There are two boxes, one red, one green. There is a treasure in one of them.
The green box has a label that says "Exactly one of the labels is false."
The red box has a label that says "The treasure is in this box."
Where is the treasure?
(Spoilers below.)
Solution and analysis: (20011202)
Most people conclude that the treasure is in the green box, often
reasoning as follows: Suppose the green label is true. Then the red
box must be false, so the treasure is in the green box. Now suppose
the green label is false. Then the red label must also be false, so
the treasure is again in the green box.
This is incorrect. The treasure is in the red box.
However, there is not enough information given in the problem to
determine that. The only way to know the answer is to look in the
boxes.
Normally, box-and-label puzzles contain, as part of the problem
statement, an affidavit about the truth or falsity of the labels. For
example, in Raymond Smullyan's _What is the Name of this Book_,
problem 67a:
Portia explained to the suitor that of the three statements,
at most one was true. Which casket should the suitor choose?
The problem statement above contains no such affidavits. In the
absence of such, *no* statements about the labels on the boxes could
possibly have been sufficient to determine the contents of the boxes,
because there is no reason to believe that the labels have anything to
do with the contents. Today I got a package in the mail that had the
label "Land's End Women's Lt. Brown Slippers"; upon opening it, I
discovered that it contained a toy octopus.
People sometimes complain that the stated solution (treasure in the
red box) entails a logical contradiction. This is not so. The proof
is that I have red and green boxes in my office, with labels as
described, and my grandfather's silver and lapis ring is in the red
box. None of the conditions of the problem statement are contradicted
by this model, and since observed fact can't contain a logical
contradiction, this proves that there is no contradiction. Since the
ring is in the red box, the label on the red box is true, and the
label on the green box is neither true nor false.
People sometimes complain that the puzzle is a trick. I agree that it
is a trick, but I deny that it was an unfair trick. A *careful*
analysis would have revealed that the treasure could be in either box
without contradicting the stated conditions. People sometimes object
that the labels are lying, but since they inevitably do this
immediately after considering the possibility that the labels are
lies, they shouldn't then complain that they were unfairly surprised.
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Post Script added 20020110: I dug up my copy of _What is the Name of
This Book_ because I wanted to show that carefully constructed logic
puzzles contain affidavits about the truth or falsity of the
statements on the boxes. I knew that Smullyan was very careful and
would include such affidavits, so I looked to him for an example, and
as you see above, I was not disappointed.
But Smullyan did even better. After I had hunted up my copy of the
book, I reread the section that deals with labeled box puzzles.
(Chapter 5, "The Mystery of Portia's Caskets".) And I found to my
delight that problem #47 is almost exactly the same as the box puzzle I
gave above: The caskets are gold and silver; the silver one is
labelled "Exactly one of these two statments is true" and the gold one
is labelled "The portrait is not in here."
So the suitor triumphantly exclaimed, "The portrait must be in
the gold casket" and opened the lid. To his utter horror the
gold casket was empty! . . . Now, what on earth was wrong
with the suitor's reasoning.
Smullyan explains the solution more clearly than I was able to:
The suitor should have reaslized that without any information
given about the truth or falsity of any of the sentences, nor
any information given about the relation of their
truth-values, the sentences could say anything, and the object
could be anywhere. Good heavens, I can take any number of
caskets that I please and put an object in one of them and
then write any inscriptions at all on the lids; these
sentences won't convey any information whatsoever.
Smullyan goes on at some length explaining the error, and the
discussion is very much worth reading. (ISBN 0-13-955063-3.)