Problem

Source: 45th International Tournament of Towns, Junior O-Level P5, Fall 2023

Tags: combinatorics



5. Alice and Bob have found 100 bricks of the same size, 50 white and 50 black. They came up with the following game. A tower will mean one or several bricks standing on top of one another. At the start of the game all bricks lie separately, so there are 100 towers. In a single turn a player must put one of the towers on top of another tower (no flipping towers allowed) so that the resulting tower has no same-colored bricks next to each other. The players make moves in turns, Alice starts first. The one unable to make the next move loses the game. Who can guarantee the win regardless of the opponent's strategy?