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 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:
 c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 c15 c16 c17 c18 c19 c20 c21 c22 c23 c24 c25 c26 c27 c28 c29 c30 c31 c32 c33 c34 c35 c36
 Pr3 - c36 s6 - d1 - c32 s6 - d2 - c33 s6 - d3 - c34 s6 - d4 - c35 Pr3 - c31 s6 - d13 - c12 Pr3 - c8 Pr3 - c9 Pr3 - c10 Pr3 - c11 s6 - d9 - c7 s6 - d14 - c18 Pr3 - c14 Pr3 - c15 Pr3 - c16 Pr3 - c17 s6 - d10 - c13 s6 - d15 - c24 Pr3 - c20 Pr3 - c21 Pr3 - c22 Pr3 - c23 s6 - d11 - c19 s6 - d16 - c30 Pr3 - c26 Pr3 - c27 Pr3 - c28 Pr3 - c29 s6 - d12 - c25 Pr3 - c6 s6 - d5 - c2 s6 - d6 - c3 s6 - d7 - c4 s6 - d8 - c5 Pr3 - c1
 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:
 c1 c7 c13 c19 c25 c31 c37 c43 c49 c55 c61 c67 c73 c79 c85 c91 c97 c103 c109 c115 c121 c127 c133 c139 c145 c151 c157 c163 c169 c175 c181 c187 c193 c199 c205 c211
 Pr3 - c211 Pr3 - c187 Pr3 - c193 Pr3 - c199 Pr3 - c205 Pr3 - c181 Pr3 - c67 Pr3 - c43 Pr3 - c49 Pr3 - c55 Pr3 - c61 Pr3 - c37 Pr3 - c103 Pr3 - c79 Pr3 - c85 Pr3 - c91 Pr3 - c97 Pr3 - c73 Pr3 - c139 Pr3 - c115 Pr3 - c121 Pr3 - c127 Pr3 - c133 Pr3 - c109 Pr3 - c175 Pr3 - c151 Pr3 - c157 Pr3 - c163 Pr3 - c169 Pr3 - c145 Pr3 - c31 Pr3 - c7 Pr3 - c13 Pr3 - c19 Pr3 - c25 Pr3 - c1
 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:
Plane 1 (Top)
 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16) a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24) a(25) a(26) a(27) a(28) a(29) a(30) a(31) a(32) a(33) a(34) a(35) a(36)
```
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:
 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16) a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24) a(25) a(26) a(27) a(28) a(29) a(30) a(31) a(32) a(33) a(34) a(35) a(36)
=
 c(1) c(2) c(3) c(4) c(5) c(6) c(37) c(38) c(39) c(40) c(41) c(42) c(73) c(74) c(75) c(76) c(77) c(78) c(109) c(110) c(111) c(112) c(113) c(114) c(145) c(146) c(147) c(148) c(149) c(150) c(181) c(182) c(183) c(184) c(185) c(186)
 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:
 a(1) a(2) a(3) a(4) a(5) a(6) a(7) a(8) a(9) a(10) a(11) a(12) a(13) a(14) a(15) a(16) a(17) a(18) a(19) a(20) a(21) a(22) a(23) a(24) a(25) a(26) a(27) a(28) a(29) a(30) a(31) a(32) a(33) a(34) a(35) a(36)
=
 c(1) c(7) c(13) c(19) c(25) c(31) c(37) c(43) c(49) c(55) c(61) c(67) c(73) c(79) c(85) c(91) c(97) c(103) c(109) c(115) c(121) c(127) c(133) c(139) c(145) c(151) c(157) c(163) c(169) c(175) c(181) c(187) c(193) c(199) c(205) c(211)
 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 6th 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 6th 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:
 Order Main Characteristics Subroutine Results - - - - 6 Perfect Magic Cubes, Concentric Perfect Magic Cubes, Plane Permutations - - - - -
 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.