A note on marking lines in [k]ⁿ

Abstract

In ${[k]^n=[k]{\times}[k]{\times}\cdots{\times}[k]}$, a coordinate line consists of the collection of points where all but one coordinate is fixed and the unfixed coordinate varies over all possibilities. We consider the problem of marking (or designating) one point on each line in [k]ⁿ so that each point in [k]ⁿ is marked either a or b times, for some fixed a or b. This is equivalent to forming a strategy for a hat guessing game for n players with k different colors of hats where the number of correct guesses, regardless of hats placed, is either a or b. If we let s ≥ 0 and t ≥ 0 denote the number of vertices marked a and b times respectively, then we have the following obvious necessary conditions: s + t = k ⁿ (the number of points) and as + bt = nk ^n–1 (the number of lines). Our main result is to show for n ≤ 5, and k arbitrary, that these necessary conditions are also sufficient.