The noted computer scientist Donald Knuth has conjectured that every positive integer can be obtained by beginning with a single 3 and applying some combination of the factorial, square root, and floor functions.

Recall that the factorial function is n! = n*(n-1)*(n-2)*...3*2*1, and for this problem, we insist that n be an integer. floor(x) gives the greatest integer less than or equal to x. For example, we can write 

floor(sqrt((3!)!))
   = floor(sqrt(6!))
   = floor(sqrt(720))
   = floor(26.83...)
   = 26

[download]
to obtain a Knuth expression for 26.

One could spend a pleasant afternoon manually creating Knuth expressions for small integers with a hand calculator. But it is much more fun to write a perl program to do this for you. The challenge is to write an elegant, fast, or golfish program that finds Knuth expressions for the integers 1 to 10 inclusive.

I'll post my solution in a day or so.

-Mark

In reply to Challenge: Chasing Knuth's Conjecture by kvale

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