Question 167
Johan and Quintijn play the following game.
Before the start of the game, the integers $1$, $2$, $\ldots$, $2024$ are written on a board. The players then each take turns, starting with Johan. On their turn a player must wipe out two integers $a$ and $b$ from the board and write their (possibly negative) difference $a - b$ on the board. The game ends when only one integer is left on the board. If this integer is divisible by $3$, Johan wins, otherwise Quintijn wins.
Determine which of the two players has a winning strategy.
Before the start of the game, the integers $1$, $2$, $\ldots$, $2024$ are written on a board. The players then each take turns, starting with Johan. On their turn a player must wipe out two integers $a$ and $b$ from the board and write their (possibly negative) difference $a - b$ on the board. The game ends when only one integer is left on the board. If this integer is divisible by $3$, Johan wins, otherwise Quintijn wins.
Determine which of the two players has a winning strategy.