On a conjecture of Butler and Graham
Motivated by a hat guessing problem proposed by Iwasawa, Butler and Graham made the following conjecture on the existence of a 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 nonnegative 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.
hat guessing games marking coordinate lines characteristic function