## [ttx[TT TT

These improvements reduce the total number of pi's to 50. John W. Gosling was the first of many readers to achieve 50, but it is only fair to add that the problem did not specifically allow exponention and that many who wrote earlier than Gosling would probably have achieved 50 had they used exponents for integers 7 and 20. (Without exponents, 7 requires three pi's and 20 requires four.) Numerous readers lowered the number of pi's below 50 by adding other symbols, such as the factorial sign or the "unary negative operator," which has the effect of rounding up instead of down. Bernard Wilde and Carl Thune, Mark T. Longley-Cook, V. E. Hoggatt, Jr., Robert L. Caswell and others conjectured that by using nested radical signs to reduce a divisor, any positive integer can be expressed with three pi's.

Cheney and John Leech each pointed out that if - [ - tt] is interpreted in a standard way to mean 4, then further reductions are possible:

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