Emptying Sets: The Cookie Monster Problem

Given a set of integers $S = \{k_1,\ k_2,...,\ k_n\}$, the Cookie Monster Problem is the problem of making all elements of the set equal 0 in the minimum number of moves. Consider the analogy of cookie jars with distinct numbers of cookies, such that $k_i$ is the number of cookies in the $i$th jar. The "Cookie Monster" wants to eat all the cookies, but at each move he must choose some subset of the jars and eat the same amount from each jar. The \emph{Cookie Monster Number of $S$}, $CM(S)$, is the minimum number of such moves necessary to empty the jars. It has been shown previously that $\lceil \log_2(|S|+1) \rceil \leq CM(S) \leq |S|$. In this paper we classify sets by determining what conditions are necessary for $CM(S)$ to equal 2 or 3 and what effect certain restrictions have on $CM(S)$. We also provide an alternative interpretation of the problem in the form of a combinatorial game and analyze the losing positions.