Q1.
Draw a 6 by 6 grid. Two players take turns covering two adjacent squares by drawing a line. The line can be placed either way: horizontally or vertically. The lines cannot overlap. The game goes on till a player is not able to place any more lines. The player who is not able to place any more lines loses. With what strategy can one play to win this game?
Answer
The winning idea is symmetry itself — copy your opponent.
Strategy for the second player
- Look at the centre of the 6 × 6 grid (the point where the middle horizontal and middle vertical lines cross).
- Whatever your opponent covers, you cover the pair of squares that is its half-turn (180°) image about that centre.
Why it always works: the 6 × 6 board has rotational symmetry of order 2 about its centre, and the centre is not on any square's edge. So the mirror-move is always empty and legal at the moment you play it — if it had been taken, its own partner (the opponent's move) would have been taken as well. Since you can always answer, you are never the one who runs out of moves. The player who cannot move loses, so the second player wins by copying.
Try This: play it on grids of other sizes. On a board with an odd number of squares in a row the first player can play in the middle first and then use the same copying strategy — so the first player wins there. Symmetry is a powerful tool in games, not just in geometry!