perplexus.info :: Logic : The girls, the boxes and the balls

The girls, the boxes and the balls (Posted on 2008-06-19)

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?

Submitted by pcbouhid

Rating: 4.6000 (5 votes)

Solution: (Hide)

If Girl #1 had the box labeled BBB, she would have known the remaining ball in that box was white. (The ball could not be black, otherwise the box would be correctly labeled.)

Also, if she had the box labeled BBW, she would have known the remaining ball in that box was black for the same reason.

If she had the other two labeled boxes, she could not have known the color of the remaining ball since it could have been either one.

However, since she did know the color of the remaining ball, she must have the box labeled either BBB or BBW. We don't yet know which is correct.
girl balls Incorrect Ball # drawn box label remaining ---------------------------------------- 1 BB BBB W, if BBB BBW B, if BBW ----------------------------------------
If Girl #2 had the box labeled BBW, she would have known the remaining ball in her box was white. (The remaining ball could not be black, otherwise the box would be correctly labeled.)

Also, if she had the box labeled BWW, she would have known the remaining ball in that box was black for the same reason.

If she had the other two labeled boxes, she could not have known the color of the remaining ball since it could have been either one.

And again, since she did know the color of her remaining ball, her box label must be either BBW or BWW. We don't yet know which is correct.
girl balls Incorrect Ball # drawn box label remaining ---------------------------------------- 1 BB BBB W, if BBB BBW B, if BBW ---------------------------------------- 2 BW BBW W, if BBW BWW B, if BWW ----------------------------------------
Girl #3 drew two white balls from her box and was not able to deduce the color of the remaining ball. This means she could not have had the box labeled WWW otherwise she would have known that the remaining ball could only be black.
girl balls Incorrect Ball # drawn box label remaining ---------------------------------------- 1 BB BBB W, if BBB BBW B, if BBW ---------------------------------------- 2 BW BBW W, if BBW BWW B, if BWW ---------------------------------------- 3 WW not WWW ----------------------------------------
Girl #3 could also not have had the box labeled BWW since she then would have known her remaining ball could only have been white.
girl balls Incorrect Ball # drawn box label remaining ---------------------------------------- 1 BB BBB W, if BBB BBW B, if BBW ---------------------------------------- 2 BW BBW W, if BBW BWW B, if BWW ---------------------------------------- 3 WW not WWW not BWW ----------------------------------------
So, if Girl #3 could not have had the box labeled WWW or BWW, she must have had either the box labeled BBB or BBW.
girl balls Incorrect Ball # drawn box label remaining ---------------------------------------- 1 BB BBB W, if BBB BBW B, if BBW ---------------------------------------- 2 BW BBW W, if BBW BWW B, if BWW ---------------------------------------- 3 WW BBB BBW ----------------------------------------
So Girl #4 knows, since no one else can possibly have the box labeled WWW, then she must have it.
girl balls Incorrect Ball # drawn box label remaining ---------------------------------------- 1 BB BBB W, if BBB BBW B, if BBW ---------------------------------------- 2 BW BBW W, if BBW BWW B, if BWW ---------------------------------------- 3 WW BBB BBW ---------------------------------------- 4 -- WWW -- ----------------------------------------
Since Girl #1's box was not the box that contained the three white balls, nor was Girl #2's and since it can't be Girl #4's, (otherwise the label would be correct) then it can only be Girl #3's box that contains the three white balls.
girl balls Incorrect Ball # drawn box label remaining ---------------------------------------- 1 BB BBB W, if BBB BBW B, if BBW ---------------------------------------- 2 BW BBW W, if BBW BWW B, if BWW ---------------------------------------- 3 WW BBB W BBW ---------------------------------------- 4 -- WWW -- ----------------------------------------
Since Girl #2 is the only box that could possible have the label BWW, then she must have it.
girl balls Incorrect Ball # drawn box label remaining ---------------------------------------- 1 BB BBB W, if BBB BBW B, if BBW ---------------------------------------- 2 BW BWW B ---------------------------------------- 3 WW BBB W BBW ---------------------------------------- 4 -- WWW -- ----------------------------------------
Since Girl #2's box contains the Black-Black-White ball combination, this combination can't be in Girl #1's box. Thus, Girl #1's box must have the Black-Black-Black ball combination, and consequentially, the BBW label.
girl balls Incorrect Ball # drawn box label remaining ---------------------------------------- 1 BB BBW B ---------------------------------------- 2 BW BWW B ---------------------------------------- 3 WW BBB W BBW ---------------------------------------- 4 -- WWW -- ----------------------------------------
Since we know Girl #1 now has the Box Label of BBW, Girl #3 cannot have it.
girl balls Incorrect Ball # drawn box label remaining ---------------------------------------- 1 BB BBW B ---------------------------------------- 2 BW BWW B ---------------------------------------- 3 WW BBB W ---------------------------------------- 4 -- WWW -- ----------------------------------------
Girl #4's box obviously then contains two white balls and one black ball since no other box has that combination in it.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Thoughts	K Sengupta	2023-12-26 07:26:31
re: answer [spoiler]	Riley Schuppe	2021-11-15 10:51:19
Solution step by step	George	2008-06-28 13:36:56
re(2): The third girl didn't apply logic.... (Spoiler)	Dej Mar	2008-06-27 01:24:48
re: The third girl didn't apply logic.... (Spoiler)	pcbouhid	2008-06-20 10:32:49
No Subject	nitin	2008-06-20 08:55:48
The third girl didn't apply logic.... (Spoiler)	Dej Mar	2008-06-20 07:27:26
Even 3d girl can guess all others labels	Praneeth	2008-06-19 22:56:38
Not logical	Steve Herman	2008-06-19 20:48:29
answer [spoiler]	Vernon Lewis	2008-06-19 19:08:22
Solution	Robby Goetschalckx	2008-06-19 10:51:09

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