MathNet 2025-part-02 — 186 problems (EN–VI) Question 72

Let $N$ be a positive integer. Ali and Hadi play a game in which they start by writing the numbers $1, 2, \dots, N$ on a board. They then take turns to make a move, starting with Ali. Each move consists of choosing a pair of integers $(k, n)$, where $k \ge 0$ and $n$ is one of the integers on the board, and then erasing every integer $s$ on the board such that $2^k|n - s$. 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 Ali can ensure that he wins, no matter how Hadi plays.