Three Jars

Date: 11/01/2000 at 20:17:51
From: Steven
Subject: Logic puzzle

We have three jars: one contains red marbles, one contains black, and 
one contains both, but we don't know which is which. The jars are 
labeled Red, Black, and Mixed, but each of the labels is wrong. Nancy 
says she can put the labels on the right jars by picking one marble 
from one container. How can she do this?

I think I know the answer, but it's hard to explain. Can you give me 
an answer so I can compare?

Date: 11/02/2000 at 15:38:33
From: Doctor Ian
Subject: Re: Logic puzzle

Hi Steven,

This kind of question can drive you crazy without a picture to help 
you keep track of what's going on. Here are the containers:

     +---+  +---+  +---+
     |   |  |   |  |   |
     +---+  +---+  +---+

We'll use the letters "r," "b," and "m" to indicate whether a 
container _actually_ contains red, black, or mixed marbles. Without 
knowing anything else, we have to assume that each container can 
contain any kind of marbles:

       r      r      r
       b      b      b
       m      m      m
     +---+  +---+  +---+
     |   |  |   |  |   |
     +---+  +---+  +---+

Now let's label the containers:

       r      r      r
       b      b      b
       m      m      m
     +---+  +---+  +---+
     | R |  | B |  | M |
     +---+  +---+  +---+

Since we know all the labels are wrong, we can eliminate some 
possibilities right off the bat:

              r      r
       b             b
       m      m       
     +---+  +---+  +---+
     | R |  | B |  | M |
     +---+  +---+  +---+

Now, let's say we take a marble out of the container marked "M", and 
let's say it's red.

              r      r
       b             b
       m      m       
     +---+  +---+  +---+
     | R |  | B |  | M |
     +---+  +---+  +---+
                    red

Since we know the label is wrong, we know that it must contain _only_ 
red marbles:

              r      r
       b             
       m      m       
     +---+  +---+  +---+
     | R |  | B |  | M |
     +---+  +---+  +---+
                    red

And since it contains only red marbles, no other container can contain 
only red marbles:

                     r
       b             
       m      m       
     +---+  +---+  +---+
     | R |  | B |  | M |
     +---+  +---+  +---+
                    red

So we know which one is red, and which is mixed. Which means that the 
other must be black:

                     r
       b             
              m       
     +---+  +---+  +---+
     | R |  | B |  | M |
     +---+  +---+  +---+
                    red


Now you need to see whether this still works if you pull a black 
marble instead of a red one out of the container marked "M"...

I hope this helps. Write back if you'd like to discuss this some more, 
or if you have any other questions.

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