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Problem 261

AMC 12 late, AIME early

Combinatorics Difficulty 4.0 Find the answer BMO Round 1 · United Kingdom · 2023

Alex and Katy play a game on an $8 \times 8$ square grid made of $64$ unit cells. They take it in turns to play, with Alex going first. On Alex's turn, he writes 'A' in an empty cell. On Katy's turn, she writes 'K' in two empty cells that share an edge. The game ends when one player cannot move. Katy's score is the number of Ks on the grid at the end of the game. What is the highest score Katy can be sure to get if she plays well, no matter what Alex does?

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