[math-fun] Unlucky thirteen

#1. Puzzle: Show that for every positive integer n less than thirteen, there exists a Laurent polynomial p(x) with coefficients in {0,1,2,3,4} satisfying p(1/2) = p(1/3) = n. (Hint: Use an iterative construction n -> n+1.) #2. Problem: Show that this NOT the case for thirteen or any larger positive integer. (I don't know how to prove this.) If #1 seems tricky, here's a simpler version: Show that for every positive integer n there exists a Laurent polynomial p(x) with coefficients in {0,1,2} satisfying p(1) = p(2) = n. Equivalently, show that every positive integer n can be written as the sum of n powers of 2, each of which may appear at most twice (where "power of 2" means "2^k for some k in Z"). Jim Propp