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 6.0   Construction Methods (Higher Order) 6.5   Composition by means of Trenkler Cubes 6.5.1 Introduction Comparable with the Medjig Method as discussed in Section 6.6.1 of 'Magic Squares', it is possible to construct Magic Cubes C of order 2n based on Magic Cubes B of order n. This method, as published by Professor M. Trenkler (ref. Math. Gaz. 82, March 1998), combines a Magic Cube B of order n with an Auxiliary Cube T composed of n3 order 2 Cubelets, each containing the numbers 0, 1 ... 7. 6.5.2 Simple Magic Cubes of order 6 For Simple Magic Cubes of order 6 the Trenkler Method can be summarised as follows: Construct a Trenkler Cube T composed of 27 order 2 Cubelets, such that all rows, columns, pillars and the four space diagonals sum to 21; Construct a 3 x 3 x 3 Magic Cube B (ref. Attachment 1); Construct the 6 x 6 x 6 Simple Magic Cube C by adding 27 * t(i,j) to b(i) for i = 1 ... 27 and j = 1 ... 8. A numerical example is shown below:
Magic Cube B Trenkler Cube T Magic Cube C
Plane 1
 10 8 24 5 21 16 27 13 2
Plane 1
 0 5 5 0 6 5 3 6 3 6 3 0 3 5 5 0 3 5 6 0 3 6 6 0 6 0 0 6 3 6 3 5 5 3 0 5
Plane 2
 7 2 2 7 1 2 4 1 4 1 4 7 4 2 2 7 4 2 1 7 4 1 1 7 1 7 7 1 4 1 4 2 2 4 7 2
Plane 1
 10 145 143 8 186 159 91 172 89 170 105 24 86 140 156 21 97 151 167 5 102 183 178 16 189 27 13 175 83 164 108 162 148 94 2 137
Plane 2
 199 64 62 197 51 78 118 37 116 35 132 213 113 59 75 210 124 70 32 194 129 48 43 205 54 216 202 40 110 29 135 81 67 121 191 56
Plane 2
 6 19 17 25 14 3 11 9 22
Plane 3
 0 5 5 6 0 5 3 6 0 3 3 6 6 0 3 6 6 0 3 5 5 0 3 5 6 0 3 6 6 0 3 5 5 0 3 5
Plane 4
 7 2 2 1 7 2 4 1 7 4 4 1 1 7 4 1 1 7 4 2 2 7 4 2 1 7 4 1 1 7 4 2 2 7 4 2
Plane 3
 6 141 154 181 17 152 87 168 19 100 98 179 187 25 95 176 165 3 106 160 149 14 84 138 173 11 90 171 184 22 92 146 144 9 103 157
Plane 4
 195 60 73 46 206 71 114 33 208 127 125 44 52 214 122 41 30 192 133 79 68 203 111 57 38 200 117 36 49 211 119 65 63 198 130 76
Plane 3
 26 15 1 12 7 23 4 20 18
Plane 5
 5 0 5 0 6 5 3 6 3 6 3 0 0 3 3 6 3 6 5 6 5 0 0 5 5 0 5 3 3 5 3 6 0 6 6 0
Plane 6
 2 7 2 7 1 2 4 1 4 1 4 7 7 4 4 1 4 1 2 1 2 7 7 2 2 7 2 4 4 2 4 1 7 1 1 7
Plane 5
 161 26 150 15 163 136 107 188 96 177 82 1 12 93 88 169 104 185 147 174 142 7 23 158 139 4 155 101 99 153 85 166 20 182 180 18
Plane 6
 80 215 69 204 28 55 134 53 123 42 109 190 201 120 115 34 131 50 66 39 61 196 212 77 58 193 74 128 126 72 112 31 209 47 45 207

Notes: The order 2 Cubelets applied above are plane symmetrical.
An example of this method was previously published by A. Sayles (ref. The Monist 20, 1910 pp 299-303).

An optimized guessing routine (MgcCube6a) counted, based on the properties defined above, the following number of Trenkler Cubes per Unique Center Cubelet:

 a(94) a(93) a(88) a(87) j94 j93 j88 n9 n/Plane Total 0 3 5 6 1 2 3 552 28368 15659136 0 3 6 5 1 2 4 208 2880 599040 0 5 3 6 1 3 2 496 28368 14070528 0 5 6 3 1 3 4 2216 1440 3191040 0 6 3 5 1 4 2 192 2880 552960 0 6 5 3 1 4 3 2192 1440 3156480 Total 5856 - 37229184

resulting in a total of 8 * 37229184 = 2,98 108 Trenkler Cubes, of which a few are shown in Attachment 6.5.1.

Attachment 6.5.2 contains the resulting Simple Magic Cubes, based on the order 3 Magic Cube B shown above.

6.5.3 Associated Magic Cubes of order 6

With some minor modifications the Trenkler Method can be used for the construction of Associated Magic Cubes:

• Construct an Associated Trenkler Cube T composed of 27 order 2 Cubelets, such that:
- all rows, columns and pillars sum to 21 and
- the associated pairs sum to 7;
• Construct a 3 x 3 x 3 Magic Cube B (ref. Attachment 1);
• Construct the 6 x 6 x 6 Associated Magic Cube C by adding 27 * t(i,j) to b(i) for i = 1 ... 27 and j = 1 ... 8.

A numerical example is shown below:

Magic Cube B Associated Trenkler Cube T Associated Magic Cube C
Plane 1
 10 8 24 5 21 16 27 13 2
Plane 1
 7 1 1 7 4 1 2 4 4 2 2 7 1 4 1 4 7 4 2 7 7 2 2 1 7 4 1 4 4 1 2 1 7 2 2 7
Plane 2
 0 5 5 6 0 5 3 6 0 3 6 3 3 5 5 0 3 5 6 0 3 6 6 0 6 5 5 0 0 5 3 0 3 6 6 3
Plane 1
 199 37 35 197 132 51 64 118 116 62 78 213 32 113 48 129 205 124 59 194 210 75 70 43 216 135 40 121 110 29 81 54 202 67 56 191
Plane 2
 10 145 143 170 24 159 91 172 8 89 186 105 86 140 156 21 97 151 167 5 102 183 178 16 189 162 148 13 2 137 108 27 94 175 164 83
Plane 2
 6 19 17 25 14 3 11 9 22
Plane 3
 7 4 4 1 4 1 2 1 7 2 2 7 4 1 4 7 4 1 2 7 1 2 2 7 4 1 1 7 7 1 2 7 4 2 2 4
Plane 4
 3 5 5 3 0 5 6 0 0 6 6 3 0 5 5 6 0 5 6 3 0 3 6 3 0 5 5 0 6 5 6 3 6 3 3 0
Plane 3
 195 114 127 46 125 44 60 33 208 73 71 206 133 52 122 203 111 30 79 214 41 68 57 192 119 38 36 198 211 49 65 200 117 63 76 130
Plane 4
 87 141 154 100 17 152 168 6 19 181 179 98 25 160 149 176 3 138 187 106 14 95 165 84 11 146 144 9 184 157 173 92 171 90 103 22
Plane 3
 26 15 1 12 7 23 4 20 18
Plane 5
 4 1 1 4 7 4 2 7 7 2 2 1 7 1 1 4 7 1 2 4 7 2 2 4 4 1 4 7 1 4 2 7 1 2 2 7
Plane 6
 0 5 5 0 6 5 6 3 3 6 3 0 6 5 5 0 0 5 3 0 3 6 3 6 0 5 5 3 3 5 6 3 0 6 6 0
Plane 5
 134 53 42 123 190 109 80 215 204 69 55 28 201 39 34 115 212 50 66 120 196 61 77 131 112 31 128 209 45 126 58 193 47 74 72 207
Plane 6
 26 161 150 15 163 136 188 107 96 177 82 1 174 147 142 7 23 158 93 12 88 169 104 185 4 139 155 101 99 153 166 85 20 182 180 18
 Attachment 6.5.3 shows the first occurring 48 Associated Trenkler Cubes (ref. MgcCube6b). Attachment 6.5.4 shows the resulting Associated Magic Cubes, based on the order 3 Magic Cube B shown above. 6.5.4 Associated Magic Cubes of order 6       Magic Center Planes The Trenkler Method can also be used for the construction of Associated Magic Cubes with Magic Center Planes: Construct an Associated Trenkler Cube T composed of 27 order 2 Cubelets, such that: - all rows, columns, pillars and the main diagonals of the six center planes sum to 21 and - the associated pairs sum to 7; Construct a 3 x 3 x 3 Magic Cube B (ref. Attachment 1); Construct the 6 x 6 x 6 Associated Magic Cube C by adding 27 * t(i,j) to b(i) for i = 1 ... 27 and j = 1 ... 8. The diagonals of the 6 center planes will sum to the magic sum. A numerical example is shown below:
Magic Cube B Associated Trenkler Cube T Associated Magic Cube C
Plane 1
 10 8 24 5 21 16 27 13 2
Plane 1
 6 7 0 1 7 0 3 0 6 3 4 5 7 1 3 7 2 1 2 0 6 4 3 6 3 6 6 4 0 2 0 7 0 2 5 7
Plane 2
 1 4 5 4 1 6 5 2 7 2 2 3 5 3 0 2 4 7 4 6 5 1 5 0 5 4 3 5 3 1 1 2 1 7 6 4
Plane 1
 172 199 8 35 213 24 91 10 170 89 132 159 194 32 102 210 70 43 59 5 183 129 97 178 108 189 175 121 2 56 27 216 13 67 137 191
Plane 2
 37 118 143 116 51 186 145 64 197 62 78 105 140 86 21 75 124 205 113 167 156 48 151 16 162 135 94 148 83 29 54 81 40 202 164 110
Plane 2
 6 19 17 25 14 3 11 9 22
Plane 3
 7 6 7 0 1 0 0 5 2 6 5 3 0 7 2 6 2 4 6 2 4 0 3 6 3 1 2 6 4 5 5 0 4 3 6 3
Plane 4
 4 1 4 3 7 2 2 3 1 5 6 4 1 4 7 3 5 1 3 5 1 5 0 7 4 2 1 5 2 7 7 6 7 0 1 0
Plane 3
 195 168 208 19 44 17 6 141 73 181 152 98 25 214 68 176 57 111 187 79 122 14 84 165 92 38 63 171 130 157 146 11 117 90 184 103
Plane 4
 114 33 127 100 206 71 60 87 46 154 179 125 52 133 203 95 138 30 106 160 41 149 3 192 119 65 36 144 76 211 200 173 198 9 49 22
Plane 3
 26 15 1 12 7 23 4 20 18
Plane 5
 3 1 0 6 5 6 6 4 2 4 3 2 7 2 6 2 1 3 0 3 5 7 4 2 4 5 5 0 5 2 1 6 3 2 3 6
Plane 6
 0 2 5 7 0 7 5 7 3 1 1 4 1 4 3 1 7 5 6 5 0 4 6 0 2 3 4 1 7 4 7 0 6 7 0 1
Plane 5
 107 53 15 177 136 163 188 134 69 123 82 55 201 66 169 61 50 104 12 93 142 196 131 77 112 139 155 20 153 72 31 166 101 74 99 180
Plane 6
 26 80 150 204 1 190 161 215 96 42 28 109 39 120 88 34 212 158 174 147 7 115 185 23 58 85 128 47 207 126 193 4 182 209 18 45
 Note: The deduction of the applied guessing routine MgcCube6c is described in Exhibit V. Attachment 6.5.31 shows the first occurring 48 Associated Trenkler Cubes with Magic Center Planes (generated within 40 sec). Attachment 6.5.41 shows the resulting Associated Magic Cubes, based on the order 3 Magic Cube B shown above. 6.5.5 Associated Magic Cubes of order 6       Magic Border Planes (s-Magic) Associated Magic Cubes with Magic Center Planes as deducted in Section 6.5.4 above can be transformed into Associated Magic Cubes with Magic Border Planes (s-Magic). Attachment 6.5.42 shows the s-Magic Associated Magic Cubes, based on the Associated Magic Cubes shown in Attachment 6.5.41. 6.5.6 Associated Magic Cubes of order 12 Based on the results of Section 6.5.3 above, Associated Magic Cubes of order 12 can be constructed as described below: Construct an Associated Trenkler Cube T composed of 216 order 2 Cubelets, such that all rows, columns and pillars sum to 42 and the associated pairs sum to 7; Select a 6 x 6 x 6 Associated Magic Cube B e.g. from the cubes constructed in Section 6.5.3 (ref. Attachment 6.5.4); Construct the 12 x 12 x 12 Associated Magic Cube C by adding 216 * t(i,j) to b(i) for i = 1 ... 216 and j = 1 ... 8. A numerical example is shown in Attachment 6.5.5. Note: The applied Associated Trenkler Cube (MC = 42) is composed out of 8 identical Associated Trenkler Cubes (MC = 21). 6.5.7 Associated Magic Cubes of order 18 Based on the results of Section 6.5.3 above, Associated Magic Cubes of order 18 can be constructed as described below: Construct an Associated Trenkler Cube T composed of 729 order 2 Cubelets, such that all rows, columns and pillars sum to 63 and the associated pairs sum to 7; Select a 9 x 9 x 9 Associated Magic Cube B e.g. from the cubes constructed in Section Section 6.3.2; Construct the 18 x 18 x 18 Associated Magic Cube C by adding 729 * t(i,j) to b(i) for i = 1 ... 216 and j = 1 ... 8. A numerical example is shown in Attachment 6.5.6. Note: The applied Associated Trenkler Cube (MC = 63) is composed out of 27 identical Associated Trenkler Cubes (MC = 21). 6.5.8 Summary The obtained results regarding the miscellaneous types of Magic Cubes as deducted and discussed in previous sections are summarized in following table:
 Type Characteristics Subroutine Results Simple Trenkler Cubes Simple Magic Cubes (6 x 6 x 6) Associated Associated Trenkler Cubes Associated Magic Cubes (6 x 6 x 6) Associated Trenkler Cube Associated Magic Cube (12 x 12 x 12) - Associated Trenkler Cube Associated Magic Cube (18 x 18 x 18) - Attachment 6.5.6 page 1 Attachment 6.5.6 page 2 Associated Mgc Cntr Planes Mgc Brdr Planes Associated Trenkler Cubes Associated Magic Cubes (6 x 6 x 6)
 Next section will provide some examples of the construction of Magic Cubes based on Trenkler Cubes of order 8 and 16.