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351 logic/smullyan/stamps.p




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This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

351 logic/smullyan/stamps.p


The moderator 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 moderator's
pocket and the two on her own head. He asks them in turn
if they know the colors of their own stamps:
A: "No"
B: "No"
C: "No"
A: "No
B: "Yes"
What are the colors of her stamps, and what is the situation?

logic/smullyan/stamps.s

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.)

 

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