a(36)
a(7)  = s6 - a(12) - a(9)  - a(10) - a(11) - a(8)

with a(i) independent for i = 26 ... 30,  20 ... 24 and i = 15, 17, 18
and  a(i) defined     for i = 1 ... 6 and i = 31 ... 36

Solutions for these equations can be generated quite fast by calculating sequentially the main diagonals and the remaining rows and columns.

7.8.5 Vertical Magic Border Planes (Left/Right)

Based on a comparable substitution:

the defining equations of the Magic Left Square can be written as:

a(8)  = s6 -  a(1)  - a(15) - a(22) - a(29) - a(36)
a(16) = s6 -  a(21) - a(15) - a(22) +
           - (a(1)  + a(3)  + a(4)  + a(6)  - a(7) - a(12) - a(25) - a(30) + a(31) + a(33) + a(34) + a(36))/2
a(11) = s6 -  a(6)  - a(16) - a(21) - a(26) - a(31)
a(27) = s6 -  a(25) - a(26) - a(28) - a(29) - a(30)
a(10) = s6 -  a(16) - a(22) - a(28) - a(4)  - a(34)
a(9)  = s6 -  a(15) - a(21) - a(27) - a(3)  - a(33)
a(20) = s6 -  a(21) - a(22) - a(23) - a(19) - a(24)
a(17) = s6 -  a(11) - a(23) - a(29) - a(5)  - a(35)
a(14) = s6 -  a(15) - a(16) - a(17) - a(13) - a(18)

with a(i) independent for i = 26, 28, 29,  21 ... 23 and i = 15
and  a(i) defined     for i = 1 ... 6, i = 31 ... 36 and i = 7, 12, 13, 18, 19, 24, 25, 30

Solutions for these equations can be generated quite fast by calculating sequentially the main diagonals and the remaining rows and columns.

Based on the equations listed above, guessing routines can be written to generate Prime Number Perfect Concentric Magic Cubes of order 6 within a reasonable time (PrimeCubes78).

Attachment 7.8.1 shows, for a few Magic Sums, the first occurring 6^th order Prime Number Perfect Concentric Magic Cubes.

7.8.6 Transformations

Comparable with order 6 Magic Squares, Perfect Magic Cubes of order 6 might be subject to following transformations:

• Any plane n can be interchanged with plane (7 - n), as well as the combination of these permutations.
  The possible number of unique transformations is 2³ / 2 = 4.
  
  
• Any permutation can be applied to the planes 1, 2, 3 provided that the same permutation is applied to the planes 6, 5, 4. The possible number of transformations is 3! = 6.
  
  
• Combination of abovementioned transformations will result in 24 unique solutions, which are shown in Attachment 7.8.2 for the magic sum MC = 19530 .

Note: Secondary properties, like the applied symmetry, are not invariant to the transformations described above.

Based on these 24 transformations and the 48 cubes which can be found by means of rotation and/or reflection any 6^th order (Perfect) Magic Cube corresponds with a Class of 24 * 48 = 1152 (Perfect) Magic Cubes.

7.8.7 Summary

The obtained results regarding the miscellaneous Prime Number Perfect Concentric Magic Cubes as deducted and discussed in previous sections are summarized in following table:

Following sections will describe and illustrate how Prime Number (Simple) Magic Cubes can be constructed based on the sum of three suitable selected Latin Cubes.