An Extension of the Perplexing Polynomial Puzzle

Abstract

Summary Originally published as a puzzle in 2005 [3], the Perplexing Polynomial Puzzle indeed is perplexing: any given polynomial p ( x ) with nonnegative integer coefficients can be completely determined by just two evaluations. In this article, an extension is made to polynomials with arbitrary integer coefficients, by considering a simple translation x ↦ x + k with k ∈ N such that the result is a new polynomial with only nonnegative integer coefficients on which the original solution can be used. A proof is given that this is indeed always possible, and a method is constructed to determine a suitable k to do so.