Problem

Source: ELMO 2023/6

Tags: Elmo, combinatorics



For a set \(S\) of positive integers and a positive integer \(n\), consider the game of \((n,S)\)-nim, which is as follows. A pile starts with \(n\) watermelons. Two players, Deric and Erek, alternate turns eating watermelons from the pile, with Deric going first. On any turn, the number of watermelons eaten must be an element of \(S\). The last player to move wins. Let \(f(S)\) denote the set of positive integers \(n\) for which Deric has a winning strategy in \((n,S)\)-nim. Let \(T\) be a set of positive integers. Must the sequence \[T, \; f(T), \; f(f(T)), \;\ldots\]be eventually constant? Proposed by Brandon Wang and Edward Wan