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By David Wells

A better half to a similar author's "Dictionary of Curious and fascinating Numbers" and "Dictionary of Curious and fascinating Geometry", this publication covers mathematical and logical puzzles from the traditional Greeks to the current day

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5 0 5 0 0 0 5 0 oz. ... ... ... Second Example† . The next of these is a not uncommon game, played by two people, say A and B. A begins by mentioning some number not greater than (say) six, B may add to that any number not greater than six, A may add to that again any number not greater than six, and so on. He wins who is the first to reach (say) 50. Obviously, if A calls 43, then whatever B adds to that, A can win next time. Similarly, if A calls 36, B cannot prevent A’s calling 43 the next time.

143. 1. ‘even’ inserted as per errata sheet 32 ARITHMETICAL RECREATIONS. [CH. I It may be shown that 2m + 1 is composite if m is not a power of 2, but of course it does not follow that 2m + 1 is a prime if m is a power of 2. As a matter of fact the theorem is not true. In 1732 Euler* showed that if n = 5 the formula gives 4294, 967297, which is equal to 641 × 6, 700417: curiously enough, these factors can be deduced at once from Fermat’s remark on the possible factors of numbers of the form 2m ± 1, from which it may be shown that the prime factors (if any) of 232 + 1 must be primes of the form 64n + 1.

Second Example† . The next of these is a not uncommon game, played by two people, say A and B. A begins by mentioning some number not greater than (say) six, B may add to that any number not greater than six, A may add to that again any number not greater than six, and so on. He wins who is the first to reach (say) 50. Obviously, if A calls 43, then whatever B adds to that, A can win next time. Similarly, if A calls 36, B cannot prevent A’s calling 43 the next time. In this way it is clear that the key numbers are those forming the arithmetical progression 43, 36, 29, 22, 15, 8, 1; and whoever plays first ought to win.

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