MathNet 2024-part-02 — 168 problems (EN–VI) Question 148

Let $N$ be a positive integer. Geoff and Ceri play a game in which they start by writing the numbers $1, 2, \ldots, N$ on a board. They then take turns to make a move, starting with Geoff. Each move consists of choosing a pair of integers $(k, n)$, where $k \geqslant 0$ and $n$ is one of the integers on the board, and then erasing every integer $s$ on the board such that $2^{k} \mid n-s$. The game continues until the board is empty. The player who erases the last integer on the board loses.

Determine all values of $N$ for which Geoff can ensure that he wins, no matter how Ceri plays.

(Indonesia)