Download e-book for iPad: Magic Cubes: New Recreations by William H. Benson

By William H. Benson

The magic sq. specialists observe their wits to magic cubes. you wish no complex mathematical wisdom to build those third-dimensional brain bogglers; together with pandiagonal and ideal cubes — many totally new structures, too. Hours of pleasure for starting and complicated magic-cube buffs. 111 figures.

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Extra info for Magic Cubes: New Recreations

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But in each of these three combinations two of the characteristics add to 5. It follows that no possible selection of generating characteristics can generate a set of controlling characteristics that will make it possible to construct a perfect fifth-order magic cube. This proves Lemma 1. Chapter 5 The First Perfect Magic Cube Now let us turn to Dr. Barnard's article, which we mentioned in Chapter 1. The article is remarkable for its thoroughness, especially when you consider that it was published in 1888.

ToP ORTHOGONAL SECTION dBg hAa dHc hGe dFg hEa dDc hCe cDc gCe cBg gAa cHc gGe cFg gEa bFg fEa bDc fCe bBg fAa bHc fGe aHc eGe aFg eEa aDc eCe aBg eAa hBg dAa hHc dGe hFg dEa hDc dCe gDc cCe gBg cAa gHc cGe gFg cEa fFg bEa fDc bCe fBg bAa fHc bGe eHc aGe eFg aEa eDc aCe eBg aAa C=O FIGURE 6-14. LEFT SIDE ORTHOGONAL SECTION Figures 6-12 through 6-14 show that whenever the controlling characteristic is 1, each letter appears once and only once; when it is 2 (or 6), four letters appear twice each; when it is 4, two letters appear four times each.

Notice that, as predicted, the "hundreds," "tens," and "units" of all orthogonals consist of 0, 1, 2, 3, and 4. Hence the 414 320 231 323 234 140 232 141 143 142 23 434 3 1 412 432 343 204 4 410 321 341 202 2 413 324 230 200 111 144 114 0 411 322 233 43 404 41 402 400 311 222 314 220 131 223 134 133 130 21 24 430 22 433 344 342 203 241 102 13 424 330 100 11 422 14 420 331 44 423 42 403 40 401 110 112 D= I 310 221 313 224 113 20 431 Front D = 0 132 340 201 312 334 240 332 243 104 333 244 242 103 101 12 10 421 D=3 D=2 300 211 214 120 122 33 444 31 442 303 I 123 34 440 301 212 I 32 443 304 210 121 441 302 213 124 30 Back D= 4 FIGURE 4-2.

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