I'll put all correct solutions received by FRIDAY September 28 into a hat, and draw a winner, who will get a small prize.
The puzzle: Ann, Brian and Cora hold a three-person paintball duel. The rules are as follows: Each stands at one corner of a triangle. Initially, Ann chooses a target (either Brian or Cora) and shoots. If she has a hit, the hit target leaves the game, otherwise both stay. The paint gun then moves to Brian (if he's still in the game), who repaeats the process, and then hands the gun to Cora (if she's still in the game). This cyclic rotation of the paint gun among (whoever is still left in the game of) Ann, Brian, Cora, Ann, Brian, Cora, ... goes on until only only person is left unhit; that person is the winner.
Ann hits her targets 30% of the time, independently from shot to shot; Brian hits his target 100% of the time; and Cora hits hers half the time.
There's one extra rule: initially, Ann has the option of not shooting, i.e., of allowing Brian the first shot.
Here's the puzzle: What strategy does Ann adopt to maximize her chances of winning the duel? Assume that all other players are also attempting to maximize their chances of winning, and no player holds a grudge: choices of who to shoot at are made purely with an eye to maximizing chances of winning, and never to help or hinder another player.
Solutions, together with the list of correct solvers and the prize winner, are here.