Office Applications and Entertainment, Magic Cubes

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10.0  Magic Cubes (7 x 7 x 7)

10.1  Introduction

The first known Pandiagonal Associated Magic Cube of order 7 was constructed by A. H. Frost (1866).

It was constructed based on a generalisation of the method developed by Leonhard Euler to construct Magic Squares based on Sudoku Comparable Squares (ref. e.g. Section 7.3 of 'Magic Squares').

10.2  Sudoku Comparable Cubes

Any number n = 0 ... 342 can be written as n = b1 + 7 * b2 + 49 * b3 with bi = 0, 1, 2, 3, 4, 5, 6 for i = 1, 2, 3.

Consequently any Magic Cube C of order 7 with the numbers 1 ... 343 can be written as C = B1 + 7 * B2 + 49 * B3 + [1] where the matrices B1, B2 and B3 contain only the integers 0, 1, 2, 3, 4, 5 and 6.

Such cubes for which the rows, columns and pillars contain each the seven integers 0, 1, 2, 3, 4, 5 and 6 are referred to as Sudoku Comparable Cubes.

The rows, columns and pillars of order 7 Sudoku Comparable Cubes sum to 21.

Solutions based on Sudoku Comparable Cubes can be found for:

Type

Author

Year

Pandiagonal, Associated (Perfect)

A. H. Frost

1866

Pantriagonal, Associated

John R. Hendricks

1973

Simple, Associated, 2 Sets Pandiagonals

Abhinav Soni

2001

The Historical order 7 Magic Cubes listed above are shown in Attachment 10.2.1.

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

10.3  Pandiagonal Associated Magic Cubes (Perfect)

10.3a Construction (A. H. Frost)

The decomposition of A. H. Frost's 7th order Pandiagonal Associated Magic Cube is shown below:

C = B1 + m * B2 + m2 * B3 + [1]
327 41 98 99 156 213 270
52 109 166 223 280 330 44
169 226 283 340 5 62 119
293 301 8 65 122 179 236
18 75 132 189 239 247 304
135 192 200 257 314 28 78
210 260 317 31 88 145 153
B1
4 5 6 0 1 2 3
2 3 4 5 6 0 1
0 1 2 3 4 5 6
5 6 0 1 2 3 4
3 4 5 6 0 1 2
1 2 3 4 5 6 0
6 0 1 2 3 4 5
B2
4 5 6 0 1 2 3
0 1 2 3 4 5 6
3 4 5 6 0 1 2
6 0 1 2 3 4 5
2 3 4 5 6 0 1
5 6 0 1 2 3 4
1 2 3 4 5 6 0
B3
6 0 1 2 3 4 5
1 2 3 4 5 6 0
3 4 5 6 0 1 2
5 6 0 1 2 3 4
0 1 2 3 4 5 6
2 3 4 5 6 0 1
4 5 6 0 1 2 3

113 170 227 284 341 6 63
237 294 295 9 66 123 180
305 19 76 133 183 240 248
79 136 193 201 258 315 22
154 204 261 318 32 89 146
271 328 42 92 100 157 214
45 53 110 167 224 274 331

0 1 2 3 4 5 6
5 6 0 1 2 3 4
3 4 5 6 0 1 2
1 2 3 4 5 6 0
6 0 1 2 3 4 5
4 5 6 0 1 2 3
2 3 4 5 6 0 1

2 3 4 5 6 0 1
5 6 0 1 2 3 4
1 2 3 4 5 6 0
4 5 6 0 1 2 3
0 1 2 3 4 5 6
3 4 5 6 0 1 2
6 0 1 2 3 4 5

2 3 4 5 6 0 1
4 5 6 0 1 2 3
6 0 1 2 3 4 5
1 2 3 4 5 6 0
3 4 5 6 0 1 2
5 6 0 1 2 3 4
0 1 2 3 4 5 6

249 306 20 77 127 184 241
23 80 137 194 202 259 309
147 148 205 262 319 33 90
215 272 329 36 93 101 158
332 46 54 111 168 218 275
57 114 171 228 285 342 7
181 238 288 296 10 67 124

3 4 5 6 0 1 2
1 2 3 4 5 6 0
6 0 1 2 3 4 5
4 5 6 0 1 2 3
2 3 4 5 6 0 1
0 1 2 3 4 5 6
5 6 0 1 2 3 4

0 1 2 3 4 5 6
3 4 5 6 0 1 2
6 0 1 2 3 4 5
2 3 4 5 6 0 1
5 6 0 1 2 3 4
1 2 3 4 5 6 0
4 5 6 0 1 2 3

5 6 0 1 2 3 4
0 1 2 3 4 5 6
2 3 4 5 6 0 1
4 5 6 0 1 2 3
6 0 1 2 3 4 5
1 2 3 4 5 6 0
3 4 5 6 0 1 2

91 141 149 206 263 320 34
159 216 273 323 37 94 102
276 333 47 55 112 162 219
1 58 115 172 229 286 343
125 182 232 289 297 11 68
242 250 307 21 71 128 185
310 24 81 138 195 203 253

6 0 1 2 3 4 5
4 5 6 0 1 2 3
2 3 4 5 6 0 1
0 1 2 3 4 5 6
5 6 0 1 2 3 4
3 4 5 6 0 1 2
1 2 3 4 5 6 0

5 6 0 1 2 3 4
1 2 3 4 5 6 0
4 5 6 0 1 2 3
0 1 2 3 4 5 6
3 4 5 6 0 1 2
6 0 1 2 3 4 5
2 3 4 5 6 0 1

1 2 3 4 5 6 0
3 4 5 6 0 1 2
5 6 0 1 2 3 4
0 1 2 3 4 5 6
2 3 4 5 6 0 1
4 5 6 0 1 2 3
6 0 1 2 3 4 5

220 277 334 48 56 106 163
337 2 59 116 173 230 287
69 126 176 233 290 298 12
186 243 251 308 15 72 129
254 311 25 82 139 196 197
35 85 142 150 207 264 321
103 160 217 267 324 38 95

2 3 4 5 6 0 1
0 1 2 3 4 5 6
5 6 0 1 2 3 4
3 4 5 6 0 1 2
1 2 3 4 5 6 0
6 0 1 2 3 4 5
4 5 6 0 1 2 3

3 4 5 6 0 1 2
6 0 1 2 3 4 5
2 3 4 5 6 0 1
5 6 0 1 2 3 4
1 2 3 4 5 6 0
4 5 6 0 1 2 3
0 1 2 3 4 5 6

4 5 6 0 1 2 3
6 0 1 2 3 4 5
1 2 3 4 5 6 0
3 4 5 6 0 1 2
5 6 0 1 2 3 4
0 1 2 3 4 5 6
2 3 4 5 6 0 1

13 70 120 177 234 291 299
130 187 244 252 302 16 73
198 255 312 26 83 140 190
322 29 86 143 151 208 265
96 104 161 211 268 325 39
164 221 278 335 49 50 107
281 338 3 60 117 174 231

5 6 0 1 2 3 4
3 4 5 6 0 1 2
1 2 3 4 5 6 0
6 0 1 2 3 4 5
4 5 6 0 1 2 3
2 3 4 5 6 0 1
0 1 2 3 4 5 6

1 2 3 4 5 6 0
4 5 6 0 1 2 3
0 1 2 3 4 5 6
3 4 5 6 0 1 2
6 0 1 2 3 4 5
2 3 4 5 6 0 1
5 6 0 1 2 3 4

0 1 2 3 4 5 6
2 3 4 5 6 0 1
4 5 6 0 1 2 3
6 0 1 2 3 4 5
1 2 3 4 5 6 0
3 4 5 6 0 1 2
5 6 0 1 2 3 4

191 199 256 313 27 84 134
266 316 30 87 144 152 209
40 97 105 155 212 269 326
108 165 222 279 336 43 51
225 282 339 4 61 118 175
300 14 64 121 178 235 292
74 131 188 245 246 303 17

1 2 3 4 5 6 0
6 0 1 2 3 4 5
4 5 6 0 1 2 3
2 3 4 5 6 0 1
0 1 2 3 4 5 6
5 6 0 1 2 3 4
3 4 5 6 0 1 2

6 0 1 2 3 4 5
2 3 4 5 6 0 1
5 6 0 1 2 3 4
1 2 3 4 5 6 0
4 5 6 0 1 2 3
0 1 2 3 4 5 6
3 4 5 6 0 1 2

3 4 5 6 0 1 2
5 6 0 1 2 3 4
0 1 2 3 4 5 6
2 3 4 5 6 0 1
4 5 6 0 1 2 3
6 0 1 2 3 4 5
1 2 3 4 5 6 0

Cube B2 is one of the aspects of Cube B1 (Back to Front Planes B1 are Top to Bottom Planes B2) and Cube B3 is a reflection of Cube B1 (Back/Front).

The center planes of the Sudoku Comparable Cubes B1, B2 and B3 are Ulta Magic (ref. Attachment 7.4.3) and can be used as a starting point for the construction of more Pandiagonal Associated Magic Cubes.

Attachment 10.3.1 shows a few order 7 Perfect Pandiagonal Associated Magic Cubes based on the abovementioned Ultra Magic Squares (ref. CnstrCbs7).

10.3b Transformations

Comparable with order 6 Magic Cubes (ref. Section 8.6.3), Perfect Magic Cubes of order 7 might be subject to following transformations:

  • Any plane n can be interchanged with plane (8 - 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 7, 6, 5. The possible number of transformations is 3! = 6.

  • Combination of abovementioned transformations will result in 24 unique solutions, which are shown in Attachment 10.3.2.

Note: Secondary properties, like pandiagonal diagonals, 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 7th order (Perfect) Magic Cube corresponds with a Class of 24 * 48 = 1152 (Perfect) Magic Cubes.

10.4  Pantriagonal Associated Magic Cubes

10.4a Construction (John Hendricks)

Based on a decomposition of John Hendricks 7th order Associated Pantriagonal Magic Cube as shown in Attachment 6.3.1, procedure AssPntr21 could be built, which generates the Factor Cubes B1, B2, B3 and the resulting cube C.

Higher order Associated Pantriagonal Magic Cubes, which can be constructed based on this and comparable procedures, are discussed in Section 6.3.2.

Attachment 10.4.1 shows a few additional Pantriagonal Associated Magic Cubes which could be obtained based on the method described in Section 10.3 above.

10.4b Transformations

Although the associated property is not invariant to planar shifts, 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 an order 7 Pantriagonal Magic Cube belongs to a collection of 73 * 48 = 16464 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 an order 7 Pantriagonal Magic Cube is shown in Attachment 10.4.2.

The Class of 343 elements which can be obtained by planar shifts of an order 7 Pantriagonal Magic Cube is shown in Attachment 10.4.3. Each cube of Attachment 10.4.2 can be used as a Base Cube for Attachment 10.4.3.

It should be noted that the the planar shifts are from right to left (L1 ... L6), from front to back (B1 ... B6) and from bottom to top (T1 ... T6).

10.5  Summary

The obtained results regarding the miscellaneous types of order 7 Magic Cubes as deducted and discussed in previous sections are summarized in following table:

Type

Characteristics

Subroutine

Results

Pandiagonal

Associated (Perfect)

CnstrCbs7

Attachment 10.3.1

Perfect

Diagonal Magic (Plane Permutations)

-

Attachment 10.3.2

Pantriagonal

Associated

CnstrCbs7

Attachment 10.4.1

Pantriagonal

Rotation/Reflection
Planar Shifts

-

Attachment 10.4.2
Attachment 10.4.3

Next section will provide some methods for the construction and generation of higher order Magic Cubes.

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