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?

Analyzing Girl 1 cases:
Case 1:
Girl 1 has Label BWW, so box has BBB or BBW, can't determine color, rejected case.

Case 2:
Girl 1 has Label BWW, so box has BBB or BBW, can't determine color, rejected case.

Case 3:
Girl 1 has Label BBB, so box has BBW, she can determine color, case accepted.

Case 4:
Girl 1 has Label BBW, so box has BBB, shw can determine color, case accepted.

So far we have 2 cases
Case 1:
    Girl 1: Label=BBB, Box=BBW
Case 2:
    Girl 1:Label=BBW, Box=BBB


Analyzing Girl 2 cases:
Case 1:
Girl 2 has Label BBB, so
    Girl 1 has Label BBW and box BBB,
and Girl 2 box has BBW or BWW, can't determine color, rejected case.

Case 2:
Girl 2 has Label WWW, so box has BBW or BWW, can't determine color, rejected case.

Case 3:
Girl 2 has Label BBW, so
    Girl 1 has Label BBB and box BBW,
and Girl 2 box has BWW, can determine color, case accepted.

Case 4:
Girl 2 has Label BWW, so box has BBW
    and Girl 1 has Label BBW and box BBB
can determine color, case accepted

So far we have 2 cases

Case 1:
    Girl 1: Label=BBB, Box=BBW
    Girl 2: Label=BBW, Box=BWW
Case 2:
    Girl 1: Label=BBW, Box=BBB
    Girl 2: Label=BWW, Box=BBW
   

Analyzing Girl 3 cases:
Case 1:
Girl 3 has Label WWW, so box is BWW, can determine color, rejected case.

Case 2:
Girl 3 has Label BWW, so box is WWW, can determine color, rejected case.

Case 3:
Girl 3 has Label BBW, so, neither of the cases so far stands, so we reject case.

Case 4: (the only one left)
Girl 3 has Label BBB, so
    Girl 1 has Label=BBW, Box=BBB
    Girl 2 has Label=BWW, Box=BBW
so Girl 3 box can be BWW, or WWW, can't determine color, as we could expect so.

So, the Girl 4 has these conclusions
Girl 1 has Label=BBW, Box=BBB
Girl 2 has Label=BWW, Box=BBW
Girl 3 has Label=BBB, Box=BWW or WWW

So, Girl 4 has Label WWW, and box BWW (the boxes BBB,BBW are already taken, and she cant have WWW).

So, she knows the final labels-boxes distribution
Girl 1 has Label=BBW, Box=BBB
Girl 2 has Label=BWW, Box=BBW
Girl 3 has Label=BBB, Box=WWW
Girl 4 has Label=WWW, Box=BWW

 

Posted by George on 2008-06-28 13:36:56