Alice and Bob are playing the following game: They have a 4×4 empty grid and take turns coloring one square each, starting with Alice. Assume that both players will colour the squares in with the same colour. Whoever completes any 2×2 area on the grid after having made their move, is the loser.
Is there any winning strategy for either of the two players?
First of all, I'm assuming that Bob and Alice are painting with different colors and that we are only concerned with a $2\times 2$ square of the same color. Now I realize that I may have interpreted the problem incorrectly.
Bob
can always force a win or draw with the following strategy:
Whatever square Alice colors, Bob should color the square symmetric with respect to the origin (rotate 180 degrees). This way, Bob can never complete a $2\times 2$ square unless Alice has just completed a rotated version of that square. In fact, Bob can win/draw this way on any $2n\times 2n$ grid.
Bob wins if
both Alice and Bob only colour the edges. Then Alice is eventually forced to play in the middle.
If Alice colours an inner cell for the first time:
Bob colours the diagonally opposite, and reverts to the edge strategy ignoring the extremal corners of the extremal sub-grids with inner cells. We either end up with a saturated grid (Bob wins), or Alice colours one, and so Bob colours the opposite corner with the same effect.
Alice colours a second inner cell:
Bob colours the opposite extremal corner, and the same strategy gives Bob the win.
Summary:
Bob always wins.
Who wins?
Bob.
Why?
If the game goes beyond $12$ moves, one of the possible $2 \times 2$ squares be filled and Alice will lose. It's trivial that there can be at least $4$ empty tiles that each create a square once filled, but it's at most $5$ like so (yellow means one of the connected yellow tiles is enough):
This means if Alice really is to win, she must make sure the end result is one of these colorings (or their rotations), and all Bob has to do is prevent their formation:
A pattern common to both of them is when you take an edge (but not corner) tile and move like a knight to the next edge, at least one of the four tiles is empty. Bob can try and fill all of them based on Alice's moves and by making sure to do so before Alice can protect one by creating an almost-filled square with it as the missing tile.
