F. A. P. Barnard's Magic Cubes

His paper considers magic squares (some quite unusual), circles and cyclovolutes, as well as magic cubes. The cubes he shows are an order 4, which is not magic by our present standards, and 3 perfect magic cubes (new definition), an order 8 and two order 11. As far as I can determine, the first perfect magic
cube was constructed by Frost. However, it was not normal. It used
nonconsecutive numbers from 1 to 889. the next one to be published (after
Barnard's) was by Ian Howard in a JRM paper
[3] in 1976 (he gave instructions on building a normal order 11). Other early publication of cubes (that I am aware
of) that were magic by present standards (rows, columns, pillars, and
triagonals all correct) was an order 3 by Hugel in 1876
[5], and orders 4 and 5 cubes
published in 1899 by Hermann Schubert [6]. As this page is dedicated to Dr. Barnard, I will show several unusual squares and other magic objects, as well as two of his perfect cubes. [1] A. H. Frost, On the General Properties of Nasik
Cubes, Quarterly Journal of Mathematics 15, 1878 pp 110116 Contents
Inlaid Squares
Reassemble the magic square on the left, without the colored cells, and you get the magic square on the right. Here we have an order 14 magic square that reduces
to an order 10. Then reduces to orders 8, 6, and finally order 4.
Cylinder & Sphere
Order8 Perfect cubeBarnard's paper contained 3 magic cubes of the type
we now call perfect. An order 8 which is not associated, and two order 11
cubes which are associated. (Order 9 is the smallest perfect cube that can
be associated.) In
order to verify the properties ascribed to this cube we select for
addition the terms which, in this arrangement, are brought in any
direction into line. Observing that the value of S must in general be
equal to the sum of an arithmetical series of which the first term is 1,
the last term n3, and the number of terms n, we have
S=1/2(n3+l)=1/2(n4+n) (68) And so of others. It is impossible to exhibit magic cubes to the eye (except those of small numbers, (which are necessarily imperfect) otherwise than by presenting, as here, their component squares separately. In Fig. 50 is shown the cube of 4 arranged in solid form. It is magical, except in the rows parallel to z, and in the diagonals of the faces xz, yz, and those of the solid. In this figure every cubic tessera of eight terms, however taken, will be found to give the same sum; and the cubes of the higher powers of 2 may be made to possess the same property. It will be found to be true of the cube of 8, Fig. 48, and this cube possesses other still more remarkable properties. The eight numbers, for example, which mark the solid angles of any cube less than the cube of 8, which can be made within this magic cube, will give invariably the same sum, viz, 2052. And in any right parallelopipedon, whose terminal planes are squares of the even numbers 2 or 4, and in whose lateral edges the number of terms is even, the sum of the numbers marking the solid angles will still be the same, 2052. If the terminal planes be squares of 6, the same will be true whatever the number of terms on the lateral edges; if the terminal planes be squares of the odd numbers 3, 5, or 7, and the lateral edges contain either three or seven terms, the proposition will be true of these parallelopipedons also. This enumeration does not exhaust all the peculiar properties of this remarkable cube. He also says Perfectly magic cubes may be formed on all orders from 8 upward, except the unevenly even. Horizontal plane 1  Top plane 2 1 490 59 468 8 495 62 469 251 276 200 303 254 277 193 298 144 359 182 349 137 354 179 348 438 93 393 98 435 92 400 103 465 2 491 60 472 7 494 61 299 252 280 199 302 253 273 194 352 143 358 181 345 138 355 180 102 437 89 394 99 436 96 399 57 466 3 492 64 471 6 493 195 300 256 279 198 301 249 274 184 351 142 357 177 346 139 356 398 101 433 90 395 100 440 95 489 58 467 4 496 63 470 5 275 196 304 255 278 197 297 250 360 183 350 141 353 178 347 140 94 397 97 434 91 396 104 439 Plane 3 Plane 4 328 175 382 149 321 170 379 148 126 405 65 426 123 404 72 431 9 482 51 476 16 487 54 477 243 284 208 295 246 285 201 290 152 327 174 381 145 322 171 380 430 125 401 66 427 124 408 71 473 10 483 52 480 15 486 53 291 244 288 207 294 245 281 202 384 151 326 173 377 146 323 172 70 429 121 402 67 428 128 407 49 474 11 484 56 479 14 485 203 292 248 287 206 293 241 282 176 383 150 325 169 378 147 324 406 69 425 122 403 68 432 127 481 50 475 12 488 55 478 13 283 204 296 247 286 205 289 242 Plane 5 Plane 6 449 42 507 20 456 47 510 21 315 212 264 239 318 213 257 234 336 167 374 157 329 162 371 156 118 413 73 418 115 412 80 423 17 450 43 508 24 455 46 509 235 316 216 263 238 317 209 258 160 335 166 373 153 330 163 372 422 117 409 74 419 116 416 79 505 18 451 44 512 23 454 45 259 236 320 215 262 237 313 210 376 159 334 165 369 154 331 164 78 421 113 410 75 420 120 415 41 506 19 452 48 511 22 453 211 260 240 319 214 261 233 314 168 375 158 333 161 370 155 332 414 77 417 114 411 76 424 119 Plane 7 Plane 8  Bottom 136 367 190 341 129 362 187 340 446 85 385 106 443 84 392 111 457 34 499 28 464 39 502 29 307 220 272 231 310 221 265 226 344 135 366 189 337 130 363 188 110 445 81 386 107 444 88 391 25 458 35 500 32 463 38 501 227 308 224 271 230 309 217 266 192 343 134 365 185 338 131 364 390 109 441 82 387 108 448 87 497 26 459 36 504 31 462 37 267 228 312 223 270 229 305 218 368 191 342 133 361 186 339 132 86 389 105 442 83 388 112 447 33 498 27 460 40 503 30 461 219 268 232 311 222 269 225 306All 24 planar squares are pandiagonal magic as are also the 6 oblique squares and the seven broken squares parallel to each of these[2][3] 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) subcubes also sum correctly to 2052. Also, many other shapes of parallelopipeds. A perfect magic cube is a combination pantriagonal
and pandiagonal magic cube. However, all 6 oblique must be pandiagonal
magic as well. This last condition is a natural consequence of “…all lower
dimension hypercubes are perfect”. [1] F. A. P. Barnard, Theory of
Magic Squares and Cubes, Nat. Academy of Sciences, Vol. IV, Sixth Memoir,
1888, pp 207270 Order11 Perfect cubeI decided not to put either of the two order 11 associated perfect cubes of Barnard on this page. For anyone interested in seeing them, Barnard11.doc is available for downloading. I do show other order 11 perfect cubes on this site at: Howard
1976 not associated
