Given a square of side length as least 2, there are two players who must in turn place a unit radius circle into the square without overlapping any previously placed circle.
The loser is the first player who can't place a circle.
For example, if the square has side length 2 then the first player wins. If the square is just big enough to fit two circles with side length 2+2√2, then again the first player wins by putting a circle into the center.
What is the perfect way to play? This will depend on the size of the square.