Magic Cube Update3

Material about magic cubes continues to appear. This update contains material I have received during the last nine months of 2004.
An Order 16 Bordered Magic Cube On July 16, 2004, I received the following email from Mitsutoshi Nakamura [1]: I have made an order16 'bordered' diagonal magic cube (Bordered16.xls). It consists of numbers 1 to 4096. S = 32776. This cube is a diagonal magic cube and contains within it the following subcubes that consist of consecutive integers: an order4 simple magic cube (from 2017 to 2080), S = 8194 an order6 diagonal magic cube (from 1941 to 2156), S = 12291 an order8 diagonal magic cube (from 1793 to 2304), S = 16388 an order10 diagonal magic cube (from 1549 to 2548), S = 20485 an order12 diagonal magic cube (from 1185 to 2912), S = 24582 an order14 diagonal magic cube (from 677 to 3420). S = 28679 The cube also contains a lot of order4 magic squares. I have confirmed we can construct such a cube for any even order greater than 4. With best regards, Mitsutoshi Nakamura I have analyzed these cubes using my standard test
spreadsheets. This is a bordered (or concentric) magic cube
because the middle numbers of the series are all in the center cube. The
lowest and highest numbers are in the outside shell! Good work Mitsutoshi! [1]
Mitsutoshi
Nakamura's website is
http://homepage2.nifty.com/googol/magcube/en/
Proper – Refers to a cube that contains
exactly the minimum requirements for that class of cube. i.e. a proper
simple or pantriagonal magic cube would contain no magic squares, a proper
diagonal magic cube would contain exactly 3m plus 6 simple magic squares,
etc. smagic cube – A magic cube in which the six
surfaces are magic squares. All diagonal, pandiagonal and perfect cubes
are smagic, but cubes exist in which the interior planes are not magic
squares. See the Worthington cube of 1910. Pan2,3agonal – used by Nakamura on his site
to designate what I refer to as Perfect. i.e. a combination Pantriagonal
and Pandiagonal magic cube. That way he completely avoids the confusion
over the term perfect. (Historically, the term perfect was
used by different authors to define cubes with a variety of
characteristics.) PantriagDiag – A magic cube that is a
combination Pantriagonal and Diagonal cube. All main and broken triagonals
must sum correctly. 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 one constructed so far (that I am aware of) is by Mitsutoshi Nakamura (see below). It is as order 8 and not associated. An enhanced feature of this particular cube is that
all 2x2 and corners of 5x5 cubes sum to S. This particular cube is also
complete (every pantriagonal contains m/2 complement pairs, spaced
m/2 apart). My Summary Page table of First Cube of Each Order for Each Class is now outofdate. It should have another column for the PantriagDiag cubes. However, at present there is only one such cube known (for order 8) so I will leave the table as is. The First Order 8 Pantriagonal Diagonal Magic Cube I II 1 352 300 117 422 251 143 466 492 181 193 416 79 274 358 59 254 419 471 138 345 8 116 301 279 74 62 355 180 493 409 200 349 4 120 297 250 423 467 142 184 489 413 196 275 78 58 359 418 255 139 470 5 348 304 113 75 278 354 63 496 177 197 412 303 114 6 347 140 469 417 256 198 411 495 178 353 64 76 277 468 141 249 424 119 298 350 3 57 360 276 77 414 195 183 490 115 302 346 7 472 137 253 420 410 199 179 494 61 356 280 73 144 465 421 252 299 118 2 351 357 60 80 273 194 415 491 182 III IV 230 443 463 146 321 32 108 309 271 82 38 379 172 501 385 224 25 328 308 109 446 227 151 458 500 173 217 392 87 266 382 35 442 231 147 462 29 324 312 105 83 270 378 39 504 169 221 388 325 28 112 305 226 447 459 150 176 497 389 220 267 86 34 383 460 149 225 448 111 306 326 27 33 384 268 85 390 219 175 498 311 106 30 323 148 461 441 232 222 387 503 170 377 40 84 269 152 457 445 228 307 110 26 327 381 36 88 265 218 391 499 174 107 310 322 31 464 145 229 444 386 223 171 502 37 380 272 81 V VI 373 44 96 257 210 399 507 166 160 449 437 236 315 102 18 335 394 215 163 510 45 372 264 89 99 318 330 23 456 153 237 436 41 376 260 93 398 211 167 506 452 157 233 440 103 314 334 19 214 395 511 162 369 48 92 261 319 98 22 331 156 453 433 240 91 262 370 47 512 161 213 396 434 239 155 454 21 332 320 97 168 505 397 212 259 94 42 375 333 20 104 313 234 439 451 158 263 90 46 371 164 509 393 216 238 435 455 154 329 24 100 317 508 165 209 400 95 258 374 43 17 336 316 101 438 235 159 450 VII VIII 402 207 187 486 53 364 288 65 123 294 338 15 480 129 245 428 365 52 72 281 202 407 483 190 136 473 429 244 291 126 10 343 206 403 487 186 361 56 68 285 295 122 14 339 132 477 425 248 49 368 284 69 406 203 191 482 476 133 241 432 127 290 342 11 192 481 405 204 283 70 50 367 341 12 128 289 242 431 475 134 67 286 362 55 488 185 205 404 426 247 131 478 13 340 296 121 484 189 201 408 71 282 366 51 9 344 292 125 430 243 135 474 287 66 54 363 188 485 401 208 246 427 479 130 337 16 124 293 Other News and Acknowledgements Other news Acknowledgements Mitsutoshi Nakamura has been mentioned above. He supplied cubes for the last 7 vacant cells of the table (the last one in conjunction with Abhinav Soni) His site includes cubes definitions, algorithms, theorems, etc., and is at http://homepage2.nifty.com/googol/magcube/en/ Abhinav Soni supplied an amazing 16 magic cubes for
the above table.
