Player 1 and player 2 play a game which consists of a rectangular grid with 3 rows and 20 columns.

During each players turn they can colour a square in the grid (either a 1 x 1, 2 x 2 or a 3 x 3 square) a coloured square can not be recoloured. The person who no longer is able to colour a square loses.

The question that follows is that if player 1 makes the first move, which of the players has a winning strategy. In other words, which of the players has a strategy that will have them win 100% of the times.

What I have noticed and which seems quite obvious is that the player who places the last 2x2 square wins, but i don't know what to do with that information and where to take it from there.

  • 1
    $\begingroup$ Welcome to stackexchange. Please edit the question to show us what you have tried and where you are stuck. You might try to analyze the game when there are fewer columns - say $3$, $4$ and $5$ - to get an idea about what might be going on. $\endgroup$ – Ethan Bolker Feb 3 '19 at 1:36

The first player can win by coloring a $2\times 2$ square that is centered horizontally. That leaves two single square plus two $3 \times 9$ rectangles. The first player can then mirror the second player’s moves and win.

Mirror strategies are often available for games like this. If the rectangle is odd by odd the first player can color the center square. If the rectangle is odd by even with the odd smaller the first player can color a square like this, one less than the width of the rectangle. One less is even so there will be an even number of squares in the border. I don't know about odd by even where the even is smaller.

  • $\begingroup$ Motor = mirror? $\endgroup$ – Andrés E. Caicedo Feb 3 '19 at 3:37
  • $\begingroup$ @AndrésE.Caicedo : Yes. Autocorrect $\endgroup$ – Ross Millikan Feb 3 '19 at 4:05

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