Solution
Cram on a Four-by-Four Board
Show the problem again
Two players alternately place a 1×2 domino, horizontally or vertically, on a 4×4 board so that it covers exactly two empty squares. Dominoes may not overlap. The player who cannot place a domino loses. Which player wins with optimal play? Answer 'the first player' or 'the second player'.
Worked solution
The second player wins by a symmetry strategy: answer every domino of the first player with its image under the 180 degree rotation about the centre of the board. Suppose the position is rotation-symmetric before the first player moves (it is at the start). The first player covers two empty squares, so by symmetry their images are empty as well. The image domino cannot overlap the original: that would need a square of the domino to be the image of one of its own two squares, but a square and its image are never adjacent (the centre of the board is a lattice point, not a square, and the two squares nearest to it, which are images of each other, touch only at a corner) and a square is never its own image. So the reply is legal, the position is symmetric again, and the second player always has a move. The first player therefore runs out of moves first. A full game-tree search agrees.
Source: Cram is a standard impartial domino-placing game; the symmetry argument is classical. Statement written for AxiomIQ.