IrMO 2018 Problem 10

The game of Greed starts with an initial configuration of one or more piles of stones. Player $1$ and Player $2$ take turns to remove stones, beginning with Player $1$ . At each turn, a player has two choices: • take one stone from any one of the piles (a simple move); • take one stone from each of the remaining piles (a greedy move). The player who takes the last stone wins. Consider the following two initial configurations: (a) There are $2018$ piles, with either $20$ or $18$ stones in each pile. (b) There are four piles, with $17, 18, 19$ , and $20$ stones, respectively. In each case, find an appropriate strategy that guarantees victory to one of the players.