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# One Blind Man

## Question

Four identical boxes, each box containing 3 balls, each ball being either black or white, were placed before four men: Al, Bert, Cal and Don. No two boxes contained the same selection of ball colors. Each box bore a label that correctly identified the contents of one and only one of the boxes, but no individual box bore its correct label.

After each man, at random, was given one of the boxes, they were given the following test. Each in turn, and out of sight of the others, was to blindly remove two of the balls from his box, read the label on his box, and then endeavor to tell to the others the color of the third ball, which remained in the box.

It did not seem a difficult task, but the results were a bit surprising:

1. Al:
I've drawn two black balls.
I can tell you the color of the third ball.

2. Bert:
I've drawn one white ball and one black ball.
I can tell you the color of the third ball.

3. Cal:
I've drawn two white balls.
I'm not able to tell you the color of the third ball.

4. Don: (who was blind, and could not read his box's label)
I don't need to draw.
I can tell you the colors of the balls in my box.
I can tell you the color of the third ball in each of the others' boxes.

And then he proceeded to do so. How did he tell?

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Spoiler
 Al must have opened a box containing either BBB or WWB. Since the label was wrong, the true content was either WWB or BBB. Therefore, he knew the color of his third ball. Bert opened a box with either WBB or WWB, leaving him the choice of either WWB or WBB (according to the label) and he could guess the color of his third ball. Cal's box could have had either WWW or WWB and could thus  not know the color of the third ball. So, the labels that stated "WBB", "WWB" and "BBB" could only be interpreted as having the combinations BBB, WBB and another (yet unknown) combination. Don followed the conversations and then knew for sure that his box contained the combination WWB (Cal could not have had that combination - only WWW was possible for Cal).   Al: BBB Bert: WBB Cal: WWW Don: WWB

Edited by rocdocmac
fitting
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Spoiler

There are only 4 combinations for the black and white balls, and for the labels:

w w w

w w b

w b b

b b b

If you draw two balls then you can tell your box is 1 of 2 possible combinations. If your box's label is 1 of those 2, then, since the 3rd colour on your label is wrong, you know the colour of your third ball.

Since Al drew 2 black balls he has wbb or bbb. Since he can tell the colour of his 3rd ball, his box must have been labelled bbb or wbb, respectively.

Since Bert drew 1 black and 1 white, he has wwb or wbb. Again, since he can tell the colour of the 3rd ball, his box is labelled wbb or wwb.

Cal drew 2 white balls, so he has www or wwb. However, his box does not have one of those two labels (or else he too would be able to tell the colour of his 3rd ball). So his label is either wbb or bbb.

Since Al and Cal have the bbb and the wbb labels (not sure who has which), then Bert must have the wwb label, and so, Don knows he has the www label.  Also, since Bert's label is wwb, his balls must be wbb. Taking the wbb combination from Al leaves only bbb for his ball combination (and thus, wbb for his label). That leaves only bbb for Cal's label. Don is left with only www or wwb combinations for his balls. He knows www isn't right, so his balls are wwb. That leaves Cal with www for his balls. Making it:

Al: balls =  bbb   label = wbb

Bert: balls = wbb   label = wwb

Cal: balls = www    label = bbb

Don: balls = wwb   label = www

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They're out of sight but can they hear each other?

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1 hour ago, bonanova said:

After each man, at random, was given one of the boxes, they were given the following test. Each in turn, and out of sight of the others, was to blindly remove two of the balls from his box, read the label on his box, and then endeavor to tell to the others the color of the third ball, which remained in the box.

Yes. The restriction is only that they can't read each others' labels.

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