The Universe of Discourse


Fri, 03 Jul 2015

The annoying boxes puzzle: solution
I presented this logic puzzle on Wednesday:

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."

Can you figure out which box contains the treasure?

It's not too late to try to solve this before reading on. If you want, you can submit your answer here:

The treasure is in the red box
The treasure is in the green box
There is not enough information to determine the answer
Something else:

Results

There were 506 total responses up to Fri Jul 3 11:09:52 2015 UTC; I kept only the first response from each IP address, leaving 451. I read all the "something else" submissions and where it seemed clear I recoded them as votes for "red", for "not enough information", or as spam. (Several people had the right answer but submitted "other" so they could explain themselves.) There was also one post attempted to attack my (nonexistent) SQL database. Sorry, Charlie; I'm not as stupid as I look.

	 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:

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."

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.

Solution

The 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 A

Many people erroneously conclude that the treasure is in the red box, using reasoning something like the following:

  1. Suppose the red label is true. Then exactly one label is true, and since the red label is true, the green label is false. Since it says that the treasure is in the green box, the treasure must really be in the red box.
  2. Now suppose that the red label is false. Then the green label must also be false. So again, the treasure is in the red box.
  3. Since both cases lead to the conclusion that the treasure is in the red box, that must be where it is.

What's wrong with argument A?

Here are some responses people commonly have when I tell them that argument A is fallacious:

"If the treasure is in the green box, the red label is lying."

Not quite, but argument A explicitly considers the possibility that the red label was false, so what's the problem?

"If the treasure is in the green box, the red label is inconsistent."

It could be. Nothing in the puzzle statement ruled this out. But actually it's not inconsistent, it's just irrelevant.

"If the treasure is in the green box, the red label is meaningless."

Nonsense. The meaning is plain: it says “exactly one of these labels is true”, and the meaning is that exactly one of the labels is true. Anyone presenting argument A must have understood the label to mean that, and it is incoherent to understand it that way and then to turn around and say that it is meaningless! (I discussed this point in more detail in 2007.)

"But the treasure could have been in the red box."

True! But it is not, as you can see in the pictures. The puzzle does not give enough information to solve the problem. If you said that there was not enough information, then congratulations, you have the right answer. The answer produced by argument A is incontestably wrong, since it asserts that the treasure is in the red box, when it is not.

"The conditions supplied by the puzzle statement are inconsistent."

They certainly are not. Inconsistent systems do not have models, and in particular cannot exist in the real world. The photographs above demonstrate a real-world model that satisfies every condition posed by the puzzle, and so proves that it is consistent.

"But that's not fair! You could have made up any random garbage at all, and then told me afterwards that you had been lying."

Had I done that, it would have been an unfair puzzle. For example, suppose I opened the boxes at the end to reveal that there was no treasure at all. That would have directly contradicted my assertion that "One [box] contains a treasure". That would have been cheating, and I would deserve a kick in the ass.

But I did not do that. As the photograph shows, the boxes, their colors, their labels, and the disposition of the treasure are all exactly as I said. I did not make up a lie to trick you; I described a real situation, and asked whether people they could diagnose the location of the treasure.

(Two respondents accused me of making up lies. One said:

There is no treasure. Both labels are lying. Look at those boxes. Do you really think someone put a treasure in one of them just for this logic puzzle?
What can I say? I did put a treasure in a box just for this logic puzzle. Some of us just have higher standards.)

"But what about the labels?"

Indeed! What about the labels?

The labels are worthless

The labels are red herrings; the provide no information. Consider the following version of the puzzle:

There are two boxes on a table, one red and one green. One contains a treasure.

Which box contains the treasure?

Obviously, the problem cannot be solved from the information given.

Now consider this version:

There are two boxes on a table, one red and one green. One contains a treasure. The red box is labelled "gcoadd atniy fnck z fbi c rirpx hrfyrom". The green box is labelled "ofurb rz bzbsgtuuocxl ckddwdfiwzjwe ydtd."

Which box contains the treasure?

One is similarly at a loss here.

(By the way, people who said one label was meaningless: this is what a meaningless label looks like.)

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."

But then the janitor happens by. "Don't be confused by those labels," he says. "They were stuck on there by the previous owner of the boxes, who was an illiterate shoemaker who only spoke Serbian. I think he cut them out of a magazine because he liked the frilly borders."

Which box contains the treasure?

The point being that in the absence of additional information, there is no reason to believe that the labels give any information about the contents of the boxes, or about labels, or about anything at all. This should not come as a surprise to anyone. It is true not just in annoying puzzles, but in the world in general. A box labeled “fresh figs” might contain fresh figs, or spoiled figs, or angry hornets, or nothing at all.
Order
What is the Name of this Book?
What is the Name of this Book?
with kickback
no kickback

Why doesn't every logic puzzle fall afoul of this problem?

I said as part of the puzzle conditions that there was a treasure in one box. For a fair puzzle, I am required to tell the truth about the puzzle conditions. Otherwise I'm just being a jerk.

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:
GoldSilverLead
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]?

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.

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.

But why is this so surprising?

I don't know.

Final notes

16 people correctly said that the treasure was in the green box. This has to be counted as a lucky guess, unacceptable as a solution to a logic puzzle.

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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