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The girls, the boxes and the balls (Posted on 2008-06-19) Difficulty: 3 of 5
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?

See The Solution Submitted by pcbouhid    
Rating: 4.3333 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts The third girl didn't apply logic.... (Spoiler) | Comment 5 of 9 |
Steve is correct in that there is a problem in the 3rd girl's ability to logically determine the color of her 3rd ball.

Assuming each of the 4 girls knew that each box had a different label, the 3rd girl, if she applied logic, would have been able to determine the color of her own 3rd ball, and that of the 4th girl's 3 balls.

Since the 1st girl knows the color of her 3rd ball, a logician would be able to deduce that the 1st girl's box was labelled either "BBB" or "BBW".

Since the 2nd girl knows the color of her 3rd ball, with that knowledge, coupled with the knowledge gained from the 1st girl, a logician, assuming no girl was lying or failed to apply reason, would be able to deduce that one of the following two conditions was true:

  • 1st girl's box is labelled "BBB" and her 3rd ball is White
  • 2nd girl's box is labelled "BBW" and her 3rd ball is White
  • box labelled "WWW" contains 3 Black balls; and
  • box labelled "BWW" contains 3 White balls
or
  • 1st girl's box is labelled "BBW" and her 3rd ball is Black
  • 2nd girl's box is labelled "BWW" and her 3rd ball is Black
  • box labelled "WWW" contains 1 Black and 2 White balls
  • box labelled "BBB" contains 3 White balls

Thus, since she drew 2 white balls, if her box were labelled "WWW" she should have known that her 3rd ball was Black, and, if it were labelled either "BWW" or "BBW" she should have known that her 3rd ball was White.

Edited on June 27, 2008, 1:13 am
  Posted by Dej Mar on 2008-06-20 07:27:26

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