Battlehip game theory solution: find payoff matrix and game value. For simplicity, consider a 3 by 3 board and suppose that Player I hides a destroyer (length 2 squares) horizontally or vertically on this board. Then Player II shoots by calling out squares of the board, one at a time. After each shot, Player I says whether the shot was a hit or a miss. Player II continues until both squares of the destroyer have been hit. The payoff to Player I is the number of shots that Player II has made. Let us label the squares from 1 to 9. The problem is invariant under rotations and reflections of the board. In fact, of the 12 possible positions for the destroyer, there are only two distinct invariant choices available to Player I: the strategy, [1,2]∗, that chooses one of [1,2], [2,3], [3,6], [6,9], [8,9], [7,8], [4,7], and [1,4], at random with probability 1/8 each, and the strategy, [2,5]∗, that chooses one of [2,5], [5,6], [5,8], and [4,5], at random with probability 1/4 each. This means that invariance reduces the game to a 2 by n game where n is the number of invariant strategies of Player II. Domination may reduce it somewhat further. Solve the game.