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Addendum: January 2005. Some changes and additions have been made to this page due to the discovery of a new class (Pantriagonal Diagonal) of magic cubes.
This page is intended as a supplement to my primary perfect
magic cubes page, which was written before I obtained a lot of additional
material on magic cubes.
Frost referred to this general type (pantriagonal, diagonal, pandiagonal and perfect) as
Nasik [1].
Barnard referred to it as perfect and perfectly magic. [2]
Rosser and Walker referred to the perfect cube as Diabolic [3].
Boyer and Trump refer to the diagonal cube as perfect (higher classes as perfect
with enhancements). [4]
Nakamura refers to the perfect cube (my definition) as Pan-2,3-agonal, thus
avoiding the entire controversy over the definition of perfect. [5]
I am including the dates of publication along with the cubes I show here. However, please do not conclude that these are the first published of that particular type of cube. For example, Rev. A. H. Frost published examples of all these types in 1866 and 1878! Others, in many cases, also have published prior to the date of the cubes I show. The cubes I choose for examples on this page was determined by variety and a desire not to use a cube that had been shown on another page on this site.
As a review, I will include a brief description of the features of a particular type of cube when I first show that type.
All the cubes shown on this page are available in my test spreadsheets, for download from my Downloads page.
[1] A. H. Frost,
Invention of Magic Cubes. Quarterly Journal of Mathematics, 7, 1866, pp
92-103
[2] F. A. P. Barnard,
Theory
of Magic Squares and Cubes, Nat. Academy of Sciences, Vol. IV, Sixth Memoir,
1888, pp 207-270
[3] B. Rosser and R. J. Walker, 1939,
A continuation
of The Algebraic Theory of Diabolic Magic Squares on typewritten pages
numbered 729 – 753.
[4] Christian Boyer's Multimagic Web site at
www.multimagie.com/index.htm
[5]
Mitsutoshi Nakamura's
Web site at
http://homepage2.nifty.com/googol/magcube/en/
Order 3 |
The lowest possible order for a magic cube. I show 2 non-normal cubes from Dr. Planck (1917). |
Order 4 |
The lowest possible order for a pantriagonal magic cube. I show 2 simple cubes (1922 and 1956 ) and a pantriagonal cube (1980). |
Order 5 |
The lowest possible order for a pantriagonal associated magic cube. I show the one where Hendricks introduces the term pantriagonal(1972). Also a simple associated cube. (1899) |
Order 6 |
I show a simple cube (1917), a semi-pantriagonal cube (1922), and a diagonal cube (2003). |
Order 7 |
The lowest possible order for a pandiagonal magic cube. I show Langman's (1962). |
Order 8 |
The lowest possible order for a perfect magic cube. Shown
here is Myer's (1970, called perfect) and Benson and Jacoby's true perfect
cube (1988). See an article on Nasik, the suggested alternative to the confusing term perfect. |
Order 9 |
The lowest possible order for an associated perfect magic cube. I show one (1977), and also a pantriagonal magic cube (2000). |
Order 10 |
A simple J. R. Hendricks order 10 cube with inlaid order 5 semi-magic cubes in each octant. |
Order 11 |
An order 11 perfect magic cube from instructions published in 1976. |
The simple magic cube
There are (3m2)+4 lines that
sum to m(m3+1)/2. These are the rows, columns, pillars
and the 4 triagonals. Note that no diagonals are required to sum
correctly, although some may. Order-3 is the smallest possible magic cube.
The associated magic cube
All pairs of numbers that are diametrically
equidistant on each side of the center point of the cube sum to m3 +
1. The minimum order for associated Nasik type cubes (pantriagonal,
pandiagonal, and perfect) is one more then if the that type of cube is not
associated.
Dr. C. Planck – simple - 1905
Because all four simple associated magic cubes of order 3 all already shown on
other pages on this site, I will not repeat them here. Instead, I will start
this tour through the types of magic cubes, from simple to perfect, with two
unorthodox examples.
These two order 3 cubes are taken from Dr. Planck’s order 3
Octahedron (Tesseract) so do not use consecutive numbers. [1]
The first cube is one of the four central cubes. It is fully magic, but does not
use consecutive numbers, so is not considered normal. It is associated, as are
all central hyperplanes in a higher dimension odd order hypercube. For the same
reason, the central square in this cube is associated magic.
Because this cube is associated magic, it is also semi-pantriagonal (even though
it is not normal). This can be confirmed by adding up each of the four opposite
short triagonals. We find that 6 + 46 + 36 + 76 = 164, 36 + 38 + 44 + 46 = 164,
38 + 14 + 68 + 44 = 164, and 6 + 68 + 14 + 76 = 164. In each case, when we
subtract the center cell value, the answer is123, the magic sum for this cube.
Top Middle Bottom Top Middle Bottom 65 6 52 33 79 11 25 38 60 34 74 15 65 6 52 24 43 56 36 73 14 19 41 63 68 9 46 23 45 55 36 73 14 64 5 54 22 44 57 71 3 49 30 76 17 66 4 53 22 44 57 35 75 13 A Central cube An outside cube
Now look at the second cube. It is one of the outside cubes
from his order 3 tesseract. It is not magic by present standards because only 1
of the 4 triagonals is correct. This condition for the tesseract is consistent
with the definition for a magic cube, which does not require the squares within
it have correct diagonals.
These two cubes, as part of a greater hypercube, have some numbers in common. In
this case, one of the three horizontal planes is common to both cubes.
[1] C. Planck, The
Theory of Path Nasiks, Printed for private circulation by A. J. Lawrence,
Printer, Rugby, 1905 (This is fig. 11, page 14)
Also in W. S. Andrews, Magic
Squares and Cubes, page 367, fig. 688
Pantriagonal magic cube
As well as the normal requirements of a simple
magic cube , the additional requirement is that all pantriagonals sum
correctly. There are 7m2 lines that sum to the constant. This
cube is analogous to a pandiagonal magic
square, but instead of being able to move a row or column to the opposite side,
you may move a plane. Order-4 is the smallest possible pantriagonal magic cube.
Weidemann – simple - 1922
This simple magic cube was constructed by I. Weidemann in 1922. It is not associated and has no special features, except it is semi-pantriagonal is the equivalent to the Group IV magic squares of order 4.
Top II III bottom 1 60 56 13 64 5 9 52 62 7 11 50 3 58 54 15 48 21 25 36 17 44 40 29 19 42 38 31 46 23 27 34 32 37 41 20 33 28 24 45 35 26 22 47 30 39 43 18 49 12 8 61 16 53 57 4 14 55 59 2 51 10 6 63
Weidemann, Ingenieur, Zauberquadrate und andere magische Zahlenfiguren der Ebene und des Raumes, Oscar Leiner, 1922, p. 60 Translated title is Magic squares and other plane and solid magic figures.
Meloc – simple – associated - 1956
This is a different aspect of a cube shown in Encyclopedia Brittanica 1911 edition. It is simple magic and is associated. The editor refers to this as a ‘Jupiter’ cube because it is order 4. (I do not know the significance of the word ‘meloc’.)
Top II III bottom 64 2 3 61 33 31 30 36 17 47 46 20 16 50 51 13 5 59 58 8 28 38 39 25 44 22 23 41 53 11 10 56 9 55 54 12 24 42 43 21 40 26 27 37 57 7 6 60 52 14 15 49 45 19 18 48 29 35 34 32 4 62 63 1
From a mimeographed magic newsletter called Treasury of Folklore – Fantasies in Figures, Mathematic Mysteries and Magic, edited by Stanley J. Coleman, 1956.
Hendricks - pantriagonal - not associated - 1980
Order 4 is the smallest order that can have a pantriagonal magic cube. John Hendricks published this one in 1980 [1] and established that there are 160 basic cubes of this type. Each of these can appear in 48 aspects (as can any order magic cube). He published an earlier pantriagonal magic cube in 1972 [2] when he introduced the term ‘pan-3-agonal’ (later modified to pantriagonal) for this type of magic cube.
Top II III bottom 48 7 57 18 10 33 31 56 51 28 38 13 21 62 4 43 54 29 35 12 20 59 5 46 41 2 64 23 15 40 26 49 27 52 14 37 61 22 44 3 8 47 17 58 34 9 55 32 1 42 24 63 39 16 50 25 30 53 11 36 60 19 45 6
[1] John R. Hendricks, the
Pan-3-agonal Magic Cube of Order 4, JRM 13:4, 1980-81, pp274-281
[2]
John R. Hendricks, The Pan-3-Agonal Magic Cube of Order 5, JRM 5:3:1972,
pp 205-206
Hendricks - pantriagonal – associated - 1972
This cube is pantriagonal and associated. It has no other special features except, of course, the 3 central planes are associated magic squares (because the cube is associated). [1][2]
This is the smallest order pantriagonal that can also be associated. Dr. Planck stated [3] that the smallest Nasik order in k dimensions is always 2k , (or 2k + 1 if we require association). I believe this applies to pantriagonal, pandiagonal and perfect cubes because Planck cited Rev. Frost [4][5], who produced cubes of all three of these types and lumped them all together as Nasik cubes!
Top II III 50 66 87 108 4 54 100 116 12 33 83 104 25 41 62 69 90 106 2 48 98 119 15 31 52 102 23 44 65 81 88 109 5 46 67 117 13 34 55 96 21 42 63 84 105 107 3 49 70 86 11 32 53 99 120 45 61 82 103 24 1 47 68 89 110 35 51 97 118 14 64 85 101 22 43 IV V 112 8 29 75 91 16 37 58 79 125 6 27 73 94 115 40 56 77 123 19 30 71 92 113 9 59 80 121 17 38 74 95 111 7 28 78 124 20 36 57 93 114 10 26 72 122 18 39 60 76
[1]
John R. Hendricks, The Pan-3-Agonal Magic Cube of Order 5, JRM 5:3:1972,
pp 205-206
[2]
John R. Hendricks, Magic Square Course, self-published, 1991, page 360.
[3] W. S. Andrews, Magic
Squares & Cubes, 2nd edition, Dover Publ. 1960, page 366 .
[4] A. H. Frost, Invention of Magic Cubes. Quarterly Journal of
Mathematics, 7, 1866, pp 92-103
[5] A. H. Frost, On the General Properties of Nasik Cubes, QJM 15, 1878,
pp 93-123
Schubert – simple - semi-pantriagonal - associated - 1899
The center orthogonal plane in each orientation is magic (a
feature of associated magic cubes). 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.
Because all the pantriagonals in 2 of the 4 directions sum correctly, perhaps we
could call this a semi-pantriagonal cube?
No! Because that is not the traditional definition. However, this cube is semi-pantriagonal
because it is equivalent to the definition for a semi-pandiagonal magic square.
Opposite short triagonals (one of 4 such pairs is shown here in red) plus the
center cell (green) sum to S. Also, shown here in blue, one of 4 opposite
long diagonal pairs minus the center cell sums to S. I have more details on a
semi-pantriagonal page.
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 Essays and Recreations, Open Court, 1899, page 62.
Weidemann - associated and semi-pantriagonal - 1922
I have included this cube because singly-even cubes are relatively rare.
Top II III 186 35 213 184 32 1 139 113 106 105 74 114 175 38 40 147 71 180 30 206 190 9 191 25 120 101 135 118 98 79 43 47 171 172 152 66 199 197 22 21 14 198 96 128 124 87 125 91 54 164 51 58 167 157 193 23 15 16 200 204 85 122 94 129 131 90 168 161 57 52 158 55 7 188 28 207 209 12 102 80 117 136 83 133 61 173 154 153 62 48 36 2 183 214 5 211 109 107 75 76 140 144 150 68 178 69 41 145 IV V VI 72 176 148 39 149 67 73 77 141 142 110 108 6 212 3 34 215 181 169 155 64 63 44 156 84 134 81 100 137 115 205 8 10 189 29 210 162 59 165 160 56 49 127 86 88 123 95 132 13 17 201 202 194 24 60 50 159 166 53 163 126 92 130 93 89 121 19 203 196 195 20 18 151 65 45 46 170 174 138 119 99 82 116 97 192 26 208 27 11 187 37 146 70 177 179 42 103 143 112 111 104 78 216 185 33 4 182 31
Weidemann, Ingenieur, Zauberquadrate und andere magische Zahlenfiguren der Ebene und des Raumes, Oscar Leiner, 1922, page 62.
Sayles - simple - 1910
This is a simple magic cube but has the unique feature that if the cube is divided into 27 2x2x2 cubelets, the six faces of each cubelet and 2 of the 6 diagonal planes will each sum the same value. These 27 values step from 382 to 486. For example, the top left 2x2x2 cubelet
The six faces: Two diagonal planes 4 4 4 193 139 85 4 85 85 85 139 112 166 166 166 139 166 112 58 31 31 31 31 58 139 193 193 58 58 112 193 112 394 394 394 394 394 394 394 394 Top II III 4 139 161 26 174 147 193 58 80 215 39 66 18 153 136 163 23 158 85 166 107 188 93 12 112 31 134 53 120 201 99 180 1 82 104 185 98 152 138 3 103 157 125 71 57 192 130 76 181 19 95 176 171 9 179 17 84 165 184 22 44 206 111 30 49 211 100 154 149 14 90 144 183 21 13 175 89 170 48 210 202 40 116 35 167 5 108 189 172 10 102 156 148 94 8 143 129 75 67 121 197 62 86 140 162 27 91 145 IV V VI 207 72 55 28 212 77 155 20 150 15 169 142 74 209 69 204 34 61 126 45 190 109 131 50 101 182 96 177 88 7 128 47 123 42 115 196 46 208 122 41 36 198 6 87 106 187 92 173 195 114 133 52 119 38 127 73 68 203 117 63 141 168 160 25 11 146 60 33 79 214 200 65 32 194 135 54 37 199 151 16 137 83 105 159 70 205 56 110 132 78 113 59 81 216 118 64 97 178 2 164 186 24 124 43 191 29 51 213
W. S. Andrews, Magic Squares & Cubes, 2nd edition, Dover Publ. 1960 (1917), page 197 (The Monist, ,20,1910, pp299-303).
Diagonal type magic cubes
As well as the normal requirements of a simple magic cube, the additional
requirement is that all orthogonal planes of the cube are simple magic squares.
This is the popular definition of the 'Perfect' magic cube. The smallest
diagonal magic cube possible is order 5.
Trump - diagonal - 2003
This cube contains 18 orthogonal simple magic squares of order 6. This qualifies
it as a diagonal magic cube. The six oblique squares are also simple magic.
Until I received this cube from Walter in late summer of 2003, I had seen only
two order 8 cubes and an order 12 cube of this type!
Walter sent me an order 7 and Christian Boyer sent an order 9 diagonal cube at about the same time.
I - Top II III 109 143 76 123 88 112 137 48 157 68 158 83 103 101 159 36 119 133 87 156 49 170 63 126 155 2 198 27 207 62 162 196 201 8 29 55 140 174 52 150 53 82 34 187 212 13 22 183 46 206 28 197 3 171 75 66 182 51 139 138 147 32 1 208 193 70 163 15 191 18 210 54 136 40 148 65 176 86 44 213 23 186 12 173 93 17 14 211 192 124 104 72 144 92 132 107 134 169 60 149 59 80 84 116 58 181 98 114 IV V VI - Bottom 90 56 57 184 175 89 102 178 117 95 38 121 110 125 85 145 73 113 118 21 16 209 188 99 50 215 19 190 10 167 79 61 168 47 154 142 180 11 189 20 214 37 120 30 5 204 195 97 131 43 165 67 164 81 64 202 26 199 7 153 111 185 216 9 24 106 91 151 35 166 78 130 71 200 203 6 25 146 172 4 194 31 205 45 135 177 69 152 41 77 128 161 160 33 42 127 96 39 100 122 179 115 105 94 129 74 141 108
ADDENDUM: Walter Trump discovered an order 5 cube of this type on Nov. 12, 2003. See it on my order 5 page. More from Walter himself at http://www.trump.de/magic-squares/magic-cubes/cubes-1.html.
Other cube types
Due to space restraints, I will not show examples of a simple, a pantriagonal,
and a diagonal magic cube of this order.
Pandiagonal magic cube
As well as the normal requirements of a simple magic cube, the additional
requirement is that all orthogonal planes of the cube are pandiagonal magic
squares. The 6 oblique squares are simple magic (1, 2, or 3 may be pandiagonal
magic) [1].
There are at least (9m2)+4 lines that sum correctly. This is
one of the original definitions of a ‘perfect magic cube’.
Order-7 is the smallest possible pandiagonal magic cube.
Are there any order 8
pandiagonal magic cubes?
I have examined 10 order 7 pandiagonal magic cubes. All are order 7 and all
are associated. Is this always the case?
No! On Mar. 9/03 I received six pandiagonal magic cubes from Abhinav
Soni. They were orders 9, 11, 13, 15, 17, (and 19 which I have not yet tested).
Orders 11, 13, 15, and 17 were all NOT associated.
Langman - pandiagonal -1962 [2]
This is a pandiagonal magic cube because all 21 planar square arrays
are pandiagonal magic. All 6 diagonal square arrays are simple magic (3 of the
six have all the pandiagonals in one direction correct).
All main triagonals are correct (of course) but all the pantriagonals are
correct only for one of the 4 sets (i.e. only 1 of 4 directions). This cube has
sometimes been referred to as perfect. It was published in 1962.
Rev. A. H. Frost published an order 7 pandiagonal magic cube in 1866 [3]
Top II 322 87 153 261 33 141 207 100 215 323 95 161 269 41 29 144 210 318 90 149 264 157 272 37 103 211 326 98 86 152 260 32 147 206 321 214 329 94 160 268 40 99 143 209 317 89 148 263 35 271 36 102 217 325 97 156 151 266 31 146 205 320 85 328 93 159 267 39 105 213 208 316 88 154 262 34 142 42 101 216 324 96 155 270 265 30 145 204 319 91 150 92 158 273 38 104 212 327 III IV 277 49 108 223 331 54 162 62 170 285 1 116 231 339 334 50 165 280 45 111 219 119 227 342 58 173 281 4 48 107 222 330 53 168 276 169 284 7 115 230 338 61 56 164 279 44 110 218 333 226 341 57 172 287 3 118 106 221 336 52 167 275 47 283 6 114 229 337 60 175 163 278 43 109 224 332 55 340 63 171 286 2 117 225 220 335 51 166 274 46 112 5 113 228 343 59 174 282 V VI 232 298 70 178 293 9 124 17 132 240 306 71 186 252 289 12 120 235 301 66 181 74 189 248 20 128 243 302 297 69 177 292 8 123 238 131 239 305 77 185 251 16 11 126 234 300 65 180 288 188 247 19 127 242 308 73 68 176 291 14 122 237 296 245 304 76 184 250 15 130 125 233 299 64 179 294 10 246 18 133 241 307 72 187 182 290 13 121 236 295 67 303 75 183 249 21 129 244 bottom 194 253 25 140 199 314 79 202 310 82 190 256 28 136 259 24 139 198 313 78 193 309 81 196 255 27 135 201 23 138 197 312 84 192 258 80 195 254 26 134 200 315 137 203 311 83 191 257 22
[1] (5)
J. R. Hendricks, Magic Squares to Tesseracts by Computer, Self-published 1999,
0-9684700-0-9
[2] Harry Langman, Ph. D.,
Play Mathematics, Hafner Publ. 1962, p. 75-76.
[3] A. H. Frost, Invention of Magic
Cubes. Quarterly Journal of Mathematics, 7, 1866, pp 92-103
Diagonal magic cubes
As well as the normal requirements of a simple magic cube, the additional
requirement is that all orthogonal planes of the cube are simple magic squares.
In Sept. 2003 I obtained orders 6 and 7 diagonal cubes from Walter Trump and an
order 9 from Christian Boyer.
On Nov. 13, 2003, Trump and Boyer announced the discovery of an order 5 cube of
this type.
This cube was called perfect by an early definition
(M. Gardner and others) because all squares are magic. It was constructed by R.
L. Myers in 1970
All 30 squares are simple magic i.e. rows, columns
and main diagonals of all orthogonal and oblique square arrays sum correctly. Corner sums of all order-5 sub-cubes sum to the
constant. All orthogonal squares are simple magic so it is a diagonal cube.
There are 3m2 + 6m + 4 lines that sum correctly.
Top II 19 497 255 285 432 78 324 162 134 360 106 396 313 219 469 55 303 205 451 33 148 370 128 414 442 92 342 184 5 487 233 267 336 174 420 66 243 273 31 509 473 59 309 215 102 392 138 364 116 402 160 382 463 45 291 193 229 263 9 491 346 188 438 88 486 8 266 236 89 443 181 343 371 145 415 125 208 302 36 450 218 316 54 472 357 135 393 107 79 429 163 321 500 18 288 254 185 347 85 439 262 232 490 12 48 462 196 290 403 113 383 157 389 103 361 139 58 476 214 312 276 242 512 30 175 333 67 417 III IV 306 212 478 64 141 367 97 387 423 69 331 169 28 506 248 278 14 496 226 260 433 83 349 191 155 377 119 405 296 198 460 42 109 399 129 355 466 52 318 224 252 282 24 502 327 165 427 73 337 179 445 95 238 272 2 484 456 38 300 202 123 409 151 373 199 293 43 457 380 154 408 118 82 436 190 352 493 15 257 227 507 25 279 245 72 422 172 330 366 144 386 100 209 307 61 479 412 122 376 150 39 453 203 297 269 239 481 3 178 340 94 448 168 326 76 426 283 249 503 21 49 467 221 319 398 112 354 132 V VI 381 159 401 115 194 292 46 464 492 10 264 230 87 437 187 345 65 419 173 335 510 32 274 244 216 310 60 474 363 137 391 101 34 452 206 304 413 127 369 147 183 341 91 441 268 234 488 6 286 256 498 20 161 323 77 431 395 105 359 133 56 470 220 314 140 362 104 390 311 213 475 57 29 511 241 275 418 68 334 176 440 86 348 186 11 489 231 261 289 195 461 47 158 384 114 404 471 53 315 217 108 394 136 358 322 164 430 80 253 287 17 499 235 265 7 485 344 182 444 90 126 416 146 372 449 35 301 207 VII Bottom 96 446 180 338 483 1 271 237 01 299 37 455 374 152 410 124 356 130 400 110 223 317 51 465 501 23 281 251 74 428 166 328 259 225 495 13 192 350 84 434 406 120 378 156 41 459 197 295 63 477 211 305 388 98 368 142 170 332 70 424 277 247 505 27 425 75 325 167 22 504 250 284 320 222 468 50 131 353 111 397 149 375 121 411 298 204 454 40 4 482 240 270 447 93 339 177 246 280 26 508 329 171 421 71 99 385 143 365 480 62 308 210 458 44 294 200 117 407 153 379 351 189 435 81 228 258 16 494
Martin Gardner, Time
Travel and Other Mathematical Bewilderments, 1988, page 222
John R. Hendricks, Magic Square Course, self-published, 1991, (page 405).
W. H. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ. 1981,
0-486-24140-8, pp 43-60
Rudolf Ondrejka, Letter to the Editor, Journal of Recreational
Mathematics, 20:3, 1988, pp207-209
Pantriagonal Diagonal – A magic cube that is a
combination Pantriagonal and Diagonal cube. All main and broken triagonals must
sum correctly, In addition, it will contain 3m order m simple
magic squares in the orthogonal planes, and 6 order m pandiagonal magic
squares in the oblique planes.
For short, I will reduce this unwieldy name to PantriagDiag. This is number 4 in
what is now 6 classes of magic cubes. So far, very little is known of this class
of cube. The only ones constructed so far (both by Nakamura) are order 8 (not
associated and associated).
This cube was discovered by Mitsutoshi Nakamura and named by him in 2004. So
there are now 6 classes of magic cubes: Simple, Pantriagonal, Diagonal,
Pantriagonal Diagonal (PantriagDiag), Pandiagonal, and Perfect. There are 7m2
+ 6m lines that sum correctly.
Perfect magic cube
A perfect magic cube is a combination pantriagonal and pandiagonal magic
cube. Because of the combination of the two previous conditions, all 6 oblique
squares are pandiagonal magic as well.
In a perfect magic cube there are 9m pandiagonal magic squares. That is,
all 3m orthogonal planes, the 6 oblique planes, and the 6m-1
broken planes parallel to the oblique planes [1].
There are 13m2 lines that sum correctly. Order-8 is the
smallest possible perfect magic cube.
Corners of all orders 2, 3, 4, 5, 6, 7 and 8 (including wraparound) also sum
correctly to 2052. Both order 8 perfect cubes I have seen have this feature.
Is it always present in order 8 perfect cubes? Barnard says it is [6].
Both order 8 perfect cubes also have another feature called complete [2].
Is this also common to all order 8 perfect cubes? To all 8x
perfect cubes?
The Benson Jacoby Perfect Magic Cube - 1981
This is the 3rd known published order 8 perfect magic cube
(Barnard 1888, Planck 1905) [6][7].
Benson and Jacoby referred to this cube as pandiagonal
perfect. It appeared in octal [3] and in decimal in [4].
All 24 planar squares are pandiagonal magic as are also the 6 oblique squares
and the seven broken squares parallel to each of these[1][5] for a total of 72
pandiagonal magic squares. The 256 pantriagonals also all sum correctly.
Corners of all orders 2, 3, 4, 5, 6, 7 and 8 (including wraparound) sub-cubes
also sum correctly to 2052 making this cube complete. Also, many other shapes of parallelopipeds.
Top II 280 431 214 109 273 426 211 108 219 100 288 423 222 101 281 418 377 450 187 4 384 455 190 5 182 13 369 458 179 12 376 463 432 215 110 277 425 210 107 276 99 284 424 223 102 285 417 218 449 186 3 380 456 191 6 381 14 373 457 178 11 372 464 183 216 111 278 429 209 106 275 428 283 420 224 103 286 421 217 98 185 2 379 452 192 7 382 453 374 461 177 10 371 460 184 15 112 279 430 213 105 274 427 212 419 220 104 287 422 221 97 282 1 378 451 188 8 383 454 189 462 181 9 370 459 180 16 375 III IV 313 386 251 68 320 391 254 69 246 77 305 394 243 76 312 399 368 471 174 21 361 466 171 20 163 28 360 479 166 29 353 474 385 250 67 316 392 255 70 317 78 309 393 242 75 308 400 247 472 175 22 365 465 170 19 364 27 356 480 167 30 357 473 162 249 66 315 388 256 71 318 389 310 397 241 74 307 396 248 79 176 23 366 469 169 18 363 468 355 476 168 31 358 477 161 26 65 314 387 252 72 319 390 253 398 245 73 306 395 244 80 311 24 367 470 173 17 362 467 172 475 164 32 359 478 165 25 354 V VI 304 407 238 85 297 402 235 84 227 92 296 415 230 93 289 410 321 506 131 60 328 511 134 61 142 53 329 498 139 52 336 503 408 239 86 301 401 234 83 300 91 292 416 231 94 293 409 226 505 130 59 324 512 135 62 325 54 333 497 138 51 332 504 143 240 87 302 405 233 82 299 404 291 412 232 95 294 413 225 90 129 58 323 508 136 63 326 509 334 501 137 50 331 500 144 55 88 303 406 237 81 298 403 236 411 228 96 295 414 229 89 290 57 322 507 132 64 327 510 133 502 141 49 330 499 140 56 335 VII Bottom 257 442 195 124 264 447 198 125 206 117 265 434 203 116 272 439 344 495 150 45 337 490 147 44 155 36 352 487 158 37 345 482 441 194 123 260 448 199 126 261 118 269 433 202 115 268 440 207 496 151 46 341 489 146 43 340 35 348 488 159 38 349 481 154 193 122 259 444 200 127 262 445 270 437 201 114 267 436 208 119 152 47 342 493 145 42 339 492 347 484 160 39 350 485 153 34 121 258 443 196 128 263 446 197 438 205 113 266 435 204 120 271 48 343 494 149 41 338 491 148 483 156 40 351 486 157 33 346
This is the broken plane starting on the 2nd from top horizontal plane on the left side and going down to the bottom right side, plus the right column of the top plane. It is one of the seven pandiagonal magic squares parallel to the oblique square from the upper left to lower right sides of the cube.
219 386 305 85 230 447 272 108 182 471 360 60 139 490 345 5 99 250 393 301 94 199 440 276 14 175 480 324 51 146 481 381 283 66 241 405 294 127 208 428 374 23 168 508 331 42 153 453 419 314 73 237 414 263 120 212 462 367 32 132 499 338 33 189
[1] A continuation of
The Algebraic Theory of Diabolic Magic Squares on typewritten pages numbered
729 – 753, (diabolic cubes pp 736-753).
[2] Complete = Every pantriagonal contains m/2
complement pairs spaced m/2 apart. Kanji Setsuda’s Compact (composite)
and Complete magic Cubes Web page is
http://homepage2.nifty.com/KanjiSetsuda/pages/EnglishP1.html
[3] W. H. Benson & O. Jacoby, Magic Cubes: New Recreations, Dover Publ.
1981, 0-486-24140-8.
[4] R. Ondrejka, The Most Perfect (8x8x8)
Magic Cube? (Letter to the Editor), JRM
20:3, 1988, pp207-209
[5] F. Liao, T. Katayama, K. Takaba, Technical Report 99021, School of
Informatics, Kyoto University, 1999. Available on the Internet at
http://www.amp.i.kyoto-u.ac.jp/tecrep/TR1999.html
[6] F. A. P. Barnard,
Theory
of Magic Squares and Cubes, Nat. Academy of Sciences, Vol. IV, Sixth Memoir,
1888, pp 207-270
[7] C. Planck, The
Theory of Path Nasiks, Printed for private circulation by A. J. Lawrence,
Printer, Rugby, 1905 (This is fig. 11, page 14)
Order 9 is the smallest possible for a perfect magic cube that is also associated [1].
Soni Order-9 Pantriagonal Magic Cube - 2001?
This must be classed as a pantriagonal magic cube, although all 9 horizontal and all 9 vertical back to front planes are pandiagonal magic squares. The vertical squares parallel to the sides have incorrect diagonals, so are not magic. The 6 oblique squares are also pandiagonal magic.
Also, all 3x3 arrays of the horizontal planes and the vertical planes parallel with the front face (including wrap-around), sum to the constant.
I generated this cube from Abhinav Soni ‘s program, HyperMagic Cubes.exe. [2]
Top II 635 169 291 68 331 696 473 493 129 44 379 672 449 541 105 611 217 267 485 496 114 647 172 276 80 334 681 639 164 292 72 326 697 477 488 130 27 362 706 432 524 139 594 200 301 480 500 115 642 176 277 75 338 682 615 230 250 48 392 655 453 554 88 24 368 703 429 530 136 591 206 298 438 566 91 600 242 253 33 404 658 616 234 245 49 396 650 454 558 83 58 351 686 463 513 119 625 189 281 439 561 95 601 237 257 34 399 662 574 210 311 7 372 716 412 534 149 55 348 692 460 510 125 622 186 287 415 519 161 577 195 323 10 357 728 569 211 315 2 373 720 407 535 153 38 382 675 443 544 108 605 220 270 419 520 156 581 196 318 14 358 723 III IV 423 515 157 585 191 319 18 353 724 570 212 313 3 374 718 408 536 151 39 383 673 444 545 106 606 221 268 420 521 154 582 197 316 15 359 721 636 170 289 69 332 694 474 494 127 40 387 668 445 549 101 607 225 263 481 504 110 643 180 272 76 342 677 637 165 293 70 327 698 475 489 131 25 363 707 430 525 140 592 201 302 478 501 116 640 177 278 73 339 683 613 231 251 46 393 656 451 555 89 20 364 711 425 526 144 587 202 306 434 562 99 596 238 261 29 400 666 617 232 246 50 394 651 455 556 84 59 349 687 464 511 120 626 187 282 440 559 96 602 235 258 35 397 663 575 208 312 8 370 717 413 532 150 63 344 688 468 506 121 630 182 283 V VI 60 350 685 465 512 118 627 188 280 436 567 92 598 243 254 31 405 659 571 216 308 4 378 713 409 540 146 61 345 689 466 507 122 628 183 284 421 516 158 583 192 320 16 354 725 568 213 314 1 375 719 406 537 152 37 384 674 442 546 107 604 222 269 416 517 162 578 193 324 11 355 729 632 166 297 65 328 702 470 490 135 41 385 669 446 547 102 608 223 264 482 502 111 644 178 273 77 340 678 638 163 294 71 325 699 476 487 132 26 361 708 431 523 141 593 199 303 486 497 112 648 173 274 81 335 679 621 227 247 54 389 652 459 551 85 21 365 709 426 527 142 588 203 304 435 563 97 597 239 259 30 401 664 618 233 244 51 395 649 456 557 82 VII VIII 619 228 248 52 390 653 457 552 86 19 366 710 424 528 143 586 204 305 433 564 98 595 240 260 28 402 665 614 229 252 47 391 657 452 553 90 56 346 693 461 508 126 623 184 288 437 565 93 599 241 255 32 403 660 572 214 309 5 376 714 410 538 147 62 343 690 467 505 123 629 181 285 422 514 159 584 190 321 17 352 726 576 209 310 9 371 715 414 533 148 45 380 670 450 542 103 612 218 265 417 518 160 579 194 322 12 356 727 633 167 295 66 329 700 471 491 133 42 386 667 447 548 100 609 224 262 483 503 109 645 179 271 78 341 676 634 171 290 67 333 695 472 495 128 22 369 704 427 531 137 589 207 299 484 498 113 646 174 275 79 336 680 Bottom 479 499 117 641 175 279 74 337 684 23 367 705 428 529 138 590 205 300 620 226 249 53 388 654 458 550 87 441 560 94 603 236 256 36 398 661 57 347 691 462 509 124 624 185 286 573 215 307 6 377 712 411 539 145 418 522 155 580 198 317 13 360 722 43 381 671 448 543 104 610 219 266 631 168 296 64 330 701 469 492 134
Seimiya Order 9 'Golden' Magic Cube - Associated – Perfect - 1977
This cube [3] uses the numbers from 0 to 728 (instead of 1
to 729) so the magic constant is 3276 (instead of 3285).
It is perfect, so 27 orthogonal square arrays, 6 oblique arrays, and 48 oblique
broken arrays are pandiagonal magic. It is the smallest possible order for a
cube that is both perfect and associated [1]
A.H. Frost published a Perfect order 9 magic
cube in 1878. However, it did not use consecutive numbers.
The first normal perfect magic cube seems to be the order 17 constructed
by Gabriel Arnoux in 1887.
Top II 0 706 521 617 267 397 475 119 174 402 469 124 173 3 702 517 620 266 530 581 249 379 439 128 183 72 715 133 182 75 711 526 584 248 384 433 321 388 448 92 165 54 679 539 590 57 675 535 593 320 393 442 97 164 457 101 237 63 688 503 572 303 352 499 575 302 357 451 106 236 66 684 219 27 697 512 644 312 361 421 83 311 366 415 88 218 30 693 508 647 661 494 626 276 370 430 155 228 36 424 160 227 39 657 490 629 275 375 635 285 334 412 137 192 45 670 566 191 48 666 562 638 284 339 406 142 343 484 146 201 9 652 548 599 294 648 544 602 293 348 478 151 200 12 110 210 18 724 557 608 258 325 466 611 257 330 460 115 209 21 720 553 III IV 705 513 616 269 401 474 118 178 2 473 123 172 7 704 516 612 265 404 580 251 383 438 127 187 74 714 522 181 79 713 525 576 247 386 437 132 392 447 91 169 56 678 531 589 323 677 534 585 319 395 446 96 163 61 100 241 65 687 495 571 305 356 456 567 301 359 455 105 235 70 686 498 29 696 504 643 314 365 420 82 223 368 419 87 217 34 695 507 639 310 486 625 278 374 429 154 232 38 660 159 226 43 659 489 621 274 377 428 287 338 411 136 196 47 669 558 634 52 668 561 630 283 341 410 141 190 483 145 205 11 651 540 598 296 347 543 594 292 350 482 150 199 16 650 214 20 723 549 607 260 329 465 109 256 332 464 114 208 25 722 552 603 V VI 515 615 261 400 476 122 177 1 709 125 176 6 703 520 614 264 396 472 243 382 440 131 186 73 718 524 579 78 712 529 578 246 378 436 134 185 449 95 168 55 682 533 588 315 391 538 587 318 387 445 98 167 60 676 240 64 691 497 570 297 355 458 104 300 351 454 107 239 69 685 502 569 700 506 642 306 364 422 86 222 28 418 89 221 33 694 511 641 309 360 624 270 373 431 158 231 37 664 488 230 42 658 493 623 273 369 427 161 337 413 140 195 46 673 560 633 279 667 565 632 282 333 409 143 194 51 149 204 10 655 542 597 288 346 485 596 291 342 481 152 203 15 649 547 19 727 551 606 252 328 467 113 213 324 463 116 212 24 721 556 605 255 VII VIII 619 263 399 468 121 179 5 708 514 175 8 707 519 613 268 398 471 117 381 432 130 188 77 717 523 583 245 716 528 577 250 380 435 126 184 80 94 170 59 681 532 592 317 390 441 586 322 389 444 90 166 62 680 537 68 690 496 574 299 354 450 103 242 353 453 99 238 71 689 501 568 304 505 646 308 363 414 85 224 32 699 81 220 35 698 510 640 313 362 417 272 372 423 157 233 41 663 487 628 44 662 492 622 277 371 426 153 229 405 139 197 50 672 559 637 281 336 564 631 286 335 408 135 193 53 671 206 14 654 541 601 290 345 477 148 295 344 480 144 202 17 653 546 595 726 550 610 254 327 459 112 215 23 462 108 211 26 725 555 604 259 326 Bottom 262 403 470 120 171 4 710 518 618 434 129 180 76 719 527 582 244 385 162 58 683 536 591 316 394 443 93 692 500 573 298 358 452 102 234 67 645 307 367 416 84 216 31 701 509 376 425 156 225 40 665 491 627 271 138 189 49 674 563 636 280 340 407 13 656 545 600 289 349 479 147 198 554 609 253 331 461 111 207 22 728
[1] W. S. Andrews, Magic
Squares & Cubes, 2nd edition, Dover Publ. 1960, page 366.
[2] Abhinav Soni HyperMagicCube.exe program.
Obtainable from
soni_abhinav@yahoo.com
[3] Seimiya, Mathematical Sciences
(Japanese) Magazine Dec. 1977, p. 45-47
I have not seen any order 10 Nasik type cubes (pantriagonal, pandiagonal, or perfect). Are there any possible? (I do have an order 6 pantriagonal.)
J. R. Hendricks Millennium 10 x 10 x 10 simple magic cube - 2000
This is a simple magic cube with all rows, columns, pillars, and 4 main triagonals correct. It has an order-5 semi-magic cube in each octant. In each case, 1 of the 4 triagonals of the inlaid cube is incorrect.
The two diagonals for each of the 30 planar squares sum to 4130 and 5880. It follows that the 4 oblique squares with incorrect row totals have 5 rows each that total 4130 and 5 rows that total 5880. The two oblique squares with incorrect column totals have 5 columns each with each of these totals. This cube contains no magic squares.
This cube was privately distributed by John Hendricks in December 2000 as a ‘Millennium’ magic cube. Following this listing, I will include one of the inlaid semi-magic order 5 cubes.
Top II 625 377 409 686 843 343 811 284 252 375 401 683 840 617 399 274 367 340 808 276 381 413 695 847 604 354 347 820 288 256 687 844 621 378 410 285 253 371 344 812 417 699 826 608 390 265 358 326 824 292 848 605 382 414 691 816 289 257 355 348 678 835 612 394 421 296 269 362 335 803 609 386 418 700 827 327 825 293 261 359 839 616 398 405 682 807 280 273 366 339 395 422 679 831 613 363 331 804 297 270 214 491 523 530 307 182 655 648 741 714 520 547 304 206 488 738 706 179 672 645 303 210 487 519 546 671 644 737 710 178 484 511 543 325 202 702 200 668 636 734 542 324 201 483 515 640 733 701 199 667 223 480 507 539 316 191 664 632 730 723 506 538 320 222 479 729 722 195 663 631 312 219 496 503 535 660 628 746 719 187 500 502 534 311 218 718 186 659 627 750 526 308 215 492 524 649 742 715 183 651 III IV 832 614 391 423 680 805 298 266 364 332 388 420 697 829 606 356 329 822 295 263 618 400 402 684 836 336 809 277 275 368 424 676 833 615 392 267 365 333 801 299 379 406 688 845 622 372 345 813 281 254 685 837 619 396 403 278 271 369 337 810 415 692 849 601 383 258 351 349 817 290 841 623 380 407 689 814 282 255 373 341 696 828 610 387 419 294 262 360 328 821 602 384 411 693 850 350 818 286 259 352 321 203 485 512 544 669 637 735 703 196 477 509 536 318 225 725 193 661 634 727 540 317 224 476 508 633 726 724 192 665 216 498 505 532 314 189 657 630 748 716 504 531 313 220 497 747 720 188 656 629 310 212 494 521 528 653 646 744 712 185 493 525 527 309 211 711 184 652 650 743 549 301 208 490 517 642 740 708 176 674 207 489 516 548 305 180 673 641 739 707 513 545 322 204 481 731 704 197 670 638 V VI 694 846 603 385 412 287 260 353 346 819 69 971 978 10 37 787 760 228 596 569 830 607 389 416 698 823 291 264 357 330 955 982 14 41 73 573 791 764 232 580 611 393 425 677 834 334 802 300 268 361 986 18 50 52 959 584 552 800 768 236 397 404 681 838 620 370 338 806 279 272 22 29 56 963 995 245 588 556 779 772 408 690 842 624 376 251 374 342 815 283 33 65 967 999 1 751 249 592 565 783 533 315 217 499 501 626 749 717 190 658 908 940 92 124 876 126 874 467 440 158 522 529 306 213 495 745 713 181 654 647 897 904 931 88 120 870 463 431 154 147 486 518 550 302 209 709 177 675 643 736 111 893 925 927 84 459 427 175 143 861 205 482 514 541 323 198 666 639 732 705 80 107 889 916 948 448 166 139 857 455 319 221 478 510 537 662 635 728 721 194 944 96 103 885 912 162 135 853 471 444 VII VIII 13 45 72 954 981 231 579 572 795 763 957 989 16 48 55 555 798 766 239 582 49 51 958 990 17 767 240 583 551 799 993 25 27 59 961 586 559 777 775 243 60 962 994 21 28 778 771 244 587 560 4 31 63 970 997 247 595 563 781 754 966 998 5 32 64 564 782 755 248 591 40 67 974 976 8 758 226 599 567 790 977 9 36 68 975 600 568 786 759 227 71 953 985 12 44 794 762 235 578 571 102 884 911 943 100 475 443 161 134 852 946 78 110 887 919 169 137 860 453 446 91 123 880 907 939 439 157 130 873 466 915 942 99 101 883 133 851 474 442 165 935 87 119 896 903 153 146 869 462 435 879 906 938 95 122 872 470 438 156 129 924 926 83 115 892 142 865 458 426 174 118 900 902 934 86 461 434 152 150 868 888 920 947 79 106 856 454 447 170 138 82 114 891 923 930 430 173 141 864 457 IX Bottom 26 58 965 992 24 774 242 590 558 776 1000 2 34 61 968 593 561 784 752 250 62 969 996 3 35 785 753 246 594 562 6 38 70 972 979 229 597 570 788 756 973 980 7 39 66 566 789 757 230 598 42 74 951 983 15 765 233 576 574 792 984 11 43 75 952 577 575 793 761 234 53 960 987 19 46 796 769 237 585 553 20 47 54 956 988 238 581 554 797 770 964 991 23 30 57 557 780 773 241 589 895 922 929 81 113 863 456 429 172 145 89 116 898 905 932 432 155 148 866 464 109 886 918 950 77 452 450 168 136 859 928 85 112 894 921 171 144 862 460 428 98 105 882 914 941 441 164 132 855 473 917 949 76 108 890 140 858 451 449 167 937 94 121 878 910 160 128 871 469 437 881 913 945 97 104 854 472 445 163 131 901 933 90 117 899 149 867 465 433 151 125 877 909 936 93 468 436 159 127 875
Hendricks – Top Right Back Octant
This is the top right back octant of the 10x10x10
Millennium cube.
This is not a normal magic cube because the number series does not start at 1
and numbers are not consecutive.
All rows and columns of the orthogonal planes sum correctly as do all
pandiagonals in one direction.
However, only 3 of the 4 main triagonals sum correctly, so this is only a semi
(?) magic cube.
One orthogonal plane and 2 oblique planes are simple magic (some of the inlays
have 2 orthogonal planes magic).
Top II III 343 811 284 252 375 274 367 340 808 276 805 298 266 364 332 354 347 820 288 256 285 253 371 344 812 336 809 277 275 368 265 358 326 824 292 816 289 257 355 348 372 345 813 281 254 296 269 362 335 803 327 825 293 261 359 258 351 349 817 290 807 280 273 366 339 363 331 804 297 270 294 262 360 328 821 IV V 356 329 822 295 263 287 260 353 346 819 267 365 333 801 299 823 291 264 357 330 278 271 369 337 810 334 802 300 268 361 814 282 255 373 341 370 338 806 279 272 350 818 286 259 352 251 374 342 815 283
This cube is unpublished.
Howard Order 11 Perfect Magic Cube 1976 (2002) not associated
This cube was constructed by myself using instructions published in Ian
Howard’s Letters to the Editor, JRM [1]. Howard mentions that order 11 is the
smallest possible for a prime order perfect magic cube.
(we know that perfect cubes can exist for non-prime orders as small as 8).
Rev. A. H. Frost mentioned the same bottom limit in 1878 [2]. However, he
publishes an order 9 perfect cube that does not use consecutive numbers.
Because this is a perfect cube, there are 9m pandiagonal magic squares
in this cube. That is 33 orthogonal, 6 oblique, and 10 broken squares parallel
to each oblique one. There are a total of m2 each of rows,
columns and pillars, 6m2 diagonals and 4m2
triagonals for a total of 1573 lines that sum to the constant 7326.
Top II 120 231 331 442 553 664 775 886 997 1108 1219 520 631 742 853 1085 1196 1307 87 198 298 409 261 372 604 715 815 926 1037 1148 1259 39 150 782 893 1004 1115 1226 6 238 349 460 571 682 523 634 745 856 1088 1199 1299 79 190 301 412 1055 1166 1266 46 157 268 379 490 722 833 944 785 896 1007 1118 1229 9 241 352 452 563 674 1317 97 208 319 419 530 641 752 863 974 1206 1047 1158 1269 49 160 271 382 493 725 836 936 127 359 470 581 692 803 903 1014 1125 1236 16 1320 89 200 311 422 533 644 755 866 977 1209 389 500 611 843 954 1065 1176 1287 56 167 278 130 362 473 573 684 795 906 1017 1128 1239 19 651 762 873 984 1095 1327 107 218 329 440 540 392 503 614 846 957 1057 1168 1279 59 170 281 924 1024 1135 1246 26 137 248 480 591 702 813 654 765 876 987 1098 1330 110 210 321 432 543 1186 1297 77 177 288 399 510 621 732 964 1075 916 1027 1138 1249 29 140 251 483 594 694 805 117 228 339 450 561 661 772 883 994 1105 1216 1178 1289 69 180 291 402 513 624 735 967 1078 258 369 601 712 823 934 1045 1145 1256 36 147 III IV 1052 1163 1274 54 165 265 376 487 719 830 941 132 353 464 575 686 797 908 1019 1130 1241 21 1314 94 205 316 427 538 649 749 860 971 1203 394 505 616 837 948 1059 1170 1281 61 172 283 124 356 467 578 689 800 911 1022 1133 1233 13 656 767 878 989 1100 1321 101 212 323 434 545 386 497 608 840 951 1062 1173 1284 64 175 286 918 1029 1140 1251 31 142 253 474 585 696 807 659 770 870 981 1092 1324 104 215 326 437 548 1180 1291 71 182 293 404 515 626 737 958 1069 921 1032 1143 1254 23 134 245 477 588 699 810 111 222 333 444 555 666 777 888 999 1110 1221 1183 1294 74 185 296 407 507 618 729 961 1072 263 374 595 706 817 928 1039 1150 1261 41 152 114 225 336 447 558 669 780 891 991 1102 1213 525 636 747 858 1079 1190 1301 81 192 303 414 255 366 598 709 820 931 1042 1153 1264 44 144 787 898 1009 1120 1231 11 232 343 454 565 676 528 628 739 850 1082 1193 1304 84 195 306 417 1049 1160 1271 51 162 273 384 495 716 827 938 790 901 1012 1112 1223 3 235 346 457 568 679 1311 91 202 313 424 535 646 757 868 979 1200 V VI 653 764 875 986 1097 1329 109 220 320 431 542 1185 1296 76 187 287 398 509 620 731 963 1074 915 1026 1137 1248 28 139 250 482 593 704 804 116 227 338 449 560 671 771 882 993 1104 1215 1188 1288 68 179 290 401 512 623 734 966 1077 257 368 600 711 822 933 1044 1155 1255 35 146 119 230 341 441 552 663 774 885 996 1107 1218 519 630 741 852 1084 1195 1306 86 197 308 408 260 371 603 714 825 925 1036 1147 1258 38 149 792 892 1003 1114 1225 5 237 348 459 570 681 522 633 744 855 1087 1198 1309 78 189 300 411 1054 1165 1276 45 156 267 378 489 721 832 943 784 895 1006 1117 1228 8 240 351 462 562 673 1316 96 207 318 429 529 640 751 862 973 1205 1046 1157 1268 48 159 270 381 492 724 835 946 126 358 469 580 691 802 913 1013 1124 1235 15 1319 99 199 310 421 532 643 754 865 976 1208 388 499 610 842 953 1064 1175 1286 66 166 277 129 361 472 583 683 794 905 1016 1127 1238 18 650 761 872 983 1094 1326 106 217 328 439 550 391 502 613 845 956 1067 1167 1278 58 169 280 923 1034 1134 1245 25 136 247 479 590 701 812 VII VIII 254 365 597 708 819 930 1041 1152 1263 43 154 786 897 1008 1119 1230 10 242 342 453 564 675 527 638 738 849 1081 1192 1303 83 194 305 416 1048 1159 1270 50 161 272 383 494 726 826 937 789 900 1011 1122 1222 2 234 345 456 567 678 1310 90 201 312 423 534 645 756 867 978 1210 1051 1162 1273 53 164 275 375 486 718 829 940 131 363 463 574 685 796 907 1018 1129 1240 20 1313 93 204 315 426 537 648 759 859 970 1202 393 504 615 847 947 1058 1169 1280 60 171 282 123 355 466 577 688 799 910 1021 1132 1243 12 655 766 877 988 1099 1331 100 211 322 433 544 396 496 607 839 950 1061 1172 1283 63 174 285 917 1028 1139 1250 30 141 252 484 584 695 806 658 769 880 980 1091 1323 103 214 325 436 547 1179 1290 70 181 292 403 514 625 736 968 1068 920 1031 1142 1253 33 133 244 476 587 698 809 121 221 332 443 554 665 776 887 998 1109 1220 1182 1293 73 184 295 406 517 617 728 960 1071 262 373 605 705 816 927 1038 1149 1260 40 151 113 224 335 446 557 668 779 890 1001 1101 1212 524 635 746 857 1089 1189 1300 80 191 302 413 IX X 1318 98 209 309 420 531 642 753 864 975 1207 387 498 609 841 952 1063 1174 1285 65 176 276 128 360 471 582 693 793 904 1015 1126 1237 17 660 760 871 982 1093 1325 105 216 327 438 549 390 501 612 844 955 1066 1177 1277 57 168 279 922 1033 1144 1244 24 135 246 478 589 700 811 652 763 874 985 1096 1328 108 219 330 430 541 1184 1295 75 186 297 397 508 619 730 962 1073 914 1025 1136 1247 27 138 249 481 592 703 814 115 226 337 448 559 670 781 881 992 1103 1214 1187 1298 67 178 289 400 511 622 733 965 1076 256 367 599 710 821 932 1043 1154 1265 34 145 118 229 340 451 551 662 773 884 995 1106 1217 518 629 740 851 1083 1194 1305 85 196 307 418 259 370 602 713 824 935 1035 1146 1257 37 148 791 902 1002 1113 1224 4 236 347 458 569 680 521 632 743 854 1086 1197 1308 88 188 299 410 1053 1164 1275 55 155 266 377 488 720 831 942 783 894 1005 1116 1227 7 239 350 461 572 672 1315 95 206 317 428 539 639 750 861 972 1204 1056 1156 1267 47 158 269 380 491 723 834 945 125 357 468 579 690 801 912 1023 1123 1234 14 XI -Bottom 919 1030 1141 1252 32 143 243 475 586 697 808 1181 1292 72 183 294 405 516 627 727 959 1070 112 223 334 445 556 667 778 889 1000 1111 1211 264 364 596 707 818 929 1040 1151 1262 42 153 526 637 748 848 1080 1191 1302 82 193 304 415 788 899 1010 1121 1232 1 233 344 455 566 677 1050 1161 1272 52 163 274 385 485 717 828 939 1312 92 203 314 425 536 647 758 869 969 1201 122 354 465 576 687 798 909 1020 1131 1242 22 395 506 606 838 949 1060 1171 1282 62 173 284 657 768 879 990 1090 1322 102 213 324 435 546
[1] Constructed by H. Heinz
from instructions in Ian P. Howard, Letters to the Editor, JRM,9:4, 1976-77,
pp276-278
[2] A. H. Frost, On the General Properties of Nasik Cubes, QJM 15, 1878,
pp 93-123 plus plates 1 and 2.
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Harvey Heinz harveyheinz@shaw.ca
This page last updated
October 15, 2009
Copyright © 2003 by Harvey D. Heinz