By Martin Gardner
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Additional info for Sphere Packing, Lewis Carroll, and Reversi: Martin Gardner's New Mathematical Diversions
Sample text
If this cannot be done, give a proof of impossibility. If it can be done, what is the smallest number of acute triangles into which any obtuse triangle can be dissected? Figure 10 shows a typical attempt that leads nowhere. The triangle has been divided into three acute triangles, but the fourth is obtuse, so nothing has been gained by the preceding cuts. The problem (which came to me by way of Mel Stover of Winnipeg) is amusing because even the best mathematician is likely to be led astray by it and come to a wrong conclusion.
We can record successive permutations of strands by means of the network diagram, but it will not show how the strands pass over and under one another. If we take into account this complicating topological factor, is it still possible to call on group theory for a description of what we are doing? The answer is yes, and Emil Artin, a distinguished German mathematician who died in 1962, was the first to prove it. In his elegant theory of braids, the elements of the group are “weaving patterns” (infinite in number), and the operation consists, 16 Sphere Packing, Lewis Carroll, and Reversi A A′ Figure 7.
Percival, and Richard D. Thurston independently found: GRASS GRAYS GREYS GREES GREEN 46 Sphere Packing, Lewis Carroll, and Reversi Ten readers (David M. Bancroft, Robert Bauman, Frederick J. Hooven, Arthur H. Lord, Mrs. Henry A. Morss, Sidney J. Osborn, Dodi Schultz, George Starbuck, Edward Wellen, and a reader whose signature was illegible) sent the following excellent five-step change of APE to MAN: APE APT OPT OAT MAT MAN Many readers found seven-step changes of ONE to TWO, but because all contained at least one uncommon word, I award the palm to H.