Solution
Misère Tic-Tac-Toe Openings
Show the problem again
Misère tic-tac-toe is played on the usual 3×3 grid by X (moving first) and O, but a player who completes a line of three of their own marks loses. If the grid fills with no such line, the game is a draw. For each of the nine possible opening squares for X, consider the game under optimal play by both sides afterwards. For how many opening squares does X lose? Answer with an integer.
Worked solution
Exhaustive search of the misère game tree gives the following. With perfect play the game is a draw, and the only opening that preserves the draw is the centre square. Each of the other 8 openings (four corners and four edges) loses for X against best play by O. So the number of losing openings is 8. Why the centre cannot lose: after opening there, X answers every O move with the cell diametrically opposite it. Suppose such a reply ever completed an X line. If the line passes through the centre, it consists of the centre, O's last cell and its mirror image, so it is not all X. If it misses the centre, its mirror image is a line of three O cells that was already complete, so O would have lost earlier. Hence X never loses after a centre opening. The losses after the eight other openings have no comparably short argument; they come from the exhaustive search.
Source: Misère tic-tac-toe ('toe-tac-tic') is a recreational folklore variant; no traceable original source. Statement written for AxiomIQ.