By Martin Gardner

Aha! perception demanding situations the reader's reasoning strength and instinct whereas encouraging the advance of 'aha! reactions'.

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Deutsch and Kenneth C. , in Machine Intelligence 7. A somewhat similar program was developed in 1972 by Ejvind Lynning, a Danish student working with Jacques Cohen, a physicist at Brandeis University. Snug tangrams are, for obvious reasons, usually more difficult to solve (by a person or a computer) than nonsnug figures, and the difficulty tends to increase as the number of sides decreases. One might suppose a pattern with only one solution would be harder to solve than one with many, but that is not the case.

Unfortunately, this too can produce irrational decisions. The matrix in Figure 26 (left) displays the notorious voting paradox in its simplest form. The top row shows that a third of the voters prefer candidates A, B, and C in the order ABC. The middle row shows that another third rank them BCA, and the bottom row shows that the remaining third rank them CAB. Examine the matrix carefully NONTRANSITIVE PARADOXES 1 RANK ORDER 2 57 3 Figure 26 T h e voting paradox (left) and the tournament paradox based on the magic square (right) and you will find that when candidates are ranked in pairs, nontransitivity rears its head.

As J. A. Lindon has put it elegantly in one of his unpublished mathematical Clerihews: + + + + + + + T o equations simultaneously Pellian My approach is Machiavellian. Anything goes, rather than resort to such actions As covering the walls with continued fractions. I wish to thank Meally for providing the nonrecursive formulas for square stars and triangular stars, as well as other things in this chapter, and also to thank John Harris and John McKay for additional assistance. I also found A Handbook ofInteger Sequences by N.