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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)

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Solution Solution | Comment 1 of 9
After the first girls makes the announcement, we know that having two black balls and the label enables her to identify the third ball. This means the label is either BBB or BBW.
For the second girl, similarly, we know that the label must be BBW or BWW.
The third girl sees two white balls and can NOT tell the color of the third by the label. This implies that the label includes at least two B's - it is clear that it is wrong, whatever the third ball is according to the label. This means that the label is either BBB or BBW.
If we look at the possible labels for the first and third girls, these are the same ! This means that the second girl can not have label BBW - this is on either the first or third box. This box must have the label BWW, and the second girl has a white, black and black ball (I'll use the notation [bbw] for the actual balls).
The first box contains at least two black balls, but can not have the same content as the second, so this box contains [bbb] and must have the label BBW. This tells us the third box has the label BBB. The last label must be WWW - the only label left.
For the contents of the third and fourth box, there are still two options: [www] and [wwb]. The fourth box with label WWW can not contain [www], so this box contains [wwb], and the third contains [www].
girl:	label:	content:
1 BBW bbb
2 BWW bbw
3 BBB www
4 WWW wwb

  Posted by Robby Goetschalckx on 2008-06-19 10:51:09
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