A game locating a needle in a square haystack

The following hider-seeker zero-sum game is considered. The hider hides a needle of length, in the closed unit square, and the seeker tries to locate it by shooting in a straight line across the square. The payoff to the seeker is 1 if he hits the needle and 0 otherwise.A solution of the game is obtained when or whena lies in either of the intervals and; in addition, it is shown that, whenn is a positive integer anda=1/n, the value of the game is 1/2n. The properties of the solutions are in marked contrast to those for the analogous game over the closed unit disc, which the authors solved in a previous paper, and suggest that a complete solution may well be difficult. It is also shown that every member of a whole class of haystack games has a value.