# Placing dominoes on a 3x100 checkerboard

Alice and Bob play a game by alternately placing dominoes on a $3\times100$ checkerboard (3 rows, 100 columns). Alice has the first move.

• A move of Alice consists in placing a $1\times2$ (horizontal) white dominoe.
• A move of Bob consists in placing two $2\times1$ (vertical) black dominoes.

The dominoes must be placed without overlaps, and they must always cover exactly two cells of the checkerboard. The loser is the player who cannot make a complete move.

Determine the winner of the game. (As usual, we assume that Alice and Bob use optimal strategies.)

• Are the rules of standard dominoes in play as well? Feb 8, 2015 at 9:10

First, I claim that Alice will be able to play at lease 17 dominos in the middle row. To see why, note that there are initially 99 places a domino can go in the middle row. Call such a place blocked if one or both of its squares is occupied by another domino. Each domino Alice or Bob places blocks at most 2 previously unblocked places (so that Bob can block at most 4 places on each of his turns). 16 turns by Alice and 16 turns by Bob lead to at most $16\cdot 2+16\cdot 4 =96$ of the 99 places being blocked, so Alice can still find a place to play in the middle row on her 17th turn.