Date: 6 Mar 1995 22:17:31 -0500 From: Anonymous Subject: Truel Here's a puzzler for you all: You and two of your friends get into a dispute and decide to solve it with a "truel", a three way duel. Friend #1 is a crack shot, never missing his target. Friend #2 hits his target 2/3 of the time. You hit your target 1/3 of the time. It is decided that you will take the first shot, the 2/3 marksman will take the second shot (if still alive) and the 100% marksman will go last. This will continue until there is only one left alive. On your turn you get to fire one bullet. You get to go first. In order to maximize your chances of living thru this, where should you take your opening shot? And what are your chances of winning the truel if you follow this strategy? Thanks for taking the time with this, Dave Zechiel
Date: 8 Mar 1995 15:40:20 -0500 From: Dr. Ken Subject: Re: Truel Hello there! I'll give you a couple of hints about how to get started on this problem, and then see if you can take it from there. Basically, you want to build a tree of all the various possibilities. Let's call you by the variable A, the 2/3 marksman by the variable B, and the 100% marksman by the variable C. In the first move, you can either aim at B or C, and let's denote that by A>B and A>C. Let's denote a hit by a @ (bull's eye) and a miss by a ~ (a tilde, which is used to spell the word pinata, which you try to hit when you're blindfolded, and so you usually miss). So the beginning of the tree would look like this: Move 1: A>B A>C @=1/3/ \~=2/3 @=1/3/ \~=2/3 B dies/ \ B lives C dies/ \C lives C>A, A dies/ /\ B>A/ B>A/\B>C C wins B>C/ \B>A /\ @=2/3/ \~=1/3 A dies/ \A lives B wins \A>B /\ @=1/3/ \~=2/3 B dies/ \B lives A wins You can tell when any branch has terminated, because you can follow it up to the top and you'll pass exactly 2 "dies." Can you complete the chart now? One more thing to keep in mind: the other two players are trying to win, too. So for instance, in the fork in the first tree, there aren't 50/50 odds that B will aim at C. Rather, B will look at the situation and figure out what strategy is best for himself. So unless it coincidentally ends up being even 50/50 odds, that fork will be eliminated, and we'll have only B>C or B>A. For instance, if C is shooting and there are still 3 shooters left, C will certainly shoot B, because then A will only have a 1/3 chance of hitting C on the next shot, compared with B's 2/3 chance. What will save this puzzle from being hopelessly difficult is the fact that whenever C shoots, a player gets eliminated. That's a big help, because that means that the only time the shooting can conceivably go on forever is when C is the first to die and A & B duke it out. Once you've completed the tree, you can find out the total probability each player has of survival by finding all the branches that win for each player, and then multiplying together all the different probabilities down the path that led to that termination. This will be the total probability of that branch's outcome. By the way, you may want to investigate what happens if you shoot your first shot in the air, as opposed to aiming at one of the other two folks. Just for fun. Incidentally, I thought the line "you and two of your _friends_" was a funny set-up for this puzzle. -Ken "Dr." Math
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