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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?

  Submitted by pcbouhid    
Rating: 4.3333 (3 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
SolutionSolution step by stepGeorge2008-06-28 13:36:56
re(2): The third girl didn't apply logic.... (Spoiler)Dej Mar2008-06-27 01:24:48
re: The third girl didn't apply logic.... (Spoiler)pcbouhid2008-06-20 10:32:49
SolutionNo Subjectnitin2008-06-20 08:55:48
Some ThoughtsThe third girl didn't apply logic.... (Spoiler)Dej Mar2008-06-20 07:27:26
Even 3d girl can guess all others labelsPraneeth2008-06-19 22:56:38
Not logicalSteve Herman2008-06-19 20:48:29
Solutionanswer [spoiler]Vernon Lewis2008-06-19 19:08:22
SolutionSolutionRobby Goetschalckx2008-06-19 10:51:09
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