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Fri, 03 Jul 2015
The annoying boxes puzzle: solution
There are two boxes on a table, one red and one green. One contains a treasure. The red box is labelled "exactly one of the labels is true". The green box is labelled "the treasure is in this box."It's not too late to try to solve this before reading on. If you want, you can submit your answer here:
## Results
66.52% 300 red 25.72 116 not-enough-info 3.55 16 green 2.00 9 other 1.55 7 spam 0.44 2 red-with-qualification 0.22 1 attack 100.00 451 TOTAL One-quarter of respondents got
the right answer, that there is not enough information
given to solve the problem, Two-thirds of respondents said the
treasure was in the red box.
This is wrong. The treasure
is in the green box.
## What?Let me show you. I stated:
The labels are as I said. Everything I told you was literally true. The treasure is definitely not in the red box. No, it is actually in the green box. (It's hard to see, but one of the items in the green box is the gold and diamond ring made in Vienna by my great-grandfather, which is unquestionably a real treasure.) So if you said the treasure must be in the red box, you were simply mistaken. If you had a logical argument why the treasure had to be in the red box, your argument was fallacious, and you should pause and try to figure out what was wrong with it. I will discuss it in detail below.
## SolutionThe treasure is undeniably in the green box. However, correct answer to the puzzle is "no, you cannot figure out which box contains the treasure". There is not enough information given. (Notice that the question was not “Where is the treasure?” but “Can you figure out…?”)
## (Fallacious) Argument
Many people erroneously conclude that the treasure is in the red box,
using reasoning something like the following: |

Order What is the Name of this Book? with kickback no kickback |

Typically the truth or falsity of the labels
**is part of the puzzle conditions**. Here's a typical example,
which I took from Raymond Smullyan's What is the name of this
book? (problem 67a):

… She had the following inscriptions put on the caskets:Notice that the problem condition gives the suitor a certification about the truth of the labels, on which he may rely. In the quotation above, the certification is in boldface.

Gold Silver Lead THE PORTRAIT IS IN THIS CASKET THE PORTRAIT IS NOT IN THIS CASKET THE PORTRAIT IS NOT IN THE GOLD CASKET Portia explained to the suitor that of the three statements, at most one was true.Which casket should the suitor choose [to find the portrait]?

A well-constructed puzzle will always contain such a certification, something like “one label is true and one is false” or “on this island, each person always lies, or always tells the truth”. I went to What is the Name of this Book? to get the example above, and found more than I had bargained for: problem 70 is exactly the annoying boxes problem! Smullyan says:

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.(Page 65)

Had I known ahead of time that Smullyan had treated the exact same topic with the exact same example, I doubt I would have written this post at all.

One respondent referred me to a similar post on lesswrong.

I did warn you all that the puzzle was annoying.

I started writing this post in October 2007, and then it sat on the shelf until I got around to finding and photographing the boxes. A triumph of procrastination!

[ Addendum 20150911: Steven Mazie has written a blog article about this topic, A Logic Puzzle That Teaches a Life Lesson. ]

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