Solution
Placing Mutually Safe Queens
Show the problem again
Two players alternately place a queen on an empty square of a 7×7 board so that it does not attack any queen already on the board. Two queens attack each other if they share a row, a column or a diagonal, whether or not other pieces lie between them. The player who cannot place a queen loses. Which player wins with optimal play? Answer 'the first player' or 'the second player'.
Worked solution
The first player wins by a symmetry strategy. Open on the central square, which attacks every square on its row, column and both diagonals. After that, answer every move of the opponent by placing a queen on the image of that square under the 180 degree rotation about the centre. Every square on a line through the centre is already attacked, so no queen is ever placed on such a line, and consequently a square and its image never attack each other (any line joining them would pass through the centre). If the opponent's square was legal, its image is legal too, because the queens already placed form a rotation-symmetric set. So the first player always has a reply. Exhaustive search of the game confirms the first player wins.
Source: Folklore combinatorial-game variant of the n-queens problem; no traceable original source. Statement written for AxiomIQ.