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 = nkn-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.
All Science Journal Classification (ASJC) codes
- Computer Science Applications
- Applied Mathematics
- Characteristic function
- Hat guessing games
- Marking coordinate lines