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8.0   Magic Cubes (6 x 6 x 6)

8.1   Historical Background

The historical development, from the first order 6 Simple Magic Cube to the order 6 Perfect Magic Cube and later the Pantriagonal Associated Magic Cube, can be summarised as follows:

 Type Author Year Simple Magic Cube W. Firth 1889 Magic Center Planes John Worthington 1910 Simple Magic Cube, Associated Ir. Weidemann 1922 Pantriagonal Complete Gahuko Abe 1948 Magic Border Planes (s-Magic) Walter Trump 2003, Sept Perfect Magic Cube Walter Trump 2003, Sept. Perfect Magic Cube Mitsutoshi Nakamura 2004, July Pantriagonal Associated Mitsutoshi Nakamura 2008

The Magic Cubes listed above are shown in Attachment 8.1.1 and Attachment 8.1.2.

Following sections will describe and illustrate how comparable cubes can be constructed or generated.

8.2   Simple Magic Cubes

An efficient method to generate Simple Magic Cubes of order 6 is described in Section 6.5.2.

Miscellaneous examples are shown in Attachment 6.5.2.

8.3   Associated Magic Cubes

An efficient method to generate Associated Magic Cubes of order 6 is described in Section 6.5.3.

Miscellaneous examples are shown in Attachment 6.5.4.

8.4   Magic Cubes with Magic Border Planes (s-Magic)

The first Magic Cube with Magic Border Planes, as constructed by Walter Trump (2003), was based on John Worthington’s Magic Cube with Magic Center Planes (1910).

Section 6.5.4 describes an efficient method to generate Associated Magic Cubes with Magic Center Planes.

Attachment 6.5.41 contains miscellaneous examples of Associated Magic Cubes with Magic Center Planes.

Attachment 6.5.42 contains the corresponding Associated Magic Cubes with Magic Border Planes (s-Magic).

8.5   Bordered Magic Cubes

8.5.1 Border Construction

Comparable with the method discussed in Section 4.3b, order 6 Bordered Magic Cubes can be constructed based on Complementary Anti Symmetric Magic Squares of order 6.

Examples of such squares, which can be used as top squares for Bordered Magic Cubes, are shown in Attachment 8.5.1.

The relation between opposite surface squares 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 Pr3 - c32 Pr3 - c33 Pr3 - c34 Pr3 - c35 Pr3 - c31 Pr3 - c12 Pr3 - c8 Pr3 - c9 Pr3 - c10 Pr3 - c11 Pr3 - c7 Pr3 - c18 Pr3 - c14 Pr3 - c15 Pr3 - c16 Pr3 - c17 Pr3 - c13 Pr3 - c24 Pr3 - c20 Pr3 - c21 Pr3 - c22 Pr3 - c23 Pr3 - c19 Pr3 - c30 Pr3 - c26 Pr3 - c27 Pr3 - c28 Pr3 - c29 Pr3 - c25 Pr3 - c6 Pr3 - c2 Pr3 - c3 Pr3 - c4 Pr3 - c5 Pr3 - c1
 with Pr3 = s6 / 3 the pair sum for the corresponding Magic Sum s6. 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(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 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 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(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 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 Based on the equations listed above, a guessing routine can be written to generate Bordered Magic Cubes of order 6 within a reasonable time (MgcCube6i). Attachment 8.5.2 shows, for the Anti Symmetric Magic Squares enclosed in Attachment 8.5.1, the first occurring border. For each suitable top square numerous borders can be generated (n6, dependent from the integers applied in subject top square). Moreover, each border corresponds with: 48 borders which can be obtained by means of rotation and reflection;  4 borders which can be obtained by interchanging plane n with plane (7 - n) for n = 2, 3;  2 borders which can be obtained by permutating plane 2, 3 and simultaneously plane 5, 4. Consequently each border corresponds with 2 * 4 * 48 * n6 = 384 * n6 suitable borders. Attachment 8.5.3 shows for Border C002 the eight borders which can be obtained from each other by plane exchange. 8.5.2 Center Cubes Any of the following order 4 Magic Cubes, based on the integers 77, 78 ... 140 = (1, 2 ... 64) + 76, can be used as Center Cube for the borders deducted in previous section: Simple, Associated Simple, Associated and 3D-Compact Simple, Associated with Horizontal Magic Planes Simple, Horizontal Associated Magic Planes Simple, Horizontal Pan Magic Planes (3D-Compact) Simple, Plane Symmetrical Simple, Plane Symmetrical with Horizontal Magic Planes Pantriagonal, Complete Pantriagonal, Complete with Horizontal Magic Planes Pantriagonal, 2D-Compact Pantriagonal, 2D-Compact and Plane Symmetrical Pantriagonal, 2D Compact and Complete Pantriagonal, Associated In general the resulting Bordered Magic Cube will be s-Magic. For Center Cubes with Horzontal Magic Planes, the six horizontal planes will be magic. It should be noted that order 4 Almost Perfect Magic Cubes are not suitable, as the Space Diagonals don't sum to the Magic Sum s4 (ref. Section 3.3). However order 4 Plane Symmetrical Cubes with Horizontal Magic Planes can be used for the construction of Concentric Perfect Magic Cubes, which will be discussed in Section 8.6. 8.6   Perfect Magic Cubes 8.6.1 Center Cubes As mentioned above order 4 Almost Perfect Magic Cubes are not suitable for the construction of Concentric Magic Cubes. This would require a border with corner pairs which are non symmetric over the diagonals, which is not possible (ref. Exhibit VIIIa). The Perfect Concentric Magic Cubes as constructed by Walter Trump (2003) and Mitsutoshi Nakamura (2004) are based on order 4 Plane Symmetrical Cubes with Horizontal Magic Planes (ref. Attachment 8.1.1). These center cubes have the additional property that the vertical plane diagonals sum to 2 * s4 per plane, which facilitates the border construction. 8.6.2 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 over the space diagonals. This border type allows for a construction comparable with the border described in Section 8.5.1 above (ref. Exhibit VIIIb). Attachment 8.6.2 shows some additional order 6 Perfect Concentric Magic Cubes, based on a selection of order 4 Plane Symmetrical Cubes. Attachment 8.6.3 shows some additional order 6 Perfect Concentric Magic Cubes, based on miscellaneous top squares. 8.6.3 Transformations Comparable with order 6 Magic Squares (ref. 'Magic Squares' Section 6.3), 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 8.6.4. 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. 8.6.4 Enumeration (Partial) Although a complete enumeration of order 6 Perfect Concentric Magic Cubes is beyond the scope of this section, a partial enumeration can be made based on the results of previous sections. The number of surface planes which can be generated with the variables of the edge constant are: 256 Top  Squares based on the 32 remaining integers of the top and bottom squares;   2 Left Squares based on the 32 remaining integers of the left and right squares;   1 Back Square  based on the 32 remaining integers of the back and front squares. The number of suitable Plane Symmetrical Center Cubes which can be generated with the edge constant is 128. This results in 2 * 256 * 128 = 65536 Perfect Concentric Magic Cubes, not counting rotation, reflection or transformation as discussed in Section 8.6.3 above. 8.6.5 Higher Order Perfect Concentric Magic Cubes Mitsutoshi Nakamura has proven that Perfect Concentric Magic Cubes can be constructed for any even order higher than 4, and provides on his website examples of such cubes for order 6 to 40. 8.7   Pantriagonal Magic Cubes Mitsutoshi Nakamura provides on his website, amongst others, algorithms to construct Pantriagonal Magic Cubes of order m = 4x + 2 for m >= 6. 8.7.1 Pantriagonal and Complete The algorithm to construct a Pantriagonal Complete Magic Cube of order m = 6 has been incorporated in procedure CnstrPntr6. The resulting cube is shown below:
Plane 1
 1 44 132 207 164 103 41 129 7 167 106 201 135 4 38 100 204 170 198 155 121 28 71 78 158 124 192 68 75 34 118 195 161 81 31 65
Plane 2
 42 127 8 166 108 200 133 5 39 102 203 169 2 45 130 206 163 105 157 126 191 69 73 35 120 194 160 79 32 66 197 154 123 29 72 76
Plane 3
 134 6 37 101 202 171 3 43 131 205 165 104 40 128 9 168 107 199 119 193 162 80 33 64 196 156 122 30 70 77 159 125 190 67 74 36
Plane 4
 189 146 139 19 62 96 149 142 183 59 93 25 136 186 152 99 22 56 10 53 114 216 173 85 50 111 16 176 88 210 117 13 47 82 213 179
Plane 5
 148 144 182 60 91 26 138 185 151 97 23 57 188 145 141 20 63 94 51 109 17 175 90 209 115 14 48 84 212 178 11 54 112 215 172 87
Plane 6
 137 184 153 98 24 55 187 147 140 21 61 95 150 143 181 58 92 27 116 15 46 83 211 180 12 52 113 214 174 86 49 110 18 177 89 208
 A Pantriagonal Magic Cube can be transformed into another Pantriagonal Magic Cube by moving an orthogonal plane from one side of the cube to the other. Consequently a Pantriagonal Magic Cube belongs to a collection of 63 * 48 = 10368 elements which can be found by means of rotation, reflection or planar shifts. The Class of 48 elements which can be obtained by rotation/reflection of a Pantriagonal Magic Cube is shown in Attachment 8.7.2. The Class of 216 elements which can be obtained by planar shifts of a Pantriagonal Magic Cube is shown in Attachment 8.7.1. Each cube of Attachment 8.7.2 can be used as a Base for Attachment 8.7.1. It should be noted that the planar shifts are from right to left (L1 ... L5), from front to back (B1 ... B5) and from bottom to top (T1 ... T5). The cube shown above is essential different from the cube of Gahuko Abe as it can’t be obtained by any of the operations described above (rotation, reflection, planar shifts). 8.7.2 Pantriagonal and Associated The algorithm to construct a Pantriagonal Associated Magic Cube of order m = 6 has been incorporated in procedure AssPntr6. The resulting cube is shown below:
Plane 1
 1 79 191 80 192 108 102 171 47 62 100 169 193 103 194 77 6 78 105 60 104 167 49 166 196 73 8 101 198 75 54 165 107 164 106 55
Plane 2
 36 129 143 182 34 127 160 190 161 2 135 3 141 132 32 131 31 184 130 4 158 5 159 195 30 189 29 134 136 133 154 7 128 197 156 9
Plane 3
 199 97 200 71 12 72 39 180 92 179 91 70 202 67 14 95 204 69 96 177 41 68 94 175 16 64 206 65 207 93 99 66 98 173 43 172
Plane 4
 45 174 44 119 151 118 124 10 152 11 153 201 42 123 149 176 40 121 148 13 122 203 150 15 147 126 38 125 37 178 145 205 146 17 120 18
Plane 5
 208 61 20 89 210 63 84 81 83 188 28 187 22 58 212 59 213 87 33 186 86 185 85 76 214 82 215 56 27 57 90 183 35 74 88 181
Plane 6
 162 111 53 110 52 163 142 19 116 209 144 21 51 168 50 113 157 112 139 211 140 23 114 24 48 117 155 170 46 115 109 25 137 26 138 216
 Although other Pantriagonal Magic Cubes can be constructed by means of rotation, reflection and/or planar shifts, the associated property is not invariant to planar shifts. Attachment 8.7.3 shows some additional Pantriagonal Associated Magic Cubes which could be found with the method described in Exhibit VIIIc. 8.8   Summary The obtained results regarding the miscellaneous types of order 6 Magic Cubes as deducted and discussed in previous sections are summarized in following table:
 Type Characteristics Subroutine Results Simple Classic Associated Classic Magic Center Planes Magic Border Planes (s-Magic) - Bordered Symmetrical Edges Perfect Concentric, Miscellaneous Center Cubes - Concentric, Miscellaneous Top Squares - Plane Permutations - Pantriagonal Complete Associated
 Next section will provide some methods for the construction and generation of order 7 Magic Cubes.