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Who Stole Second Base?

Date: 20 Feb 1995 15:42:57 -0500
From: Anonymous
Subject: The Stolen Base

Help!!  (settle for a hint)

The umpire was convinced that either Archie, Buster, Cal, or Dusty 
had stolen (really stolen) second base.  Each player, in turn, made a 
statement, but only one of the four statements was true.
          Archie said, "I didn't  take it."
          Buster said, "Archie is lying."
          Cal said, "Buster is lying."
          Dusty said, "Buster took it."
Who told the truth?

Date: 20 Feb 1995 19:20:04 -0500
From: Dr. Ken
Subject: Re: The Stolen Base

Hello there!

I think I'll try giving you a hint first.  When you are told that only one
of the four statements is true, you can then try each statement as the true
statement to see whether it works.  For example, let's assume that the last
statement is true; this implies that the other three statements are false, so
let's see if that gives us a consistent set of statements.  To do this,
let's write out the four statements, but with only the last statement
intact, and the for the other three let's use the negative of what they said.

So we get:
Archie took it.
Archie is telling the truth (i.e. Archie didn't take it)
Buster is telling the truth (i.e. Archie is telling the truth)
Buster took it.

Is this a consistent set of statements?  I don't think so.  The most obvious
contradiction is that it claims Archie and Buster both took the base.  So
Dusty can't be the one telling the truth.

Try the same thing with the other three guys, and see which gives you a
consistent set of statements.  Hopefully there will be only one that works,
and that will tell you which one is telling the truth (although not
necessarily who took the base!).

-Ken "Dr." Math

Is it possible for Archie to be telling the truth while Buster, Cal, and 
Dusty are lying? Go through all of their statements.  If Archie is telling 
the truth, then we know:  1) he didn't take it; 2) Buster is lying (Buster 
says Archie is lying, so that means that the truth is that Archie isn't 
lying, so we are okay so far); 3) Cal is lying (Cal says Buster is lying, 
so that means that since Cal is lying, Buster really is NOT lying, so 
Buster is telling the truth).  This contradicts the given conditions 
of the problem, though, so we know that Archie isn't telling the truth.  

Now you consider the 2 other cases!
If you need any more help, feel free to write back!

--Sydney, "dr.math"
Associated Topics:
High School Logic
Middle School Logic

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