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8.0 Magic Cubes (6 x 6 x 6)
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:
The Magic Cubes listed above are shown in Attachment 8.1.1 and
Attachment 8.1.2.
An efficient method to generate Simple Magic Cubes of order 6 is described in Section 6.5.2.
An efficient method to generate Associated Magic Cubes of order 6 is described in Section 6.5.3.
8.4 Magic Cubes with Magic Border Planes (sMagic)
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).
8.5 Bordered Magic Cubes
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.

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.

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 
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
Attachment 8.5.2 shows, for the Anti Symmetric Magic Squares enclosed in Attachment 8.5.1, the first occurring border.
Consequently each border corresponds with 2 * 4 * 48 * n6 = 384 * n6 suitable borders.
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:
In general the resulting Bordered Magic Cube will be sMagic.
For Center Cubes with Horzontal Magic Planes, the six horizontal planes will be magic.
8.6 Perfect Magic Cubes
As mentioned above order 4 Almost Perfect Magic Cubes are not suitable for the construction of Concentric Magic Cubes.
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.
Comparable with order 6 Magic Squares (ref. 'Magic Squares' Section 6.3), Perfect Magic Cubes of order 6 might be subject to following transformations:
Note: Secondary properties, like the applied symmetry, are not invariant to the transformations described above.
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 suitable Plane Symmetrical Center Cubes which can be generated with the edge constant is 128.
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.
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.
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.
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 (sMagic)

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.

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