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 a, \(0<a\leq \sqrt{2}\), 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 \(a\geq \sqrt{2}/2\) or when a lies in either of the intervals [\(\sqrt{5}/4\), 2-\(\sqrt{2}]\) and [\(\sqrt{2}/3\), 1/2]; in addition, it is shown that, when n is a positive integer and \(a=1/n\), 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 [ibid. 47, 383-391 (1985; Zbl 0556.90096)], 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.
scientific article; zbMATH DE number 3936966