Hendricks' Inlaid Magic Cubes
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Hendricks' Inlaid Magic Cubes
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Over many years he has come up with many innovative ideas. These include:
On this page I will illustrate some of John’s inlaid magic cube creations. All are taken from a book [1] [2] he has published on this subject. As is normal for my web pages, my purpose is mainly to illustrate. For a detailed discussion of construction methods, and more inlaid magic cubes, please refer to his book. John Hendricks has a web site at http://members.shaw.ca/johnhendricksmath/. However, his books are now all out of print. I will also include 1 order 8 cube constructed by myself. Actually this is better described as a composition magic cube. [1] John R. Hendricks, Inlaid Magic Squares and
Cubes, self-published, 1999, 0-9684700-1-7, 188+ pages.
The 28-in-1 bent triagonal cube Earlier in his book, John described an order 8 magic cube wherein each of the 8 octants consisted of an order 4 magic cube. The cube described here does a little better then that. This is a semi-pantriagonal magic cube. It’s claim to fame
is that it contains bent triagonals. These are similar to the bent diagonals of
Franklin's famous magic squares. However, the four straight triagonals are also
correct so this cube is truly magic. Examples of bent triagonals, starting at
the top left corner are 336,177, 250, 263, then 3 sets of four numbers that
return to top corners; 218, 295, 368, 145; 506, 7, 80, 433; and 39, 474, 401,
112. The above feature insures that the main triagonals are
correct for order 4 cubes located as the 8 octants of the order 8 cube. However,
because all broken triagonals that start at all the odd numbered rows, columns
and planes also sum correctly, we actually have many more inlaid order 4 cubes
that are magic. In fact, there are 9 cubes that start with the top left corner
on the top plane at rows and columns 1, 3 and 5. Similarly, 9 that start with
that corner on the 3rd plane from the top, and 9 with that corner on the 5th
plane from the top (but see the note below regarding wrap-around). An additional feature is that the corners of all order 3 and 7 cubes (within the main cube) total to the magic constant of 2052. Corners of most of the cubes of orders 2, 4, 6, and 8 also sum correctly. Interestingly, corners of none of the cubes of order 5 sum to 2052! From Inlaid Magic Squares and Cubes,
1999 pp 130-136 Listing for the order 8 magic cube Top II 336 144 241 305 464 16 113 433 329 137 248 312 457 9 120 440 376 184 201 265 504 56 73 393 369 177 208 272 497 49 80 400 185 377 264 200 57 505 392 72 192 384 257 193 64 512 385 65 129 321 320 256 1 449 448 128 136 328 313 249 8 456 441 121 352 160 225 289 480 32 97 417 345 153 232 296 473 25 104 424 360 168 217 281 488 40 89 409 353 161 224 288 481 33 96 416 169 361 280 216 41 489 408 88 176 368 273 209 48 496 401 81 145 337 304 240 17 465 432 112 152 344 297 233 24 472 425 105 III IV 178 370 271 207 50 498 399 79 183 375 266 202 55 503 394 74 138 330 311 247 10 458 439 119 143 335 306 242 15 463 434 114 327 135 250 314 455 7 122 442 322 130 255 319 450 2 127 447 383 191 194 258 511 63 66 386 378 186 199 263 506 58 71 391 162 354 287 223 34 482 415 95 167 359 282 218 39 487 410 90 154 346 295 231 26 474 423 103 159 351 290 226 31 479 418 98 343 151 234 298 471 23 106 426 338 146 239 303 466 18 111 431 367 175 210 274 495 47 82 402 362 170 215 279 490 42 87 407 V VI 334 142 243 307 462 14 115 435 331 139 246 310 459 11 118 438 374 182 203 267 502 54 75 395 371 179 206 270 499 51 78 398 187 379 262 198 59 507 390 70 190 382 259 195 62 510 387 67 131 323 318 254 3 451 446 126 134 326 315 251 6 454 443 123 350 158 227 291 478 30 99 419 347 155 230 294 475 27 102 422 358 166 219 283 486 38 91 411 355 163 222 286 483 35 94 414 171 363 278 214 43 491 406 86 174 366 275 211 46 494 403 83 147 339 302 238 19 467 430 110 150 342 299 235 22 470 427 107 VII VIII 180 372 269 205 52 500 397 77 181 373 268 204 53 501 396 76 140 332 309 245 12 460 437 117 141 333 308 244 13 461 436 116 325 133 252 316 453 5 124 444 324 132 253 317 452 4 125 445 381 189 196 260 509 61 68 388 380 188 197 261 508 60 69 389 164 356 285 221 36 484 413 93 165 357 284 220 37 485 412 92 156 348 293 229 28 476 421 101 157 349 292 228 29 477 420 100 341 149 236 300 469 21 108 428 340 148 237 301 468 20 109 429 365 173 212 276 493 45 84 404 364 172 213 277 492 44 85 405 This is the top left back cube. Top II III IV 336 144 241 305 329 137 248 312 178 370 271 207 183 375 266 202 376 184 201 265 369 177 208 272 138 330 311 247 143 335 306 242 185 377 264 200 192 384 257 193 327 135 250 314 322 130 255 319 129 321 320 256 136 328 313 249 383 191 194 258 378 186 199 263 This is the central order 4 cube. Top II III IV 250 314 455 7 255 319 450 2 262 198 59 507 259 195 62 510 194 258 511 63 199 263 506 58 318 254 3 451 315 251 6 454 287 223 34 482 282 218 39 487 227 291 478 30 230 294 475 27 295 231 26 474 290 226 31 479 219 283 486 38 222 286 483 35 This is the bottom right front cube Top II III IV 478 30 99 419 475 27 102 422 36 484 413 93 37 485 412 92 486 38 91 411 483 35 94 414 28 476 421 101 29 477 420 100 43 491 406 86 46 494 403 83 469 21 108 428 468 20 109 429 19 467 430 110 22 470 427 107 493 45 84 404 492 44 85 405 If we include wrap-around, there are many more order 4 magic cubes within the order 8 cube. I guess, though, that we could not really call it inlaid if a cube is in two or four parts! Here I show an example that is wrapped around from left to right. It starts in the top plane, row 3, column 7. Top II III IV 392 72 185 377 385 65 142 384 122 442 327 135 127 447 322 130 448 128 129 321 441 121 136 328 66 386 383 191 71 391 378 186 97 417 352 160 104 424 345 153 415 95 162 354 410 90 167 359 89 409 360 168 96 416 353 161 423 103 154 346 418 98 159 351 All order 4 cubes shown here are also semi-pantriagonal and bent triagonal magic cubes.
A versatile magic cube This is an order 8 simple magic cube with an order 4
pantriagonal magic cube in the center and 12 4x4 pandiagonal magic squares in
the planes parallel to each face. The inner cube sums to 1026 in rows, columns,
pillars and all pantriagonals. All 12 pandiagonal magic squares sum to 1026 in
rows, columns and all pandiagonals. The cube as a whole sums to 2052 in rows,
columns pillars and the 4 triagonals. I call this a versatile magic cube because it may be easily changed to a different cube by changes to the components. The order 4 inner cube may be placed in any of its 48 aspects by rotations and/or reflections. In addition, because it is pantriagonal, any of the 64 numbers may be brought to the upper left hand corner. This gives 48 x 64 = 3072 different inner cubes. Each of the 3 sets of 4 magic squares may have their layers interchanged (4!=24 ways). In addition, there are 8 aspects due to rotations and/or reflections and each of 16 numbers may be brought to the top left corner. Note however that the 4 squares in the stack must be treated as a unit when making these changes. Finally, the stacks themselves may be interchanged in 3!=6 ways. So there are 18,432 variations involving operations to the squares. 3072 x 18432 makes 56,623,104 variations to this one order 8 magic cube. Then this complete cube also has 48 aspects. However, these are not normally considered when counting magic object variations. John R. Hendricks, Inlaid Magic Squares
and Cubes, 1999, pp 137-147 Top Top pantriagonal magic square II 2nd pantriagonal magic square 505 72 264 185 200 377 392 57 2 447 255 322 319 130 127 450 16 433 241 336 305 144 113 464 503 74 266 183 202 375 394 55 489 88 217 359 170 280 425 24 18 431 224 354 175 273 87 490 32 417 304 146 351 225 96 481 487 90 297 151 346 232 418 31 465 112 343 233 296 154 401 48 42 407 338 240 289 159 111 466 40 409 162 288 209 367 104 473 479 98 167 281 216 362 410 39 56 393 272 177 208 369 73 505 463 114 242 335 306 143 434 15 449 128 249 328 313 136 448 1 58 391 263 186 199 378 71 506 III Top plane of 4x4x4 cube IV 2nd plane of 4x4x4 cube 5 444 309 132 379 206 69 508 507 70 203 382 133 308 443 6 500 77 269 188 323 246 396 53 14 435 243 326 189 268 118 459 28 476 148 291 221 366 101 421 493 45 286 173 339 228 404 84 422 102 230 341 171 284 475 27 83 403 364 219 293 150 46 494 107 427 347 236 278 165 22 470 414 94 213 358 156 299 483 35 469 21 301 158 356 211 428 108 36 484 163 276 238 349 93 413 461 116 252 333 182 259 437 12 51 398 262 179 332 253 75 502 60 389 196 373 142 315 124 453 454 123 318 139 372 197 390 59 V 3rd plane of 4x4x4 cube VI Bottom plane of 4x4x4 cube 4 445 134 307 204 381 68 509 510 67 380 205 310 131 446 3 501 76 190 267 244 325 397 52 11 438 324 245 270 187 115 462 406 86 235 348 166 277 491 43 99 419 357 214 300 155 30 478 44 492 157 302 212 355 85 405 477 29 275 164 350 237 420 100 485 37 292 147 365 222 412 92 20 468 174 285 227 340 109 429 91 411 342 229 283 172 38 486 430 110 220 363 149 294 467 19 460 117 331 254 261 180 436 13 54 395 181 260 251 334 78 499 61 388 371 198 317 140 125 452 451 126 141 316 195 374 387 62 VII 3rd horizontal magic square VIII Bottom pantriagonal magic square 7 442 314 135 250 327 122 455 512 65 193 384 257 192 385 64 498 79 207 370 271 178 399 50 9 440 312 137 248 329 120 457 47 402 290 160 337 239 106 471 472 105 295 153 344 234 408 41 474 103 215 361 168 282 415 34 33 416 210 368 161 287 97 480 23 426 176 274 223 353 82 495 496 81 169 279 218 360 432 17 482 95 345 231 298 152 43 26 25 424 352 226 303 145 89 488 458 119 311 138 247 330 439 10 49 400 201 376 265 184 80 497 63 386 194 383 258 191 66 511 456 121 320 129 256 321 441 8
A more versatile magic cube In his book John shows an order 10 with an inlaid order 6
cube and 12 order 4 magic squares, so it is the same style as the order 8 I have
just described. The constant sum for the order 12 is 10,374 which is the required sum for a normal order 12. The sum for the order 4 cubes and squares is 3,458. Here I will just show the top horizontal layer of the cube (including the top four pandiagonal magic squares. Variations of this cube just by operations on the 8 inlaid cubes are;
This gives a total of 123,863,040 possible variations involving the inlaid cubes only. Still available are the variations involving the 48 pantriagonal magic squares! 942 1230 355 222 1651 643 1075 67 1518 1363 510 798 966 474 1400 401 1183 619 1110 174 1700 101 1483 763 751 1267 317 1340 534 1122 607 1543 41 1616 258 978 882 1328 546 1255 329 703 1026 1628 246 1555 29 847 859 389 1195 462 1412 1014 715 113 1471 186 1688 870 775 487 1386 1495 90 1098 666 1674 199 378 1207 919 811 1242 343 234 1639 1062 630 55 1530 1351 522 955 727 306 1280 569 1303 1146 583 6 1580 269 1603 1002 990 1435 437 1172 414 595 1134 1711 161 1448 138 739 835 1160 426 1423 449 1038 691 1460 126 1723 149 894 906 557 1315 294 1292 679 1050 281 1591 18 1568 823 930 499 1374 1507 78 655 1087 1662 211 366 1219 786This is the top horizontal plane. Blue numbers are the top pandiagonal magic squares. Black numbers are the top layer of the expansion shell. Following is the top left back inlaid order 4 pantriagonal magic cube. It's top left corner is row 2, column 2 of the 2nd plane from the top of the order 12 cube. Top II III IV 563 1308 422 1165 1429 446 1284 299 312 1295 433 1418 1154 409 1319 576 289 1274 456 1439 1175 432 1298 553 566 1309 419 1164 1428 443 1285 302 1296 311 1417 434 410 1153 575 1320 1307 564 1166 421 445 1430 300 1283 1310 565 1163 420 444 1427 301 1286 312 1295 433 1418 431 1176 554 1297 This complete cube complete with illustrations is also shown on my cube_12.htm page. John R. Hendricks, Inlaid Magic Squares and Cubes, 1999, pp
163-182 8 Order 4 cubes = 1 order 8 Here 8 order 4 cubes are constructed from a pattern cube so all have identical features. Each is pantriagonal, compact and complete. They were then combined as octants to form an order 8 magic cube. [1] The order 8 cube, however, is just simple magic. It has none of the special features of the small cubes, presumably because of discontinuances between the small cube interfaces.
The eight order 4 cubes were constructed by putting the first 8 numbers in pattern position 1 of each of the 8 cubes in turn. Then the next 8 numbers were inserted into pattern position two, but in reverse order. The same procedure was followed, always reversing the order for each pass, until all 512 numbers were used and the 8 order 4 cubes were completed. The eight cubes were rotated or reflected as necessary when being placed in the order 8 cube, so that the numbers from 1 to 8 appeared in the corners of the large cube. Pattern Cube Plane 1 -Top Plane 2 Plane 3 Plane 4-Bottom 1 48 49 32 60 21 12 37 13 36 61 20 56 25 8 41 63 18 15 34 6 43 54 27 51 30 3 46 10 39 58 23 4 45 52 29 57 24 9 40 16 33 64 17 53 28 5 44 62 19 14 35 7 42 55 26 50 31 2 47 11 38 59 22 Cube 1 Plane 1 -Top Plane 2 Plane 3 Plane 4 - Bottom 1 384 385 256 480 161 96 289 97 288 481 160 448 193 64 321 497 144 113 272 48 337 432 209 401 240 17 368 80 305 464 177 32 353 416 225 449 192 65 320 128 257 512 129 417 224 33 352 496 145 112 273 49 336 433 208 400 241 16 369 81 304 465 176 Cube 2 Plane 1 -Top Plane 2 Plane 3 Plane 4 - Bottom 2 383 386 255 479 162 95 290 98 287 482 159 447 194 63 322 498 143 114 271 47 338 431 210 402 239 18 367 79 306 463 178 31 354 415 226 450 191 66 319 127 258 511 130 418 223 34 351 495 146 111 274 50 335 434 207 399 242 15 370 82 303 466 175 Cube 3 Plane 1 -Top Plane 2 Plane 3 Plane 4 - Bottom 3 382 387 254 478 163 94 291 99 286 483 158 446 195 62 323 499 142 115 270 46 339 430 211 403 238 19 366 78 307 462 179 30 355 414 227 451 190 67 318 126 259 510 131 419 222 35 350 494 147 110 275 51 334 435 206 398 243 14 371 83 302 467 174 Etc Plane 1 - Top Plane 2 7 378 391 250 251 390 379 6 474 167 90 295 294 91 166 475 503 138 119 266 267 118 139 502 42 343 426 215 214 427 342 43 26 359 410 231 230 411 358 27 455 186 71 314 315 70 187 454 490 151 106 279 278 107 150 491 55 330 439 202 203 438 331 54 489 152 105 280 277 108 149 492 56 329 440 201 204 437 332 53 25 360 409 232 229 412 357 28 456 185 72 313 316 69 188 453 504 137 120 265 268 117 140 501 41 344 425 216 213 428 341 44 8 377 392 249 252 389 380 5 473 168 89 296 293 92 165 476 Plane 3 Plane 4 103 282 487 154 155 486 283 102 442 199 58 327 326 59 198 443 407 234 23 362 363 22 235 406 74 311 458 183 182 459 310 75 122 263 506 135 134 507 262 123 423 218 39 346 347 38 219 422 394 247 10 375 374 11 246 395 87 298 471 170 171 470 299 86 393 248 9 376 373 12 245 396 88 297 472 169 172 469 300 85 121 264 505 136 133 508 261 124 424 217 40 345 348 37 220 421 408 233 24 361 364 21 236 405 73 312 457 184 181 460 309 76 104 281 488 153 156 485 284 101 441 200 57 328 325 60 197 444 Plane 5 Plane 6 447 194 63 322 323 62 195 446 98 287 482 159 158 483 286 99 79 306 463 178 179 462 307 78 402 239 18 367 366 19 238 403 418 223 34 351 350 35 222 419 127 258 511 130 131 510 259 126 82 303 466 175 174 467 302 83 399 242 15 370 371 14 243 398 81 304 465 176 173 468 301 84 400 241 16 369 372 13 244 397 417 224 33 352 349 36 221 420 128 257 512 129 132 509 260 125 80 305 464 177 180 461 308 77 401 240 17 368 365 20 237 404 448 193 64 321 324 61 196 445 97 288 481 160 157 484 285 100 Plane 7 Plane 8 - Bottom 479 162 95 290 291 94 163 478 2 383 386 255 254 387 382 3 47 338 431 210 211 430 339 46 498 143 114 271 270 115 142 499 450 191 66 319 318 67 190 451 31 354 415 226 227 414 355 30 50 335 434 207 206 435 334 51 495 146 111 274 275 110 147 494 49 336 433 208 205 436 333 52 496 145 112 273 276 109 148 493 449 192 65 320 317 68 189 452 32 353 416 225 228 413 356 29 48 337 432 209 212 429 340 45 497 144 113 272 269 116 141 500 480 161 96 289 292 93 164 477 1 384 385 256 253 388 381 4 There is a lot of number manipulation required to assign
the required numbers to each order 4 cube, but this drudgery was largely
eliminated by using a spreadsheet. The concept of using sub-cubes with equal constants may be extended to other orders of cubes. Only even order sub-cubes may be used because each cube must consist of complete complement pairs. [1] H. Heinz, Jan. 6, 2003. See also my Composition cubes page.
In 1999 John Hendricks published an order 6 magic
tesseract with an order 3 inlaid magic tesseract. Just as you can have an inlaid magic square in one quadrant of a larger magic square, so too can you have an inlaid magic cubes within the larger cube. And, now the world’s first example of an inlaid magic tesseract of order 3 situated within the world’s first magic tesseract of order six. A square has quadrants, a cube has octants and a tesseract has hexadecimants. In this tesseract the inlaid order 3 tesseract is in hexidecimant 6. The magic sum for the sixth-order tesseract is 3891 and the magic sum for the inner magic sub-tesseract revealed by the red numbers, is 1824. Click on these thumbnails for a larger view. Then use your browser back button to return here.
The order 6 tesseract uses the numbers from 1 to 46 = 1 to 1296. It is not associated. However, we know that the inlaid order 3 tesseract should be associated, because all order 3 magic hypercubes are. A quick look at diametrically opposite corners show that all sum to 1216, which is the sum of the first and last numbers (570 + 646) used in this (order 3) tesseract.
The complete Inlaid Order 6 Magic
Tesseract was supplied as an 8 page chart insert with these books;
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