On a conjecture of Butler and Graham

In this paper we prove a conjecture of Bulter and Graham [2] on the existence of a certain way of marking the lines in [k] for any prime k. The conjecture states that there exists a way of marking each line of [k] one point so that every point in [k] is marked exactly a or b times as long as the parameters (a, b, n, k) satisfy the condition that there are integers s, t such that s + t = k and as + bt = nk. Moreover, we prove the conjecture for the case when a = 0 for general k. Bulter and Graham used a inflated markup method in [2]. However, the construction of the base cases of the induction is far more complicated. Here we introduce a characteristic function which classifies the points of [k] and has some desired symmetrical property. By a sophisticated using of this property, we can achieve that each point is marked either a or b times. There is a natural connection between this marking line problem and the hat guessing game problem. Our work implies the answer to a hat guessing question proposed by Iwasawa [6] for the case when the total number of hat colors is a prime, more precisely, the necessary and sufficient condition of the existence of a guessing strategy which guarantees either a or b correct guesses under the model that each player is assigned hat uniformly and independently.