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John Hendricks is well known among magic square hobbyists as a prolific designer and writer on the subject.
Over many years he has come up with many innovative ideas.
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 now 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.
[1] John R. Hendricks,
Inlaid Magic Squares and Cubes, self-published, 1999, 0-9684700-1-7, 188+
pages.
[2]
John R. Hendricks, Inlaid Magic Squares and Cubes, 2nd
edition, self-published, 2000, 0-9684700-3-3, 250+ pages. Edited and illustrated
by Holger Danielsson.
| The 28-in-1 bent triagonal cube | 27 order 4 cubes inlaid in an order 8 cube. |
| A versatile magic cube | Contains 1 pantriagonal cube and 12 pandiagonal magic squares. |
| A more versatile magic cube | Contains 8 pantriagonal cubes and 48 pandiagonal magic squares. |
| 8 Order 4 cubes = 1 order 8 | The 8 octants are each an order 4 pantriagonal, compact and complete magic cube. |
| An inlaid magic tesseract | An order 6 with inlaid order 3. |
| John Hendricks - original page | This page (1998) on my Geocites site has many examples of squares, cubes, tesseracts. Also history, etc. |
Earlier in the 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.
There are also 3 sets of four numbers that end up at bottom corners. They are
254, 259, 332, 181; 3, 510, 437, 76; and 291, 222, 149, 364. Finally we have a
set of four numbers (478, 35, 108, 405) which is the completion of the straight
triagonal.
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).
John Hendricks refers to this cube as
his 28-in-1!
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
From Inlaid Magic Squares and Cubes, 2nd edition, 2000 pp 147-152
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.
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.
Following is a list of the actual numbers shown as horizontal planes. The inlaid
cube and the horizontal magic squares are shown in blue. As with all the
examples in John’s book, he shows how to obtain a variety of magic cubes of
this type by using different solution sets.
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
John R. Hendricks, Inlaid Magic Squares and Cubes, 2nd
edition, 2000, pp 153-165
 |
To the left is the two layer ‘expansion shell’. Above is the 'insides',
In the center is the order 4 pantriagonal magic cube. It is surrounded by 12
pandiagonal magic squares. |
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.
He then describes a more elaborate cube which uses just a single layer expansion
shell. It is an order 12 simple magic cube with eight order 4 inlaid pantriagonal
magic cubes and 48 order 4 pandiagonal magic squares.
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 786
This 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
John R. Hendricks, Inlaid Magic Squares and Cubes, 2nd
edition, 2000, pp 185-207
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 first sub-cube, as it appears in the order 8. |
 This shows the placement of the sub-cubes in the octants of the order 8 cube. The cube to the left was inserted into the lower left front octant, as indicated by the 1 in the corner. |
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 individual order 4 cubes were entered manually into the order 8 test
spreadsheet , while mentally reflecting or rotating them as required to put the
numbers 1 to 8 in the corners of the large cube. The whole project, from
conception to final testing, took only about 4 hours.
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
In 1999 John Hendricks published an order 6 magic tesseract
with an order 3 inlaid magic tesseract.
I have included it on this page to illustrate a consistency between dimensions
(a tesseract is a 4 dimensional cubical object). I will show that the order 3
tesseract, cube and square are all associated.
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.
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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.
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Similarly, we can confirm that the order 3 cube shown is one of the four center cubes of the order 3 tesseract, therefore it should also be associated. Again we can confirm by summing opposite corners. In this case they should also sum to 1216, the sum of the first and last numbers used in this construction.
Finally, we can confirm that the square shown, because it is one of 3 central squares of the order 3 cube, is magic and also associated. |
The complete Inlaid Order 6 Magic Tesseract was supplied as an
8 page chart insert with these books;
John R. Hendricks, Magic Squares to Tesseracts by Computer,
self-published, 1998, 0-9684700-0-9, 142++ pages.
John R. Hendricks, Inlaid Magic Squares and Cubes, 2nd
edition, self-published, 2000, 0-9684700-3-3, 250+ pages. Edited and illustrated
by Holger Danielsson.
It may be downloaded as a PDF document here.
Please send me Feedback about my Web
site!
Harvey Heinz harveyheinz@shaw.ca
This page last updated
October 15, 2009
Copyright © 2003 by Harvey D. Heinz
