At David Butler's One Hundred Factorial last week, I played a nice space-claiming game.
I was inspired to make a computery version. I've called it "This island is big enough for the both of us"
Play at

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@christianp Might an odd x odd board be a better option? On this board an even number of squares are left at the end so draws are possible, and if Player 2 is content with a draw than P2 can force the draw and P1 has no chance to play for a win. An odd board would leave an odd number of squares at the end and so no draws possible. (Unless I've missed something obvious like usual.)

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