On a conjecture of Butler and Graham

Authors:
Tengyu Ma;Xiaoming Sun;Huacheng Yu
Affiliations:
Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China;Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China;Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China
Venue:
Designs, Codes and Cryptography
Year:
2013

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Abstract

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驴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.