Three women were seated around a table. After blindfolded, a numbered disc was pasted on each of their foreheads. The women were truthfully told "Each of you has either a 1, a 2 or a 3 on your forehead, and the sum of your numbers is either 6 or 7".

After the blindfolds were removed, each woman in turn was asked to name the number on her forehead without seeing it. The question was repeated until only one woman failed to name the number on her forehead.

When it was logically possible to name the number on her forehead, each woman did so; when it was not logically possible to name the number on her forehead, she would said "I donīt know my number", and waited until the question was repeated to her next time around.

Each woman had a 2 on her forehead. Which woman failed to name her number?

This is the method I often use to solve this type of puzzle.

I make a chart.  Each row corresponds to a different possibility (e.g. 123 means that the first woman has a 1, the second has a 2, and the third has a 3).  Of course, only one possibility is correct, but the women don't know that, and must consider every possibility.  Each column corresponds to a response.  For example, if the first column is "x", it means the first response is "I don't know."

123 x o
132 x o
133 o 
213 x x o
222 x x x o o x
223 x x x o o x
231 x o
232 x x o
312 o 
313 x o
321 o 
322 x o
331 x o

It seems the third woman never figures out her number.  This is an outcome I don't usually see, because usually I stop looking after the first person figures it out.  Interesting problem!

Edit: I realize my thinking process is not very clear when I use the chart, so I will elaborate just this once.

Basically, I work 1 column at a time.  I put myself in the woman's position, and only look at the other women's numbers.  If there are two or more possibilities where the other women have the same pair of numbers (e.g. 123 and 223), I place an x.  If there is only one, I place an o, and ignore that row in the columns afterwards.

Edited on December 15, 2005, 9:35 pm

Posted by Tristan on 2005-12-15 21:28:03