Problem #3

October 21, 2004 was Martin Gardner's 90^th 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.