On a conjecture of Butler and Graham

Abstract

Motivated by a hat guessing problem proposed by Iwasawa [6], Butler and Graham [2] made the following conjecture on the existence of certain way of marking the coordinate lines in [k] n : there exists a way to mark one point on each coordinate line in [k] n , so that every point in [k] n is marked exactly a or b times as long as the parameters (a, b, n, k) satisfies that there are non-negative integers s and t such that s + t = k n and as + bt = nk n−1 . In this paper we prove this conjecture for any prime number k. Moreover, we prove the conjecture for the case when a = 0 for general k.