Problem #3

October 21, 2004 was Martin Gardner's 90th birthday. Gardner was the author of the "Mathematical Games" column in Scientific American from 1956 to 1986. Through his columns and books he popularized both recreational and "serious" mathematics and inspired a generations of young people to pursue mathematical careers. A partial list of his mathematical books is here.

In tribute to Martin Gardner, this month's problem is a variant of one from his book "Wheels, Life, and Other Mathematical Amusements".

Using only π's, the four arithmetic operations, +, −, ×, /, the floor function (perhaps repeatedly), and parentheses, write the integers from 1 to 20 inclusive using as few π's as possible. [Recall that floor(x) is the greatest integer less than or equal to x. For example floor(3) = 3 and floor(2.3) = 2.]

For instance

2 = floor((floor(π) × π × π − π) / (floor(π) × floor(π) + π))

although this is definitely not optimal.

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