The Truel

Date: 10/13/98 at 18:19:00
From: Sabina 
Subject: Algebra: Thinking problem.

Hi Dr. Math! I have difficulty doing a thinking problem in Algebra.
Here is the problem:

The Truel.

A truel is similar to a duel, except there are three participants 
rather than two. One morning Mr. Black, Mr. Gray, and Mr. White decide 
to resolve a conflict by trueling with pistols until only one of them 
survives. Mr. Black is the worst shot, hitting his target on average 
only one time in three. Mr. Gray is a better shot, hitting his target 
two times out of three.  Mr. White is the best shot, hitting his 
target every time. To make the truel fairer, Mr. Black is allowed to 
shoot first, followed by Mr. Gray (if he is still alive), followed by 
Mr. White (if he is still alive), and around again until only one of 
them is alive. The question is this: Where should Mr. Black aim his 
first shot?

I think that participants would have the same distance between each 
other. Maybe Mr. Black should aim his first shot straight between 
Mr. Gray and Mr. White, so he would probably shoot one of them.
But I am not sure, and I need logical reasons to support my answer.

Thank you very much.

Date: 10/20/98 at 20:03:42
From: Doctor Santu
Subject: Re: Algebra: Thinking problem.

Hello, Sabina. What an interesting question. One never knows when one 
might be in a truel! Might as well be prepared.

Try working backward from the end of the Truel. You can work backward 
from the scenarios where Black gets killed, and figure what he should 
not have done, or work backward from the scenarios where he lives, and 
see what he did right. We'll start off using the latter method.

Suppose Black survives. This means he didn't get shot by Grey, and he 
didn't get shot by White.

White never misses. Why didn't Black get shot by White? He must have 
been shot by someone else. Who could it be?

Could it be Grey?

I really should leave the reasoning to you with that little hint - 
work backward from the fact that you want Black to survive.

Dr. Santu, The Math Forum
Check out our web site! http://mathforum.org/dr.math/