Pied Numbers

The old problem of expressing integers with four 4's (discussed in a Scientific American column reprinted as Chapter 5 of The Incredible Dr. Matrix) has been given many variations. In an intriguing new variant proposed by Fitch Cheney one is allowed to use only pi and symbols for addition, subtraction, multiplication, division, square root and the "round-down function." In the last operation, indicated by brackets, one takes the greatest integer that is equal to or less than the value enclosed by the brackets. Parentheses also may be used, as in algebra, but no other symbols are allowed. Each symbol and pi may be repeated as often as necessary, but the desideratum is to use as few pi symbols as possible. For example, 1 can be written [ViT] and 3 even more simply as [it].

The reader is invited to do his best to express the integers from 1 through 20 according to these rules, and to compare them with the best Cheney was able to achieve.

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