Office Applications and Entertainment, Magic Cubes

8b   Perfect Concentric Magic Cubes (6 x 6 x 6)
     Border Construction

8b-1 Introduction

This Exhibit VIIIb describes the construction of order 6 Perfect Concentric Magic Cubes based on the application of:

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

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.

8b-2 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.

8b-3 Vertical Magic Border Planes (L/R)

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

the defining equations of the Left 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.

8b-4 Vertical Magic Border Planes (B/F)

Based on a comparable substitution:

the defining equations of the Magic Back 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.

8b-5 Conclusion

The linear equations deducted in previous sections can be incorporated in a guessing routine for the Magic Sum s1 = 651, and the integers 1 ... 76 and 141 ... 216.

Solutions can be obtained by guessing the 36 independent variables. With a careful selection of the variables this appeared to be possible within a reasonable time.