Who Is the Youngest Boy?

Date: 04/28/2002 at 16:36:35
From: Kamie Oda
Subject: One of those "Ed is the oldest"

Dear Dr. Math,

This is my problem:

Al, Ed, and Tom are different ages. Exactly one of the following 
three statements is true:  

   1. Ed is the oldest.  
   2. Al is not the oldest.  
   3. Tom is not the youngest. 

Who is the youngest boy?

The answer is Ed. I tried to make a table of these but failed. If Al 
is not the oldest, then Ed or Tom is oldest, Al, Ed, or Tom could be 
in the middle, and Al, Ed, or Tom could be the youngest. There are too 
many possibilities to narrow it down!

Thanks for your help,
Kamie O

Date: 04/28/2002 at 16:56:52
From: Doctor Ian
Subject: Re: One of those "Ed is the oldest"

Hi Kamie,

Have you considered looking at the possible age orderings?  
Suppose we list the subjects in order of increasing age:

  Al,  Ed,  Tom
  Al,  Tom, Ed
  Ed,  Al,  Tom
  Ed,  Tom, Al
  Tom, Al,  Ed
  Tom, Ed,  Al

Now there aren't so many possibilities, and if we check these against 
the truth of the three statements, 
    
                   Ed is    Al is not    Tom is not
                   oldest   oldest       youngest

  Al,  Ed,  Tom     F         T              
  Al,  Tom, Ed      T         T              
  Ed,  Al,  Tom     F         T              
  Ed,  Tom, Al      F                       T
  Tom, Al,  Ed      T                       F
  Tom, Ed,  Al      F                       F

I've left some of the values for you to fill in. When you've completed 
the table, you'll find that only one row contains exactly one true 
value. 

Does this help?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/