 |
 |
 |
 |
 |
|
|
|
|
|
With order 8, we are introduced to two new classifications,
The diagonal cube, and the perfect cube.
Both of the order 8 diagonal cubes I have appear already on other pages so I
will not reproduce them here.
I have no order 8 pantriagonal magic cubes that are associated, or that contain
magic squares. Are there any of either?
I have not seen any order 8 pandiagonal magic cubes or, for that matter, such
cubes in higher orders.
The perfect cube I show is not associated, because an associated perfect magic
cube can appear first only in order 9. Much more information on perfect cubes
and listings for such cube may be seen on my perfect
and perfect2 pages.
A feature I did test for in order 8 is for corners of smaller order cube
arrays within the summing to S.
| Andrews | 1908 | Simple magic cube, no magic squares |
| Hendricks inlaid | 1993 | Simple magic cube, no magic squares. |
| Hetherington | 1997 | Diagonal magic cube, 30 magic squares |
| Hendricks inlaid 2 | 1999 | Pantriagonal magic cube, no magic squares |
| Soni | 2001 | Pantriagonal magic cube, no magic squares |
| Hendricks Perfect | 1998 | Perfect, magic cube, 30 pandiagonal magic squares |
This cube has the minimum requirements required to be magic.
This simple magic cube, first published in 1908 is associated. It contains no magic squares, but the 8 corners of all orders 3 and 7 sub cubes sum to S.
W. S. Andrews, Magic
Squares & Cubes, Open Court, 1908, 193+ pages.
W. S. Andrews, Magic Squares & Cubes, 2nd edition,
Dover Publ. 1960, 419+ pages .
Horizontal plane I - Top II 1 511 510 4 5 507 506 8 448 66 67 445 444 70 71 441 504 10 11 501 500 14 15 497 73 439 438 76 77 435 434 80 496 18 19 493 492 22 23 489 81 431 430 84 85 427 426 88 25 487 486 28 29 483 482 32 424 90 91 421 420 94 95 417 33 479 478 36 37 475 474 40 416 98 99 413 412 102 103 409 472 42 43 469 468 46 47 465 105 407 406 108 109 403 402 112 464 50 51 461 460 54 55 457 113 399 398 116 117 395 394 120 57 455 454 60 61 451 450 64 392 122 123 389 388 126 127 385 III IV 384 130 131 381 380 134 135 377 193 319 318 196 197 315 314 200 137 375 374 140 141 371 370 144 312 202 203 309 308 206 207 305 145 367 366 148 149 363 362 152 304 210 211 301 300 214 215 297 360 154 155 357 356 158 159 353 217 295 294 220 221 291 290 224 352 162 163 349 348 166 167 345 225 287 286 228 229 283 282 232 169 343 342 172 173 339 338 176 280 234 235 277 276 238 239 273 177 335 334 180 181 331 330 184 272 242 243 269 268 246 247 265 328 186 187 325 324 190 191 321 249 263 262 252 253 259 258 256 V VI 257 255 254 260 261 251 250 264 192 322 323 189 188 326 327 185 248 266 267 245 244 270 271 241 329 183 182 332 333 179 178 336 240 274 275 237 236 278 279 233 337 175 174 340 341 171 170 344 281 231 230 284 285 227 226 288 168 346 347 165 164 350 351 161 289 223 222 292 293 219 218 296 160 354 355 157 156 358 359 153 216 298 299 213 212 302 303 209 361 151 150 364 365 147 146 368 208 306 307 205 204 310 311 201 369 143 142 372 373 139 138 376 313 199 198 316 317 195 194 320 136 378 379 133 132 382 383 129 VII VIII - Bottom 128 386 387 125 124 390 391 121 449 63 62 452 453 59 58 456 393 119 118 396 397 115 114 400 56 458 459 53 52 462 463 49 401 111 110 404 405 107 106 408 48 466 467 45 44 470 471 41 104 410 411 101 100 414 415 97 473 39 38 476 477 35 34 480 96 418 419 93 92 422 423 89 481 31 30 484 485 27 26 488 425 87 86 428 429 83 82 432 24 490 491 21 20 494 495 17 433 79 78 436 437 75 74 440 16 498 499 13 12 502 503 9 72 442 443 69 68 446 447 65 505 7 6 508 509 3 2 512
This simple magic cube is not associated. S = 1026
It contains no magic squares. All order 5 sub cubes have corners summing to S.
However, this cube is inlaid. Each of the 8 octants of this cube is itself an order 4 pantriagonal magic cube with S = 1026. One of the octants is shown below the listing for this order 8 cube.
John R. Hendricks, An Inlaid Magic Cube, JRM 25:4, 1993, pp 286-288.
Horizontal plane I - Top II 375 200 314 137 73 506 8 439 249 330 184 263 455 120 394 57 257 178 336 255 63 400 114 449 143 320 194 369 433 2 512 79 192 271 241 322 386 49 463 128 306 129 383 208 16 447 65 498 202 377 135 312 504 71 441 10 328 247 265 186 122 457 55 392 210 353 159 304 496 95 417 18 352 239 273 162 98 465 47 416 168 279 233 346 410 41 471 104 298 153 359 216 24 423 89 490 281 170 344 231 39 408 106 473 151 296 218 361 425 26 488 87 367 224 290 145 81 482 32 431 225 338 176 287 479 112 402 33 III IV 272 191 321 242 50 385 127 464 130 305 207 384 448 15 497 66 378 201 311 136 72 503 9 442 248 327 185 266 458 121 391 56 199 376 138 313 505 74 440 7 329 250 264 183 119 456 58 393 177 258 256 335 399 64 450 113 319 144 370 193 1 434 80 511 169 282 232 343 407 40 474 105 295 152 362 217 25 426 88 487 223 368 146 289 481 82 432 31 337 226 288 175 111 480 34 401 354 209 303 160 96 495 17 418 240 351 161 274 466 97 415 48 280 167 345 234 42 409 103 472 154 297 215 360 424 23 489 90 V VI 131 308 206 381 445 14 500 67 269 190 324 243 51 388 126 461 245 326 188 267 459 124 390 53 379 204 310 133 69 502 12 443 332 251 261 182 118 453 59 396 198 373 139 316 508 75 437 6 318 141 371 196 4 435 77 510 180 259 253 334 398 61 451 116 294 149 363 220 28 427 85 486 172 283 229 342 406 37 475 108 340 227 285 174 110 477 35 404 222 365 147 292 484 83 429 30 237 350 164 275 467 100 414 45 355 212 302 157 93 494 20 419 155 300 214 357 421 22 492 91 277 166 348 235 43 412 102 469 VII VIII - Bottom 252 331 181 262 454 117 395 60 374 197 315 140 76 507 5 438 142 317 195 372 436 3 509 78 260 179 333 254 62 397 115 452 307 132 382 205 13 446 68 499 189 270 244 323 387 52 462 125 325 246 268 187 123 460 54 389 203 380 134 309 501 70 444 11 349 238 276 163 99 468 46 413 211 356 158 301 493 94 420 19 299 156 358 213 21 422 92 491 165 278 236 347 411 44 470 101 150 293 219 364 428 27 485 86 284 171 341 230 38 405 107 476 228 339 173 286 478 109 403 36 366 221 291 148 84 483 29 430
 |
The top back left octant of the above cube is an order 4 pantriagonal magic cube. Because it is an inlay, the numbers used are not consecutive, and so it is not a normal cube. S = 1026. Any of the eight order 4 cubes may be rotated or reflected to any of it's 48 aspects, it may be transformed to a different pandiagonal magic cube by translocation of planes, or the 8 cubes may be rearranged within the order 8 cube in any manner, without destroying the magic of the order 8 cube. I II
III IV
|
This cube is classed as a diagonal magic cube, with extra features. This cube was never published, but I constructed it from instructions by Charles Hetherington that were received by Mutsumi Suzuki in Aug. 1997.
The 8
horizontal planes and 8 vertical planes parallel with the front of the cube, are
simple magic squares.
The 8 vertical planes parallel with the sides of the cube are pandiagonal magic
squares.
The 6 oblique planes are simple magic squares.
Corners of all orders 3, 5 and 7 sub cubes sum to S. When I mention this
feature, wrap-around is always assumed to apply. That means that there are m3
sub cubes of the order specified that are correct.
Matsumi Suzuki’s excellent site is now available at http://mathforum.com/te/exchange/hosted/suzuki/MagicSquare.html
Horizontal plane I - Top II 417 31 80 498 238 340 259 189 78 500 419 29 257 191 240 338 216 362 313 135 411 37 118 460 315 133 214 364 120 458 409 39 478 100 51 397 145 303 384 194 49 399 480 98 382 196 147 301 171 277 326 252 488 90 9 439 328 250 169 279 11 437 486 92 266 184 231 345 69 507 428 22 229 347 268 182 426 24 71 505 127 449 402 48 308 142 221 355 404 46 125 451 223 353 306 144 373 203 156 294 58 392 471 105 154 296 375 201 469 107 60 390 4 446 493 83 335 241 162 288 495 81 2 448 164 286 333 243 III IV 234 344 263 185 421 27 76 502 261 187 236 342 74 504 423 25 415 33 114 464 212 366 317 131 116 462 413 35 319 129 210 368 149 299 380 198 474 104 55 393 378 200 151 297 53 395 476 102 484 94 13 435 175 273 322 256 15 433 482 96 324 254 173 275 65 511 432 18 270 180 227 349 430 20 67 509 225 351 272 178 312 138 217 359 123 453 406 44 219 357 310 140 408 42 121 455 62 388 467 109 369 207 160 290 465 111 64 386 158 292 371 205 331 245 166 284 8 442 489 87 168 282 329 247 491 85 6 444 V VI 499 77 30 420 192 258 337 239 32 418 497 79 339 237 190 260 134 316 363 213 457 119 40 410 361 215 136 314 38 412 459 117 400 50 97 479 195 381 302 148 99 477 398 52 304 146 193 383 249 327 280 170 438 12 91 485 278 172 251 325 89 487 440 10 348 230 181 267 23 425 506 72 183 265 346 232 508 70 21 427 45 403 452 126 354 224 143 305 450 128 47 401 141 307 356 222 295 153 202 376 108 470 389 59 204 374 293 155 391 57 106 472 82 496 447 1 285 163 244 334 445 3 84 494 242 336 287 161 VII VIII - Bottom 188 262 341 235 503 73 26 424 343 233 186 264 28 422 501 75 461 115 36 414 130 320 367 209 34 416 463 113 365 211 132 318 199 377 298 152 396 54 101 475 300 150 197 379 103 473 394 56 434 16 95 481 253 323 276 174 93 483 436 14 274 176 255 321 19 429 510 68 352 226 177 271 512 66 17 431 179 269 350 228 358 220 139 309 41 407 456 122 137 311 360 218 454 124 43 405 112 466 385 63 291 157 206 372 387 61 110 468 208 370 289 159 281 167 248 330 86 492 443 5 246 332 283 165 441 7 88 490
All 256 pantriagonals are correct in this cube, so this is a pantriagonal magic cube.
Planar squares are not magic because of incorrect diagonals. Because all pantriagonals are correct, all pandiagonals are correct in the 6 oblique squares. However, 4 squares have incorrect rows and 2 have incorrect columns. i.e. there are no magic squares in this cube.
Corner sums of all orders-3 and 7 cubes equal S. Also, this cube is termed complete because every pantriagonal contains m/2 complement pairs with the two members of each pair spaced m/2 apart.
This cube is similar to the first Hendricks cube shown
inasmuch as the 8 octants are inlaid order 4 magic cubes.
However, as stated above this cube is pantriagonal. This permits it to be
transformed to a different magic cube by moving planes from one side of the cube
to the opposite side. This is similar to moving rows or columns in a pandiagonal
magic square from one side to the opposite side, to obtain a different
pandiagonal magic square. By doing this, however, the inlaid feature is lost and
the octants will no longer be magic cubes.
Of course, the order 8 cube may also be transformed by translocation of planes in any of the order 4 cubes also, because they are all pantriagonal as well. Furthermore, the cube may also be changed by rotating or reflecting any (or all) of the 8 order 4 cubes to any of their 48 aspects and/or exchanging positions of the individual cubes. These transformations will, in almost all cases, destroy the pantriagonal feature of the order 8 cube.
John R. Hendricks, Inlaid Magic Squares and Cubes, self-published, 1999, 0-9684700-1-7, 188+ pages.
Horizontal plane I - Top II 348 50 461 167 92 306 205 423 15 357 154 500 271 101 410 244 503 157 354 12 247 413 98 268 164 458 53 351 420 202 309 95 138 484 31 373 394 228 287 117 477 183 332 34 221 439 76 290 37 335 180 474 293 79 436 218 370 28 487 141 114 284 231 397 380 18 493 135 124 274 237 391 47 325 186 468 303 69 442 212 471 189 322 44 215 445 66 300 132 490 21 383 388 234 277 127 170 452 63 341 426 196 319 85 509 151 364 2 253 407 108 258 5 367 148 506 261 111 404 250 338 60 455 173 82 316 199 429 III IV 482 140 375 29 226 396 119 285 181 479 36 330 437 223 292 74 333 39 476 178 77 295 220 434 26 372 143 485 282 116 399 229 52 346 165 463 308 90 421 207 359 13 498 156 103 269 242 412 159 501 10 356 415 245 266 100 460 162 349 55 204 418 93 311 450 172 343 61 194 428 87 317 149 511 4 362 405 255 260 106 365 7 508 146 109 263 252 402 58 340 175 453 314 84 431 197 20 378 133 495 276 122 389 239 327 45 466 188 71 301 210 444 191 469 42 324 447 213 298 68 492 130 381 23 236 386 125 279 V VI 352 54 457 163 96 310 201 419 11 353 158 504 267 97 414 248 499 153 358 16 243 409 102 272 168 462 49 347 424 206 305 91 142 488 27 369 398 232 283 113 473 179 336 38 217 435 80 294 33 331 184 478 289 75 440 222 374 32 483 137 118 288 227 393 384 22 489 131 128 278 233 387 43 321 190 472 299 65 446 216 467 185 326 48 211 441 70 304 136 494 17 379 392 238 273 123 174 456 59 337 430 200 315 81 505 147 368 6 249 403 112 262 1 363 152 510 257 107 408 254 342 64 451 169 86 320 195 425 VII VIII - Bottom 486 144 371 25 230 400 115 281 177 475 40 334 433 219 296 78 329 35 480 182 73 291 224 438 30 376 139 481 286 120 395 225 56 350 161 459 312 94 417 203 355 9 502 160 99 265 246 416 155 497 14 360 411 241 270 104 464 166 345 51 208 422 89 307 454 176 339 57 198 432 83 313 145 507 8 366 401 251 264 110 361 3 512 150 105 259 256 406 62 344 171 449 318 88 427 193 24 382 129 491 280 126 385 235 323 41 470 192 67 297 214 448 187 465 46 328 443 209 302 72 496 134 377 19 240 390 121 275
Following is the top right back quadrant of the above order 8 pantriagonal magic cube. Each of these octants are also pandiagonal magic and also complete. These order 4 cubes are not normal because the numbers used in each cube are not consecutive.
I II III IV 92 306 205 423 271 101 410 244 226 396 119 285 437 223 292 74 247 413 98 268 420 202 309 95 77 295 220 434 282 116 399 229 394 228 287 117 221 439 76 290 308 90 421 207 103 269 242 412 293 79 436 218 114 284 231 397 415 245 266 100 204 418 93 311
This pantriagonal magic cube is not associated and contains
no magic squares. In fact, I have not seen any order 8 pantriagonal magic cubes
that are associated or that contain any magic squares.
Corners of all cubes of orders 2, 4, 5, 6, and 8 (including wrap-around) all sum correctly,
so this cube is compact. This cube is
also complete.
In additions All planar diagonal pairs sum to two times the magic constant, S.
There are horizontal bent diagonals (V shaped)
on all horizontal planes, and vertical planes parallel to the front of the cube
starting all cells of columns 1 and 5,
and
on all vertical planes parallel to the sides, starting on all cells of columns 3 and 7.
There are no planes that have vertical bent diagonals starting on all cells of
any particular row or column.
HyperMagicCube.exe program. Obtainable from his magic cubes site.
Horizontal plane I - Top II 1 506 3 508 8 511 6 509 488 31 486 29 481 26 483 28 128 391 126 389 121 386 123 388 409 98 411 100 416 103 414 101 129 378 131 380 136 383 134 381 360 159 358 157 353 154 355 156 256 263 254 261 249 258 251 260 281 226 283 228 288 231 286 229 449 58 451 60 456 63 454 61 40 479 38 477 33 474 35 476 448 71 446 69 441 66 443 68 89 418 91 420 96 423 94 421 321 186 323 188 328 191 326 189 168 351 166 349 161 346 163 348 320 199 318 197 313 194 315 196 217 290 219 292 224 295 222 293 III IV 41 466 43 468 48 471 46 469 504 15 502 13 497 10 499 12 88 431 86 429 81 426 83 428 393 114 395 116 400 119 398 117 169 338 171 340 176 343 174 341 376 143 374 141 369 138 371 140 216 303 214 301 209 298 211 300 265 242 267 244 272 247 270 245 489 18 491 20 496 23 494 21 56 463 54 461 49 458 51 460 408 111 406 109 401 106 403 108 73 434 75 436 80 439 78 437 361 146 363 148 368 151 366 149 184 335 182 333 177 330 179 332 280 239 278 237 273 234 275 236 201 306 203 308 208 311 206 309 V VI 57 450 59 452 64 455 62 453 480 39 478 37 473 34 475 36 72 447 70 445 65 442 67 444 417 90 419 92 424 95 422 93 185 322 187 324 192 327 190 325 352 167 350 165 345 162 347 164 200 319 198 317 193 314 195 316 289 218 291 220 296 223 294 221 505 2 507 4 512 7 510 5 32 487 30 485 25 482 27 484 392 127 390 125 385 122 387 124 97 410 99 412 104 415 102 413 377 130 379 132 384 135 382 133 160 359 158 357 153 354 155 356 264 255 262 253 257 250 259 252 225 282 227 284 232 287 230 285 VII VIII - Bottom 17 490 19 492 24 495 22 493 464 55 462 53 457 50 459 52 112 407 110 405 105 402 107 404 433 74 435 76 440 79 438 77 145 362 147 364 152 367 150 365 336 183 334 181 329 178 331 180 240 279 238 277 233 274 235 276 305 202 307 204 312 207 310 205 465 42 467 44 472 47 470 45 16 503 14 501 9 498 11 500 432 87 430 85 425 82 427 84 113 394 115 396 120 399 118 397 337 170 339 172 344 175 342 173 144 375 142 373 137 370 139 372 304 215 302 213 297 210 299 212 241 266 243 268 248 271 246 269
I show a cube by John Hendricks here that contains 8 order-4 cubes so contains bent triagonals!
And another one here that contains 28 order-4 magic cubes, so contains many bent triagonals!
Addendum November 1,
2006
I received a similar cube to Soni's, hand constructed by 84 year old Arsène Durupt. His
cube differs in features only in that it is not complete, and order-5
sub-cube corners sum incorrectly. Also, bent diagonals of planes parallel to the
sides of the cube start on a different column then in the other two
orientations.
Order 8 is the lowest possible for a perfect magic cube and order 9 is the lowest possible for an associated perfect magic cube. See my perfect and perfect2 pages for more information on this class of cube.
All 24 planar squares are pandiagonal magic as are also the 6 oblique squares and the 42 broken (2 segment) oblique squares. The 264 pantriagonals also all sum correctly.
Corners of all orders 2, 3, 4, 5, 6, 7 and 8 also sum correctly to 2052. Is this always the case with 8m perfect cubes?
So 30 order-8 pandiagonal magic squares each have 8 rows, 8 columns and 16 pandiagonals sum correctly as do 256 pantriagonals. Total combinations so far = 960 lines. Corners of 7 orders of cubes = 7 times 256 (counting wrap-around) = 1792 corners. Total sums = 2752 (plus possible other combinations not yet discovered).
John R. Hendricks, Magic Squares to Tesseracts by Computer, self-published, 1998, 0-9684700-0-9
Horizontal plane I - Top II 88 185 240 449 408 377 304 1 395 350 307 38 75 158 243 486 300 5 84 189 236 453 404 381 242 487 394 351 306 39 74 159 403 382 299 6 83 190 235 454 73 160 241 488 393 352 305 40 234 455 402 383 298 7 82 191 309 36 77 156 245 484 397 348 81 192 233 456 401 384 297 8 398 347 310 35 78 155 246 483 301 4 85 188 237 452 405 380 247 482 399 346 311 34 79 154 406 379 302 3 86 187 238 451 80 153 248 481 400 345 312 33 239 450 407 378 303 2 87 186 308 37 76 157 244 485 396 349 III IV 65 152 249 496 385 344 313 48 422 331 286 51 102 139 222 499 317 44 69 148 253 492 389 340 223 498 423 330 287 50 103 138 390 339 318 43 70 147 254 491 104 137 224 497 424 329 288 49 255 490 391 338 319 42 71 146 284 53 100 141 220 501 420 333 72 145 256 489 392 337 320 41 419 334 283 54 99 142 219 502 316 45 68 149 252 493 388 341 218 503 418 335 282 55 98 143 387 342 315 46 67 150 251 494 97 144 217 504 417 336 281 56 250 495 386 343 314 47 66 151 285 52 101 140 221 500 421 332 V VI 112 129 216 505 432 321 280 57 435 358 267 30 115 166 203 478 276 61 108 133 212 509 428 325 202 479 434 359 266 31 114 167 427 326 275 62 107 134 211 510 113 168 201 480 433 360 265 32 210 511 426 327 274 63 106 135 269 28 117 164 205 476 437 356 105 136 209 512 425 328 273 64 438 355 270 27 118 163 206 475 277 60 109 132 213 508 429 324 207 474 439 354 271 26 119 162 430 323 278 59 110 131 214 507 120 161 208 473 440 353 272 25 215 506 431 322 279 58 111 130 268 29 116 165 204 477 436 357 VII VIII - Bottom 121 176 193 472 441 368 257 24 414 371 294 11 94 179 230 459 261 20 125 172 197 468 445 364 231 458 415 370 295 10 95 178 446 363 262 19 126 171 198 467 96 177 232 457 416 369 296 9 199 466 447 362 263 18 127 170 292 13 92 181 228 461 412 373 128 169 200 465 448 361 264 17 411 374 291 14 91 182 227 462 260 21 124 173 196 469 444 365 226 463 410 375 290 15 90 183 443 366 259 22 123 174 195 470 89 184 225 464 409 376 289 16 194 471 442 367 258 23 122 175 293 12 93 180 229 460 413 372
Please send me Feedback about my Web
site!
Harvey Heinz harveyheinz@shaw.ca
This page last updated
October 14, 2009
Copyright © 2003 by Harvey D. Heinz