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 6.0   Construction Methods (Higher Order) 6.5   Composition by means of Trenkler Cubes 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. 6.5.7 Simple Magic Cubes of order 8 For Simple Magic Cubes of order 8 the Trenkler Method can be summarised as follows: Construct a Trenkler Cube T composed of 64 order 2 Cubelets, such that all rows, columns, pillars and the four space diagonals sum to 28; Select a suitable 4 x 4 x 4 Magic Cube B e.g. from the cubes constructed in Section 3.16.6; Construct the 8 x 8 x 8 Simple Magic Cube C by adding 64 * t(i,j) to b(i) for i = 1 ... 64 and j = 1 ... 8. A numerical example is shown below:
Plane 1 (Pantr/Compl)
 1 32 49 48 56 41 8 25 13 20 61 36 60 37 12 21
Plane 2
 62 35 14 19 11 22 59 38 50 47 2 31 7 26 55 42
Plane 3
 4 29 52 45 53 44 5 28 16 17 64 33 57 40 9 24
Plane 4
 63 34 15 18 10 23 58 39 51 46 3 30 6 27 54 43
Plane 1 (Trenkler Cube T)
 0 6 3 5 7 1 4 2 3 5 0 6 4 2 7 1 4 2 7 1 3 5 0 6 7 1 4 2 0 6 3 5 0 6 3 5 7 1 4 2 3 5 0 6 4 2 7 1 4 2 7 1 3 5 0 6 7 1 4 2 0 6 3 5
Plane 3
 3 5 0 6 4 2 7 1 0 6 3 5 7 1 4 2 7 1 4 2 0 6 3 5 4 2 7 1 3 5 0 6 3 5 0 6 4 2 7 1 0 6 3 5 7 1 4 2 7 1 4 2 0 6 3 5 4 2 7 1 3 5 0 6
Plane 5
 0 6 3 5 7 1 4 2 3 5 0 6 4 2 7 1 4 2 7 1 3 5 0 6 7 1 4 2 0 6 3 5 0 6 3 5 7 1 4 2 3 5 0 6 4 2 7 1 4 2 7 1 3 5 0 6 7 1 4 2 0 6 3 5
Plane 7
 3 5 0 6 4 2 7 1 0 6 3 5 7 1 4 2 7 1 4 2 0 6 3 5 4 2 7 1 3 5 0 6 3 5 0 6 4 2 7 1 0 6 3 5 7 1 4 2 7 1 4 2 0 6 3 5 4 2 7 1 3 5 0 6
Plane 2
 4 2 7 1 3 5 0 6 7 1 4 2 0 6 3 5 0 6 3 5 7 1 4 2 3 5 0 6 4 2 7 1 4 2 7 1 3 5 0 6 7 1 4 2 0 6 3 5 0 6 3 5 7 1 4 2 3 5 0 6 4 2 7 1
Plane 4
 7 1 4 2 0 6 3 5 4 2 7 1 3 5 0 6 3 5 0 6 4 2 7 1 0 6 3 5 7 1 4 2 7 1 4 2 0 6 3 5 4 2 7 1 3 5 0 6 3 5 0 6 4 2 7 1 0 6 3 5 7 1 4 2
Plane 6
 4 2 7 1 3 5 0 6 7 1 4 2 0 6 3 5 0 6 3 5 7 1 4 2 3 5 0 6 4 2 7 1 4 2 7 1 3 5 0 6 7 1 4 2 0 6 3 5 0 6 3 5 7 1 4 2 3 5 0 6 4 2 7 1
Plane 8
 7 1 4 2 0 6 3 5 4 2 7 1 3 5 0 6 3 5 0 6 4 2 7 1 0 6 3 5 7 1 4 2 7 1 4 2 0 6 3 5 4 2 7 1 3 5 0 6 3 5 0 6 4 2 7 1 0 6 3 5 7 1 4 2
Plane 1 (Magic Cube C)
 1 385 224 352 497 113 304 176 193 321 32 416 305 177 496 112 312 184 489 105 200 328 25 409 504 120 297 169 8 392 217 345 13 397 212 340 509 125 292 164 205 333 20 404 317 189 484 100 316 188 485 101 204 332 21 405 508 124 293 165 12 396 213 341
Plane 3
 254 382 35 419 270 142 467 83 62 446 227 355 462 78 275 147 459 75 278 150 59 443 230 358 267 139 470 86 251 379 38 422 242 370 47 431 258 130 479 95 50 434 239 367 450 66 287 159 455 71 282 154 55 439 234 362 263 135 474 90 247 375 42 426
Plane 5
 4 388 221 349 500 116 301 173 196 324 29 413 308 180 493 109 309 181 492 108 197 325 28 412 501 117 300 172 5 389 220 348 16 400 209 337 512 128 289 161 208 336 17 401 320 192 481 97 313 185 488 104 201 329 24 408 505 121 296 168 9 393 216 344
Plane 7
 255 383 34 418 271 143 466 82 63 447 226 354 463 79 274 146 458 74 279 151 58 442 231 359 266 138 471 87 250 378 39 423 243 371 46 430 259 131 478 94 51 435 238 366 451 67 286 158 454 70 283 155 54 438 235 363 262 134 475 91 246 374 43 427
Plane 2
 257 129 480 96 241 369 48 432 449 65 288 160 49 433 240 368 56 440 233 361 456 72 281 153 248 376 41 425 264 136 473 89 269 141 468 84 253 381 36 420 461 77 276 148 61 445 228 356 60 444 229 357 460 76 277 149 252 380 37 421 268 140 469 85
Plane 4
 510 126 291 163 14 398 211 339 318 190 483 99 206 334 19 403 203 331 22 406 315 187 486 102 11 395 214 342 507 123 294 166 498 114 303 175 2 386 223 351 306 178 495 111 194 322 31 415 199 327 26 410 311 183 490 106 7 391 218 346 503 119 298 170
Plane 6
 260 132 477 93 244 372 45 429 452 68 285 157 52 436 237 365 53 437 236 364 453 69 284 156 245 373 44 428 261 133 476 92 272 144 465 81 256 384 33 417 464 80 273 145 64 448 225 353 57 441 232 360 457 73 280 152 249 377 40 424 265 137 472 88
Plane 8
 511 127 290 162 15 399 210 338 319 191 482 98 207 335 18 402 202 330 23 407 314 186 487 103 10 394 215 343 506 122 295 167 499 115 302 174 3 387 222 350 307 179 494 110 195 323 30 414 198 326 27 411 310 182 491 107 6 390 219 347 502 118 299 171
 The resulting Magic Cube C constructed above is Pantriagonal and Complete. Attachment 6.5.7 shows a few more examples of order 8 Pantriagonal Magic Cubes based on the Trenkler Cube applied above and following order 4 Base Cubes: Pantriagonal Magic Cube, Complete, Horizontal Magic Planes Pantriagonal Magic Cube, Plane Symmetrical Attachment 6.5.8 contains an order 8 Simple Magic Cube with Horizontal Pan Magic Planes based on the Trenkler Cube shown above and an order 4 Simple Magic Base Cube with Horizontal Pan Magic Planes (ref. Section 3.13.2). Notes: It should be noted that Almost Perfect Magic Cubes are not suitable as Base Cubes, as the space diagonals don't sum to the Magic Sum s4 (ref. Section 3.3). John Hendricks used the Trenkler Cube applied above for the construction of Pantriagonal, Perfect Pandiagonal, Compact and Complete Magic Cubes based on order 8 Most Perfect Base Squares. 6.5.8 Associated Magic Cubes of order 8 The Trenkler Method can also be used for the construction of Associated Magic Cubes, as described below: Construct an Associated Trenkler Cube T composed of 64 order 2 Cubelets, such that all rows, columns and pillars sum to 28 and the associated pairs sum to 7; Select an Associated 4 x 4 x 4 Magic Cube B e.g. from the cubes constructed in Section 3.16.3; Construct the 8 x 8 x 8 Associated Magic Cube C by adding 64 * t(i,j) to b(i) for i = 1 ... 64 and j = 1 ... 8. A numerical example is shown below:
Plane 1 (Ass, Hor Magic)
 8 61 1 60 11 50 14 55 62 7 59 2 49 12 56 13
Plane 2
 45 24 44 17 34 27 39 30 23 46 18 43 28 33 29 40
Plane 3
 25 36 32 37 22 47 19 42 35 26 38 31 48 21 41 20
Plane 4
 52 9 53 16 63 6 58 3 10 51 15 54 5 64 4 57
Plane 1 (Ass Trenkler Cube T)
 0 6 3 5 7 1 4 2 3 5 0 6 4 2 7 1 4 2 7 1 3 5 0 6 7 1 4 2 0 6 3 5 0 6 3 5 7 1 4 2 3 5 0 6 4 2 7 1 4 2 7 1 3 5 0 6 7 1 4 2 0 6 3 5
Plane 3
 3 5 0 6 4 2 7 1 0 6 3 5 7 1 4 2 7 1 4 2 0 6 3 5 4 2 7 1 3 5 0 6 3 5 0 6 4 2 7 1 0 6 3 5 7 1 4 2 7 1 4 2 0 6 3 5 4 2 7 1 3 5 0 6
Plane 5
 5 3 6 0 2 4 1 7 6 0 5 3 1 7 2 4 1 7 2 4 6 0 5 3 2 4 1 7 5 3 6 0 5 3 6 0 2 4 1 7 6 0 5 3 1 7 2 4 1 7 2 4 6 0 5 3 2 4 1 7 5 3 6 0
Plane 7
 6 0 5 3 1 7 2 4 5 3 6 0 2 4 1 7 2 4 1 7 5 3 6 0 1 7 2 4 6 0 5 3 6 0 5 3 1 7 2 4 5 3 6 0 2 4 1 7 2 4 1 7 5 3 6 0 1 7 2 4 6 0 5 3
Plane 2
 4 2 7 1 3 5 0 6 7 1 4 2 0 6 3 5 0 6 3 5 7 1 4 2 3 5 0 6 4 2 7 1 4 2 7 1 3 5 0 6 7 1 4 2 0 6 3 5 0 6 3 5 7 1 4 2 3 5 0 6 4 2 7 1
Plane 4
 7 1 4 2 0 6 3 5 4 2 7 1 3 5 0 6 3 5 0 6 4 2 7 1 0 6 3 5 7 1 4 2 7 1 4 2 0 6 3 5 4 2 7 1 3 5 0 6 3 5 0 6 4 2 7 1 0 6 3 5 7 1 4 2
Plane 6
 1 7 2 4 6 0 5 3 2 4 1 7 5 3 6 0 5 3 6 0 2 4 1 7 6 0 5 3 1 7 2 4 1 7 2 4 6 0 5 3 2 4 1 7 5 3 6 0 5 3 6 0 2 4 1 7 6 0 5 3 1 7 2 4
Plane 8
 2 4 1 7 5 3 6 0 1 7 2 4 6 0 5 3 6 0 5 3 1 7 2 4 5 3 6 0 2 4 1 7 2 4 1 7 5 3 6 0 1 7 2 4 6 0 5 3 6 0 5 3 1 7 2 4 5 3 6 0 2 4 1 7
Plane 1 (Ass Magic Cube C)
 8 392 253 381 449 65 316 188 200 328 61 445 257 129 508 124 267 139 498 114 206 334 55 439 459 75 306 178 14 398 247 375 62 446 199 327 507 123 258 130 254 382 7 391 315 187 450 66 305 177 460 76 248 376 13 397 497 113 268 140 56 440 205 333
Plane 3
 237 365 24 408 300 172 465 81 45 429 216 344 492 108 273 145 482 98 283 155 39 423 222 350 290 162 475 91 231 359 30 414 215 343 46 430 274 146 491 107 23 407 238 366 466 82 299 171 476 92 289 161 29 413 232 360 284 156 481 97 221 349 40 424
Plane 5
 345 217 420 36 160 288 101 485 409 25 356 228 96 480 165 293 86 470 175 303 403 19 362 234 150 278 111 495 339 211 426 42 355 227 410 26 166 294 95 479 419 35 346 218 102 486 159 287 112 496 149 277 425 41 340 212 176 304 85 469 361 233 404 20
Plane 7
 436 52 329 201 117 501 144 272 372 244 393 9 181 309 80 464 191 319 70 454 378 250 387 3 127 511 134 262 442 58 323 195 394 10 371 243 79 463 182 310 330 202 435 51 143 271 118 502 133 261 128 512 324 196 441 57 69 453 192 320 388 4 377 249
Plane 2
 264 136 509 125 193 321 60 444 456 72 317 189 1 385 252 380 11 395 242 370 462 78 311 183 203 331 50 434 270 142 503 119 318 190 455 71 251 379 2 386 510 126 263 135 59 443 194 322 49 433 204 332 504 120 269 141 241 369 12 396 312 184 461 77
Plane 4
 493 109 280 152 44 428 209 337 301 173 472 88 236 364 17 401 226 354 27 411 295 167 478 94 34 418 219 347 487 103 286 158 471 87 302 174 18 402 235 363 279 151 494 110 210 338 43 427 220 348 33 417 285 157 488 104 28 412 225 353 477 93 296 168
Plane 6
 89 473 164 292 416 32 357 229 153 281 100 484 352 224 421 37 342 214 431 47 147 275 106 490 406 22 367 239 83 467 170 298 99 483 154 282 422 38 351 223 163 291 90 474 358 230 415 31 368 240 405 21 169 297 84 468 432 48 341 213 105 489 148 276
Plane 8
 180 308 73 457 373 245 400 16 116 500 137 265 437 53 336 208 447 63 326 198 122 506 131 259 383 255 390 6 186 314 67 451 138 266 115 499 335 207 438 54 74 458 179 307 399 15 374 246 389 5 384 256 68 452 185 313 325 197 448 64 132 260 121 505
 It can be noticed that the Horizontal Magic Planes of the resulting order 8 Associated Magic Cube C are Magic. 6.5.9 Associated Magic Cubes of order 16 Based on the results of Section 6.5.8 above, Associated Magic Cubes of order 16 can be constructed as described below: Construct an Associated Trenkler Cube T composed of 512 order 2 Cubelets, such that all rows, columns and pillars sum to 56 and the associated pairs sum to 7; Select an 8 x 8 x 8 Associated Magic Cube B e.g. from the cubes constructed in Section 6.5.9 (ref. Attachment 6.5.8); Construct the 16 x 16 x 16 Associated Magic Cube C by adding 512 * t(i,j) to b(i) for i = 1 ... 512 and j = 1 ... 8. A numerical example is shown in Attachment 6.5.9. Note: The applied Associated Trenkler Cube (MC = 56) is composed out of 8 identical Associated Trenkler Cubes (MC = 28). 6.5.10 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 Property Results Pantriagonal Pantriagonal Magic Cubes (8 x 8 x 8) - Simple Simple Magic Cube (8 x 8 x 8) Hor. PM Planes Associated Trenkler Cube Associated Magic Cube (16 x 16 x 16) Hor. Magic Planes Attachment 6.5.9 page 1 Attachment 6.5.9 page 2
 Next section will provide some examples of the construction of Magic Cubes of single even order (10, 14 and 18).