## Friday, January 15, 2010

### Stamp Colors

The Grand Master takes a set of 8 stamps, 4 red and 4 green, known to the logicians, and loosely affixes two to the forehead of each logician so that each logician can see all the other stamps except those 2 in the Grand Master's pocket and the two on her own forehead. He asks them in turn if they know the colors of their own stamps:

A: "No."
B: "No."
C: "No."

Again, he asks them in turn if they know the colors of their own stamps.
A: "No."
B: "Yes."

What color stamps does B have?

1. B has 1 red, 1 green.

To work, the other two must have either:

A=GG, C=RR
or
A=RR, C=GG

Taking the first case:
-B sees A=GG, C=RR
-When A doesn't know what she has, it means B can't possibly have RR, or A would know.
-On B's first turn, she still isn't sure if she has GG or RG.
-When C doesn't know what she has, it means B can't possibly have GG, or C would know.
-So B then knows she must have one of each, RG.

The same logic works for A=RR, C=GG.

2. Bravo, Andy.

3. B says: "Suppose I have red-red. A would have said on her second turn: 'I see that B has red-red. If I also have red-red, then all four reds would be used, and C would have realized that she had green-green. But C didn't, so I don't have red-red. Suppose I have green-green. In that case, C would have realized that if she had red-red, I would have seen four reds and I would have answered that I had green-green on my first turn. On the other hand, if she also has green-green [we assume that A can see C; this line is only for completeness], then B would have seen four greens and she would have answered that she had two reds. So C would have realized that, if I have green-green and B has red-red, and if neither of us answered on our first turn, then she must have green-red.
"'But she didn't. So I can't have green-green either, and if I can't have green-green or red-red, then I must have green-red.'
So B continues:
"But she (A) didn't say that she had green-red, so the supposition that I have red-red must be wrong. And as my logic applies to green-green as well, then I must have green-red."
So B had green-red, and we don't know the distribution of the others certainly.

(Actually, it is possible to take the last step first, and deduce that the person who answered YES must have a solution which would work if the greens and reds were switched -- red-green.)

4. Oops, forgot to add my 'bravo' for Andy, too. Your explanation was a more succinct than mine.

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