Four girls were blindfolded and each was given an identical box, containing different colored balls:

One box contained 3 black balls.
One box contained 2 black balls and 1 white ball.
One box contained 1 black ball and 2 white balls.
One box contained 3 white balls.

Each box had a label on it reading "BBB" (Three Black) or "BBW" (Two Black, One White) or "BWW" (One Black, Two White) or "WWW" (Three White). The girls were told that none of the four labels correctly described the contents of the box to which it was attached.

Each girl was told to draw two balls from her box, at which point the blindfold would be removed so that she could see the two balls in her hand and the label on the box assigned to her. She was given the task of trying to guess the color of the ball remaining in her box.

As each girl drew balls from her box, her colors were announced for all the girls to hear, but the girls could not see the labels on any boxes other than their own.

The first girl, having drawn two black balls, looked at her label and announced: "I know the color of the third ball!"

The second girl drew one white and one black ball, looked at her label and similarly stated: "I too know the color of the third ball!"

The third girl drew two white balls, looked at her label, and said: "I can't tell the color of the third ball."

Finally, the fourth girl declared: "I don't need to remove my blindfold or any balls from my box, and yet I know the color of all three of them. What's more, I know the color of the third ball in each of the other boxes, as well as the labels of each of the boxes that you have."

The first three girls were amazed by the fourth girl's assertion and promptly challenged her. She proceeded to identify everything that she said she could.

Can you do the same?

Girl 4 has made a very educated guess.  It requires her to assume, not unreasonably, that girls 1, 2 and 3 all are capable of simple logical reasoning, but that girl 3 is incapable of slightly more advanced logic.

Girl 4's problem is that any inference available to her is also available to the 3rd girl.  Girl 3 has more information than Girl 4, so she has enough information to infer the color of all balls and all labels.  The fact that she cannot gives the 4th girl a contradiction.  

Girl 4 could just as well have resolved the contradiction by assuming, instead, that girl 3 was incapable of simple logical reasoning, in which case Girl 4 has no information available about the Girl 3's labels, and must declare "I can't tell the color of my third ball".  It was just sheer hubris (or a failure to detect the contradiction) that led her to predict her ball color without removing her blindfold.  Had she removed her blindfold, and seen her balls and her label, she could have have determined her third ball logically, without any guessing about which girl(s) among the first three were not logicians.

Posted by Steve Herman on 2008-06-19 20:48:29