2nd Australian 1981 Problem B3 In an archery contest, the two contestants A and B start at x = 0 and walk towards the target at x = 1 together. Each chooses the distance from which he shoots. A has probability x 2 of hitting the target from x. B has probability x of hitting the target from x. If both hit the target, then the first to shoot wins. What is A's best strategy?
If the first player to shoot hits the target then he wins. If he misses, then the other player waits until he is up against the target and cannot miss, so if the first player to shoot misses, then he loses. So if A shoots first at x, then he has a prob x 2 of winning, whereas if B shoots first at x, then A has a prob 1-x of winning. Put k = (√5-1)/2 = 0.618... . Then for x  k, then x 2 > 1-x, so it is worse for A if B shoots first. Moreover, the longer he waits the worse his position if B manages to shoot first. Thus A's strategy is as follows. If B shoots at x < k, then he waits until he reaches the target before shooting. If B has not shot at x = k, then A shoots. s are also available in Australian Mathematical Olympiads 1979-1995 by H Lausch and P J Taylor, ISBN 0858896451, published by Australian Mathematics Trust, can be ordered by email. 2nd Aus 1981 © John Scholes jscholes@kalva.demon.co.uk 28 Oct 2003 Last updated/corrected 28 Oct 03