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Early magic square development took place in Asia, notably
in China, India, and the Arab world. The idea of extending this concept to 3
dimensions, however, seems to have originated in the West.
The first occurrence I have found of magic cubes was in France, by Pierre de Fermat (1601 –
1665), followed by Joseph Sauveur (1653 – 1716) and B. Violle (cc1838).
Then the emphasis seems to shift to England, with writings by Rev. A. H. Frost, followed by W. Firth (? – 1889), Dr. C. Planck (?), etc. By 1900 the investigation of magic cubes had progressed to many other countries.
It goes without saying, that this collection of early magic
cubes is probably not complete.
I would be very interested in hearing from anyone who can provide additional
information on this subject.
Fermat 1640 |
This order 4 cube has no correct triagonals but does contain 8 simple magic squares. |
Sauveur 1710 |
All diagonals and triagonals are correct but only 1 row and 1 column of each orthogonal array. 21 arrays sum 1575. |
Violle order 4 1838 |
An order 4 cube with 14 of the 18 4 x 4 arrays summing to 520. |
Violle order 5 1838 |
This cube is not magic but all diagonals and triagonals sum correctly. 21 planes = 1575 |
Violle order 6 1838 |
This cube has similar characteristics to Violle's order 4. 24 planes = 3906 |
Violle order 7 1838 |
This cube has the same characteristics as Violle's order 5. 27 planes = 8428 |
Frankenstein - 8 1875 |
This magic cube contains 30 simple magic squares. Triagonals are correct. |
Hugel order 3 1876 |
This is a disguised index # 1, but earliest record I could find of the order 3. |
Frost order 4 1878 |
The 4 horizontal planes of this cube are pandiagonal magic squares. Triagonals are incorrect. |
Barnard order 4 1888 |
This cube is not magic by present standards, but does possess some unique features. |
Firth order 6 1889 |
This cube has the minimum requirements to make it simple magic. First published in 1917? |
Schubert order 4 1898 |
A simple, associated magic cube. |
Schubert order 5 1898 |
A simple, associated magic cube. |
Fourrey order 4 1899 |
Not magic but contains 8 magic squares. Same features as the Fermat cube. |
Andrews order 4 1908 |
A simple, associated magic cube. |
Andrews order 5 1908 |
This associated magic cube contains 10 pandiagonal and 5 simple magic squares. |
Worthington order 6 1910 |
The 6 faces of this simple magic cube are simple magic squares. |
 |
I II III IV 4 62 63 1 53 11 10 56 60 6 7 57 13 51 50 16 41 23 22 44 32 34 35 29 17 47 46 20 40 26 27 37 21 43 42 24 36 30 31 33 45 19 18 48 28 38 39 25 64 2 3 61 9 55 54 12 8 58 59 5 49 15 14 52 This cube is not magic by present day
definition because NO triagonals are correct. The 4 horizontal planes and
the 4 vertical planes parallel to the sides are simple magic squares. All
rows and columns sum correctly for 4 of the 6 oblique squares. This cube was
described in a letter to Marin Mersenne dated Apr. 1, 1640 [1].
[1] W. S. Andrews, Magic Squares &
Cubes, 2nd edition, Dover Publ. 1960, originally
published 1917 (footnote p. 365). |
This cube, published by Joseph Sauveur in 1710, is not magic by present day standards. However, it does have some interesting features.
This cube and the method of construction was graciously supplied by Christian Boyer [1] of France, who was successful in locating the original text of Joseph Sauveur in the Mémoires de l'Académie Royale des Sciences of 1710. He was not permitted to photocopy the document because of its age, so the following is prepared from his notes.
Definition by Joseph Sauveur (original French text) of what is a magic cube:
"Un cube magique est un cube divisé en cellules cubiques qui
So Sauveur means
that, for example in this 5th-order cube, the 25 numbers of each square (the
5+5+5 layers
parallel to each side, and the 6 diagonal squares, or 15+6=21 squares) should
have the same sum.
This definition was used by most of the magic cube constructors of the 18th and 19th century
The 15 orthogonal planes each have these features:
Only I row and 1 column sums correctly (S = 315).
The total of the 5 rows is 5S.
Both main diagonals sum correctly.
All pandiagonals in one direction sum correctly.
The 6 oblique planes have these features:
The total of the 5 rows in each plane is 5S.
Both diagonals are correct in each plane.
3 of these planes are magic squares, because all rows and
columns and 2 main diagonals sum correctly.
2 planes have all 5 rows, and all 5 pandiagonals in one direction, with correct
sums.
1 plane has all 5 columns, and all 5 pandiagonals in one direction, with correct
sums.
[1] Christian Boyer has recently made some amazing discoveries regarding multimagic squares and cubes.
Sauveur's cube [2] formed by adding the numbers in these three subsidiary squares Plane 1 - Top 62 2 32 92 122 50 0 25 75 100 10 0 5 15 20 2 2 2 2 2 114 54 9 44 99 100 50 0 25 75 10 0 5 15 20 4 4 4 4 4 90 105 60 20 50 75 100 50 0 25 10 0 5 15 20 5 5 5 5 5 36 76 106 66 21 25 75 100 50 0 10 0 5 15 20 1 1 1 1 1 13 28 83 118 73 0 25 75 100 50 10 0 5 15 20 3 3 3 3 3 Plane 2 74 14 29 84 119 50 0 25 75 100 20 10 0 5 15 4 4 4 4 4 125 65 5 35 95 100 50 0 25 75 20 10 0 5 15 5 5 5 5 5 96 111 51 6 41 75 100 50 0 25 20 10 0 5 15 1 1 1 1 1 48 88 103 58 18 25 75 100 50 0 20 10 0 5 15 3 3 3 3 3 22 37 77 107 67 0 25 75 100 50 20 10 0 5 15 2 2 2 2 2 Plane 3 70 25 40 80 110 50 0 25 75 100 15 20 10 0 5 5 5 5 5 5 116 71 11 26 81 100 50 0 25 75 15 20 10 0 5 1 1 1 1 1 93 123 63 3 33 75 100 50 0 25 15 20 10 0 5 3 3 3 3 3 42 97 112 52 7 25 75 100 50 0 15 20 10 0 5 2 2 2 2 2 19 49 89 104 59 0 25 75 100 50 15 20 10 0 5 4 4 4 4 4 Plane 4 56 16 46 86 101 50 0 25 75 100 5 15 20 10 0 1 1 1 1 1 108 68 23 38 78 100 50 0 25 75 5 15 20 10 0 3 3 3 3 3 82 117 72 12 27 75 100 50 0 25 5 15 20 10 0 2 2 2 2 2 34 94 124 64 4 25 75 100 50 0 5 15 20 10 0 4 4 4 4 4 10 45 100 115 55 0 25 75 100 50 5 15 20 10 0 5 5 5 5 5 Plane 5 - Bottom 53 8 43 98 113 50 0 25 75 100 0 5 15 20 10 3 3 3 3 3 102 57 17 47 87 100 50 0 25 75 0 5 15 20 10 2 2 2 2 2 79 109 69 24 39 75 100 50 0 25 0 5 15 20 10 4 4 4 4 4 30 85 120 75 15 25 75 100 50 0 0 5 15 20 10 5 5 5 5 5 1 31 91 121 61 0 25 75 100 50 0 5 15 20 10 1 1 1 1 1
[2] This from Christian Boyer email of March 16, 2003
B. Violle published a monumental book on Magic squares in
1838. From it I have extracted the following cubes of orders 4, 5, 6, and 7. The
method for constructing orders 8 and 9 was also presented, but I do not read
French, so leave to someone else the pleasure of reconstructing these.
The entire book of over 1000 pages is available free on the Web [1].
 |
I II III IV 13 32 48 61 57 44 28 9 53 40 24 5 1 20 36 49 2 19 35 50 54 39 23 6 58 43 27 10 14 31 47 62 3 18 34 51 55 38 22 7 59 42 26 11 15 30 46 63 16 29 45 64 60 41 25 12 56 37 21 8 4 17 33 52 This is not considered magic by present definition. Even though the 4 triagonals are correct, rows, columns and pillars are not. The total of the 16 cells in each of the 18 square arrays (the 3 x 4 orthogonal and the 6 oblique) sum to 4 x 130 = 520. Both diagonals of these 18 arrays also sum correctly to 130. All 4 triagonals sum correctly to 130. None of the 12 planar squares have any rows or columns that sum to 130! Two of the oblique squares have all rows summing correctly and four have all columns summing correctly. Violle cube 6 has exactly the same features except this one is associated and order 6 is not. |
[1] Par B. Violle, Traité complet des Carrés Magiques, 1837, (French). This book is available on the Internet at at http://gallica.bnf.fr.as scanned pages. See my Cube_biblio page for download instructions.
This cube is pantriagonal because ALL triagonals sum correctly to 315. However, it is not magic (by today's standards) because rows and columns are incorrect. It is not associated. Also, all diagonals including broken diagonal pairs, of the planar arrays sum correctly.
The 25 numbers in each of the 15 planar squares and 6 oblique squares sum to 1575 (5 times the correct constant of 315). This is a required feature of J. Sauveur's 1710 definition of a magic cube. However, a close inspection of the four Violle cubes, the Sauveur order 5 and the Fourrey order 3 reveal differences in other characteristics.
Only 1 row and 1 column of each planar square is correct. In fact the rows sum to 5 different values starting with 65 and increasing by 125. The columns also sum to 5 different values this time starting at 305 and increasing by 5 (the order of the cube). The 5 planes in each direction have different arrangements of these sums.
Four of the oblique squares have all rows summing correctly and two have all columns summing correctly. One pair of the oblique squares has the columns summing the same as the planar squares.
This order 5 cube has almost the same characteristics as the Violle order 7.
I II III 15 24 8 17 1 88 97 81 95 79 36 50 34 43 27 45 29 38 47 31 118 102 111 125 109 66 55 64 73 57 75 59 68 52 61 23 7 16 5 14 96 85 94 78 87 80 89 98 82 91 28 37 46 35 44 101 115 124 108 117 110 119 103 112 121 58 67 51 65 74 6 20 4 13 22 IV V 114 123 107 116 105 62 71 60 69 53 19 3 12 21 10 92 76 90 99 83 49 33 42 26 40 122 106 120 104 113 54 63 72 56 70 2 11 25 9 18 84 93 77 86 100 32 41 30 39 48
Par B. Violle, Traité complet des Carrés
Magiques, 1837 (French).
This is not considered magic by present definition. Even though the 4 triagonals are correct, rows, columns and pillars are not.
The total of the 36 cells in each of the 18 square arrays (the 3 x 6 orthogonal and the 6 oblique) sum to 6 x 651 = 3906. Both diagonals of these 24 arrays also sum correctly to 651.
All 4 triagonals sum correctly to 651. None of the 12 planar squares have any rows or columns that sum to 651! Two of the oblique squares have all rows summing correctly and four have all columns summing correctly. This cube has the exact same features as Violle order 4 except this one is not associated.
I II III 1 32 33 4 35 6 187 188 207 208 191 210 193 200 201 202 197 198 42 68 70 39 71 37 48 44 64 63 47 61 162 164 166 165 161 157 78 107 106 75 104 73 120 119 136 135 116 133 126 131 130 129 122 121 109 143 142 111 140 114 79 83 100 99 80 102 85 95 94 93 86 90 150 176 177 148 179 145 156 152 171 172 155 169 54 56 57 58 53 49 181 215 213 184 212 186 7 11 27 28 8 30 13 23 21 22 14 18 IV V VI 19 14 15 16 23 24 205 206 189 190 209 192 31 2 3 34 5 36 168 158 160 159 167 163 66 62 46 45 65 43 180 146 148 177 149 175 132 125 124 123 128 127 102 101 82 81 98 79 108 77 76 105 74 103 91 89 88 87 92 96 133 137 118 117 134 120 139 113 112 141 110 144 60 50 51 52 59 55 174 170 153 154 173 151 72 38 39 70 41 67 199 197 195 196 200 204 25 29 9 10 26 12 211 185 183 214 182 216
Par B. Violle, Traité complet des Carrés Magiques, 1837 (French).
This is not considered magic by present definition. Even though the 4 triagonals are correct, rows columns and pillars are not. The total of the 49 cells in each of 27 square arrays sum to 7 x 1204 = 8428.
All pantriagonals sum correctly to 1204 as do all planar diagonals. None of the 21 planar squares have any rows or columns that sum to 1204! In fact the rows sum to 7 different values starting with 1057 and increasing by 49. The columns also sum to 7 different values this time starting at 1183 and increasing by 7 (the order of the cube). The 7 planes in each direction have different arrangements of these sums.
Four of the oblique squares have all rows summing correctly
and two have all columns summing correctly. One pair of the oblique squares has
the columns summing the same as the planar squares.
All pandiagonals of all planar and oblique squares are correct.
This cube has exactly the same characteristics as the
Violle order 5.
I II 6 14 15 23 31 39 47 53 61 69 77 78 86 94 237 245 197 205 213 221 229 284 292 251 259 260 268 276 76 84 85 93 52 60 68 123 131 139 147 99 107 115 258 266 267 275 283 291 250 305 313 321 329 330 338 297 146 105 106 114 122 130 138 193 152 160 168 169 177 185 328 336 337 296 304 312 320 32 40 48 7 8 16 24 167 175 176 184 192 151 159 214 222 230 238 239 198 206 III IV 100 108 116 124 132 140 141 154 155 163 171 179 187 195 331 339 298 306 314 322 323 42 43 2 10 18 26 34 170 178 186 194 153 161 162 224 225 233 241 200 208 216 9 17 25 33 41 49 1 63 64 72 80 88 96 55 240 199 207 215 223 231 232 294 246 254 262 270 278 286 79 87 95 54 62 70 71 133 134 142 101 109 117 125 261 269 277 285 293 252 253 315 316 324 332 340 299 307 V VI 201 209 217 218 226 234 242 248 256 264 272 280 281 289 89 97 56 57 65 73 81 136 144 103 111 119 120 128 271 279 287 288 247 255 263 318 326 334 342 301 302 310 110 118 126 127 135 143 102 157 165 173 181 189 190 149 341 300 308 309 317 325 333 45 4 12 20 28 29 37 180 188 196 148 156 164 172 227 235 243 202 210 211 219 19 27 35 36 44 3 11 66 74 82 90 98 50 58 VII 295 303 311 319 327 335 343 183 191 150 158 166 174 182 22 30 38 46 5 13 21 204 212 220 228 236 244 203 92 51 59 67 75 83 91 274 282 290 249 257 265 273 113 121 129 137 145 104 112
Par B. Violle, Traité complet des Carrés Magiques, 1837
(French).
Gustavus
Frankenstein published this cube in The Commercial, a Cincinnati city daily newspaper, March 11,
1875.
The instructions for building this cube later appeared as a lengthy footnote in
Barnard's 1888 [1] paper. Because rows, columns, pillars, and triagonals all sum correctly to 2052,
this cube is magic by the current definition.
This is the
Frankenstein cube in the orientation that he published it. However, he numbered
the planes from the bottom up. I reversed the numbering to be consistent with my
other cube listings.
Benson and Jacoby [2] reconstructed this cube, presumably from instructions in a
footnote of Barnard’s paper. The version they show is a different aspect of this
cube i.e. rotated and reflected.
I am grateful
to Christian Boyer for locating the original publication that this cube appeared
in.
He has posted the complete text of the article
here.
Note that Christian refers to it as the first perfect cube. However, by the new (Hendricks) definition it is classified as a diagonal cube because it contains 3m simple magic squares. (A perfect cube contains 9m pandiagonal magic squares).
All 24 planar squares and the 6 oblique squares are simple
magic. Also, corners of all 256 (including wrap-around) 5x5x5 cubes sum correctly
to S.
Benson & Jacoby refer to this as a perfect cube. They refer to their cube, which is perfect by the new definition, as
pandiagonal perfect. (In that cube, all 30 planar squares are
pandiagonal magic and all pantriagonals sum correctly [along with other
features].)
Cubes with all planar squares simple magic, like this
Frankenstein
cube, had no name as
yet under Hendricks new definitions. For a short time, I called them myers cubes in
honor of a similar cube constructed by Richard Myers in 1970 and popularized by
M. Gardner [3]. However, Aale de Winkel suggested the name diagonal magic
cubes.
I have decided to use this name instead of myers because it is more descriptive
(main diagonals of all planar squares sum correctly to S).
If
all planar squares were pandiagonal magic, instead of simple, they would be
called pandiagonal magic cubes.
And if the cube is pandiagonal
magic AND all pantriagonals are also magic the cube is called perfect!
A perfect cube has 9m pandiagonal magic squares [4][5].
[1] F.A.P. Barnard, Theory of Magic Squares
and Magic Cubes, Memoirs of the National Academy of Science, 4, 1888, pp
209-270 (footnote on pp.
244-248).
[2]
W. H. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ. 1981,
0-486-24140-8
[3] Martin Gardner, From his mathematical Games column in Scientific American,
Jan. 1976.
[4]
B. Rosser and R. J. Walker,
A Continuation of
The Algebraic Theory of Diabolic Magic Squares on typewritten pages numbered
729 – 753.
[5] F. Liao, T. Katayama, K. Takaba, Technical Report 99021, School of
Informatics, Kyoto University, 1999.
Horizontal plane I - Top Plane II 64 450 62 452 453 59 455 57 385 127 387 125 124 390 122 392 56 458 54 460 461 51 463 49 393 119 395 117 116 398 114 400 465 47 467 45 44 470 42 472 112 402 110 404 405 107 407 105 473 39 475 37 36 478 34 480 104 410 102 412 413 99 415 97 481 31 483 29 28 486 26 488 96 418 94 420 421 91 423 89 489 23 491 21 20 494 18 496 88 426 86 428 429 83 431 81 16 498 14 500 501 11 503 9 433 79 435 77 76 438 74 440 8 506 6 508 509 3 511 1 441 71 443 69 68 446 66 448 Plane III Plane IV 321 191 323 189 188 326 186 328 256 258 254 260 261 251 263 249 329 183 331 181 180 334 178 336 248 266 246 268 269 243 271 241 176 338 174 340 341 171 343 169 273 239 275 237 236 278 234 280 168 346 166 348 349 163 351 161 281 231 283 229 228 286 226 288 160 354 158 356 357 155 359 153 289 223 291 221 220 294 218 296 152 362 150 364 365 147 367 145 297 215 299 213 212 302 210 304 369 143 371 141 140 374 138 376 208 306 206 308 309 203 311 201 377 135 379 133 132 382 130 384 200 314 198 316 317 195 319 193 Plane V Plane VI 320 194 318 196 197 315 199 313 129 383 131 381 380 134 378 136 312 202 310 204 205 307 207 305 137 375 139 373 372 142 370 144 209 303 211 301 300 214 298 216 368 146 366 148 149 363 151 361 217 295 219 293 292 222 290 224 360 154 358 156 157 355 159 353 225 287 227 285 284 230 282 232 352 162 350 164 165 347 167 345 233 279 235 277 276 238 274 240 344 170 342 172 173 339 175 337 272 242 270 244 245 267 247 265 177 335 179 333 332 182 330 184 264 250 262 252 253 259 255 257 185 327 187 325 324 190 322 192 Plane VII Horizontal plane VIII - Bottom 65 447 67 445 444 70 442 72 512 2 510 4 5 507 7 505 73 439 75 437 436 78 434 80 504 10 502 12 13 499 15 497 432 82 430 84 85 427 87 425 17 495 19 493 492 22 490 24 424 90 422 92 93 419 95 417 25 487 27 485 484 30 482 32 416 98 414 100 101 411 103 409 33 479 35 477 476 38 474 40 408 106 406 108 109 403 111 401 41 471 43 469 468 46 466 48 113 399 115 397 396 118 394 120 464 50 462 52 53 459 55 457 121 391 123 389 388 126 386 128 456 58 454 60 61 451 63 449
 |
26 15
1 6
19 17 10
8 24 This is a disguised version of index # 1. It is the earliest record I could find of the order 3 magic cube. All rows, columns, pillars and 4 main diagonals sum to 42. T. Hugel, Das Problem der magishen Systeme, 1876, Verlag von A. H. Gottschick. |
Horizontal planes of this cube [1] are pandiagonal magic squares. Rows and columns of the other planes are correct, but not the diagonals. All 4 triagonals are incorrect so this cube is not magic by present standards.
It is interesting to note that Frost published two advanced truly magic cubes 12 years earlier [2]. In that paper [1], he published 6 other magic cubes, all with correct triagonals, and one of them a perfect magic cube! See my Frost page which includes all of these cubes.
I II III IV 1 56 13 60 44 17 40 29 61 12 49 8 24 45 28 33 30 43 18 39 7 62 11 50 34 23 46 27 59 2 55 14 52 5 64 9 25 36 21 48 16 57 4 53 37 32 41 20 47 26 35 22 54 15 58 3 19 38 31 42 10 51 6 63
[1] A. H. Frost, On the General Properties of Nasik
Cubes, QJM 15, 1878, pp 93-123 plus plates 1 and 2.
[2] A. H. Frost, Invention of Magic Cubes. Quarterly
Journal of Mathematics, 7, 1866, pp 92-103
Dr. F. A. P. Barnard published a lengthy and important paper on magic squares and cube in 1888 [1]. He was 79 years old at the time.
Like Dr. A. H. Frost, he was years ahead of his time on the subject of cubes, displaying an order 8 and two order 11 perfect cubes. He seemed unaware that anyone else had worked on magic cubes, except for G. Frankenstein. For more information go to my Barnard page.
 |
This order 4 cube by Barnard is not magic by present
standards. However, it does have special magic features! Horizontal
planes rows only sum correctly to 130 Barnard, like Frost, did not seem to appreciate simple magic cubes. Almost all their energies went into the design of advanced magic cubes. |
[1] F.A.P. Barnard, Theory of Magic Squares and Magic Cubes, Memoirs of the National Academy of Science, 4, 1888, pp. 207-270.
Constructed before 1889 by W. Firth (he died in 1889 [1, p. 298 footnote].
A simple magic cube. 24 planar arrays have rows and columns and the 4 main triagonals are correct. These are the minimal requirements by the current definition [2]. Oblique arrays; 4 with columns correct, 2 with rows correct. Not associated.
Dr. Planck wrote [1, page 373 footnote], “It was by this method that Firth in the 80’s constructed what was, almost certainly, the first correct magic cube of order 6.”
Top II III 2 8 134 129 186 192 5 3 132 135 189 187 117 114 146 152 62 60 6 4 130 133 190 188 1 7 136 131 185 191 118 113 150 148 64 58 182 178 21 24 121 125 180 184 18 19 127 123 54 50 109 106 168 164 177 181 22 23 126 122 183 179 17 20 124 128 52 56 110 105 162 166 144 138 174 169 16 10 139 141 172 175 11 13 154 160 70 68 97 102 140 142 170 173 12 14 143 137 176 171 15 9 156 158 66 72 98 101 IV V VI 120 115 149 147 63 57 206 204 42 45 78 76 201 207 48 43 73 79 119 116 145 151 61 59 202 208 46 41 74 80 205 203 44 47 77 75 51 55 112 107 161 165 89 93 198 199 38 34 95 91 193 196 36 40 53 49 111 108 167 163 94 90 197 200 33 37 92 96 194 195 39 35 155 157 65 71 100 103 28 30 82 85 212 214 31 25 88 83 215 209 153 159 69 67 99 104 32 26 86 81 216 210 27 29 84 87 211 213
[1] W. S. Andrews, Magic Squares & Cubes,
2nd edition, Dover Publ. 1960, original 1917. Pages 298 & 305.
See Chapter XII The ‘Theory of Reversions pp. 295-320 (written by Dr. C.
Planck).
[2] W. S. Andrews, Magic Squares & Cubes, Open Court, 1908, page 64.
This cube is magic but has no
special features except it is associated. It is one of the first magic cubes by
the current definition to be published.
The 2 members of each of the 12 planar diagonal pairs sums to the same value.
I II III IV 1 48 32 49 63 18 34 15 62 19 35 14 4 45 29 52 60 21 37 12 6 43 27 54 7 42 26 55 57 24 40 9 56 25 41 8 10 39 23 58 11 38 22 59 53 28 44 5 13 36 20 61 51 30 46 3 50 31 47 2 16 33 17 64
Hermann Schubert, Mathematical Recreations and Essays Open Court 1899 page 62.
The center plane in each dimension is magic (a feature of
associated magic cubes).
The horizontal planes and vertical planes parallel to the front, have all diagonals in one direction correct.
The oblique squares: 2 are simple magic 2 have rows
correct and 2 have columns correct. 1 oblique square has all pandiagonals in
both directions correct 4 have all pandiagonals in one direction correct.
All the pantriagonals in 2 of the 4 directions sum correctly.
Top II III 121 27 83 14 70 2 58 114 45 96 33 89 20 71 102 10 61 117 48 79 36 92 23 54 110 67 123 29 85 11 44 100 1 57 113 75 101 32 88 19 76 7 63 119 50 53 109 40 91 22 84 15 66 122 28 115 41 97 3 59 87 18 74 105 31 118 49 80 6 62 24 55 106 37 93 IV V 64 120 46 77 8 95 21 52 108 39 98 4 60 111 42 104 35 86 17 73 107 38 94 25 51 13 69 125 26 82 16 72 103 34 90 47 78 9 65 116 30 81 12 68 124 56 112 43 99 5
Hermann Schubert, Mathematical Recreations and Essays Open Court 1899 page 62.
This cube has 8 simple magic squares, the 4 horizontal and the 4 parallel to the sides. However it is not magic by present standards because, although all orthogonal lines sum correctly, none of the 4 triagonals are do. Features are exactly the same as the Fermat cube.
I II III IV 4 62 63 1 57 7 6 60 32 34 35 29 37 27 26 40 41 23 22 44 56 10 11 53 17 47 46 20 16 50 51 13 21 43 42 24 12 54 55 9 45 19 18 48 52 14 15 49 64 2 3 61 5 59 58 8 36 30 31 33 25 39 38 28
From E. Fourrey Recreations arithmetiques (Arithmetical Recreations) 8th edition Vuibert 2001.
This is a simple magic cube, having the basic
characteristics only.
The 2 members of each of the 12 planar diagonal pairs sums to the same value.
I II III IV 1 63 60 6 48 18 21 43 32 34 37 27 49 15 12 54 62 4 7 57 19 45 42 24 35 29 26 40 14 52 55 9 56 10 13 51 25 39 36 30 41 23 20 46 8 58 61 3 11 53 50 16 38 28 31 33 22 44 47 17 59 5 2 64
W. S. Andrews, Magic Squares & Cubes,
Open Court, 1908.
W. S. Andrews Magic Squares and Cubes 2nd edition Dover
Publ. 1960 p. 80 (First published in 1917)
This is an associated magic cube. It contains 5 pandiagonal magic squares (the 5 horizontal). As well, the other 2 central orthogonal squares and 4 of the 6 oblique squares are simple magic. The 3 central orthogonal squares and the 4 oblique magic squares are associated.
The cube is not pantriagonal magic because all the triagonals in only 1 of the 4 directions is correct.
Top II III 67 98 104 10 36 106 12 43 74 80 50 51 82 113 19 110 11 42 73 79 49 55 81 112 18 88 119 25 26 57 48 54 85 111 17 87 118 24 30 56 1 32 63 94 125 86 117 23 29 60 5 31 62 93 124 69 100 101 7 38 4 35 61 92 123 68 99 105 6 37 107 13 44 75 76 IV V 89 120 21 27 58 3 34 65 91 122 2 33 64 95 121 66 97 103 9 40 70 96 102 8 39 109 15 41 72 78 108 14 45 71 77 47 53 84 115 16 46 52 83 114 20 90 116 22 28 59
W. S. Andrews Magic Squares & Cubes
Open Court 1908
W. S. Andrews Magic Squares and Cubes 2nd edition Dover
Publ. 1960 p. 73 (First published in 1917)
All 6 outside planes are simple magic squares [1]. No other planar squares have any correct diagonals.
According to Leeflang [2] this was the first even order
magic cube (by modern definition) to be published, although the Firth order 6
cube was actually constructed before 1890. Not so. Herman Schubert published
orders 4 and 5 magic cubes in 1898.
Late note. Frost published an order 8 pandiagonal magic cube in 1866, and
an order 4 pantriagonal magic cube in 1878 (along with a number of other
magic cubes, including an order 9 perfect cube.). Actually, others
that are not already shown on this page are; Huber 0rder 4 1891, Planck order 10
1894 and order 8 perfect in 1905.
Top II III 106 8 7 212 209 109 166 130 129 32 30 164 163 135 136 25 27 165 199 116 113 16 12 195 37 152 148 137 143 34 36 145 149 144 138 39 196 114 115 11 15 200 33 151 150 142 140 35 40 146 147 139 141 38 21 203 202 103 100 22 128 41 47 157 154 124 121 48 42 156 159 125 17 205 208 99 104 18 126 46 44 155 153 127 123 43 45 158 160 122 112 5 6 210 211 107 161 131 133 28 31 167 168 134 132 29 26 162 IV V VI 55 192 191 83 81 49 50 185 186 86 88 56 111 1 2 213 216 108 93 60 57 176 174 91 92 61 64 169 171 94 194 117 120 9 13 198 89 62 63 172 175 90 96 59 58 173 170 95 197 119 118 14 10 193 182 74 77 70 65 183 179 79 76 67 72 178 20 206 207 98 101 19 180 75 73 68 71 184 181 78 80 69 66 177 24 204 201 102 97 23 52 188 190 82 85 54 53 189 187 87 84 51 105 4 3 215 214 110
[1] W. S. Andrews Magic
Squares & Cubes 1960 (1917) p 202 .
[2] John Worthington, A Magic
Cube of Six, The Monist, 20, 1910, pp 303-309
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October 14, 2009
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