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A Truel

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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

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
```
Associated Topics:
High School Logic
High School Probability

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