Filling in a triangular lattice
Consider the following triangular lattice:
You and a friend take turns filling in the unit triangles.
The only rule is you can't fill in a triangle that shares an edge with a triangle that's already filled in.
The player to fill in the last triangle wins.
Who has a winning strategy?
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1 solution
With respect to symmetry of the triangular lattice, in general Player 2 can always respond to Player 1 by filling in a corresponding symmetrical blue triangle on the left or right blue sides (if Player 1 fills in a blue triangle) or filling in a pink triangle on the top or bottom pink diamond (if Player 1 fills in a pink triangle), therefore securing a win.
Let's consider 2 different cases of how Player 1 can start:
Case 1 : Player 1 fills in a triangle in one of the following green triangle locations:
If necessary, rotate the board so that Player's 1 filled triangle matches one of the pink triangles in the diagram above. Player 2 can then respond by filling in the one of the other available pink triangles on the other pink diamond. This will leave only two symmetrical sides of blue triangles left. Now whatever blue triangle Player 1 chooses to fill in, Player 2 can respond by filling in the corresponding blue triangle on the other side, which will ensure that Player 2 will fill in the last available triangle and win.
Case 2 : Player 1 fills in a triangle in one of the following green triangle locations:
If necessary, rotate and reflect the board so that Player's 1 filled triangle matches the blue triangle labelled "1" below. Player 2 can then respond by filling in the corresponding blue triangle on the other side, labelled "2" below. (Gray triangles are no longer available to fill because they share an edge with a triangle that's already filled in.)
Now if Player 1 fills in a blue triangle, Player 2 can respond by filling in the corresponding blue triangle on the other side, or if Player 1 fills in a pink triangle, Player 2 can respond by filling in the other available pink triangle, which either way will ensure that Player 2 will fill in the last available triangle and win.
In either case, Player 2 can use a strategy to win the game.