

The annoying boxes puzzle: solution - colinprince
http://blog.plover.com/math/logic/annoying-boxes-solution.html

======
ChrisLomont
That is not a logic puzzle [1], even though the author claims it is; it's a
silly outcome that is easily achieved when one breaks how logic puzzles are
presented and solved.

It's more of a bait-and-switch.

Here is a famous logic puzzle, often called a Knights and Knaves style puzzle
[2]: Knights always tell the truth, Knaves always lie.

"John and Bill are standing at a fork in the road. John is standing in front
of the left road, and Bill is standing in front of the right road. One of them
is a knight and the other a knave, but you don't know which. You also know
that one road leads to Death, and the other leads to Freedom. By asking one
yes–no question, can you determine the road to Freedom?"

Here's a "solution" in the style of the author: "There is no solution, because
the road to 'Freedom' is under construction, and the detour leads through
'Death.' Hahaha! Gotcha!"

It's a non-solution, and not a logic puzzle any more.

[1]
[https://en.wikipedia.org/wiki/Logic_puzzle](https://en.wikipedia.org/wiki/Logic_puzzle)
[2]
[https://en.wikipedia.org/wiki/Knights_and_Knaves](https://en.wikipedia.org/wiki/Knights_and_Knaves)

~~~
heinrich5991
But the difference is that in this puzzle you can deduce that something is
wrong by just using logic: In the absence of any hints, you'd just consider
everything true. This obviously does not lead to a solution, as the red box
states that one box lies. When you come to this point, you might wonder why
you were trusting the boxes in the first place and reconsider your
assumptions, i.e. not putting any trust in what the boxes say. After this step
you realise that there suddenly is not enough information to determine which
box the treasure is in.

The difference between the blog post and your example is that in your example
there's nothing that makes you recheck your assumptions. A puzzle in the style
of the author's would be:

Knights always tell the truth, Knaves always lie.

John and Bill are standing at a fork in the road. John is standing in front of
the left road, and Bill is standing in front of the right road. John says:
"One of us is a knight and the other a knave". You also know that one road
leads to Death, and the other leads to Freedom. Bill says: "My road leads to
freedom".

~~~
ajanuary
If you attacked that formulation with the normal assumptions for logic
problems, the answer would be the same. If you break the assumption that you
can trust all logical statements in the problem, regardless of their origin,
then you could get a solution like "John is a dick and tells a mixture of
truth and lies as he sees fit just to mess with you". But you could get that
solution with any problem where the source of information is an entity in the
problem itself, which is why we tend to assume that isn't the case.

~~~
peteretep
That's entirely wrong. The whole point of the article is that there are two
distinct types of information present in all problems of this type:
information _about_ the problem (which the author calls certification, and
which is assumed to be true), and then information presented in the problem,
which - in almost every problem I've ever seen of this type - has statements
that aren't true in it.

    
    
        > "If you break the assumption that you can trust all
        > logical statements in the problem"
    

That is never the assumption - you are normally told explicitly that you can
trust some and not others.

------
slavik81
This is secretly a lesson about comments in legacy code.

~~~
qznc
You just transformed the puzzle from "annoying" to "can teach a worthwhile
lesson" for me.

Edit: This message can be made more clear with a slight change to the puzzle.

There are two methods, foo and bar. One returns what you desire.

    
    
        /** Exactly one of the comments is true */
        function foo();
    
        /** @returns what you desire */
        function bar();
    

Can you figure out which method returns what you desire?

------
asQuirreL
Imagine oneself playing a game of Simon Says, where we don't know who Simon is
or which bits he's actually saying, he in fact sounds exactly like everyone
else giving us instructions: we can choose to believe what we want about the
actions we are told to make, but an adversarial Simon can always contradict
us. It's not a game I'd enjoy playing nor a lesson I think I benefited from
very much...

------
msie
Ugh, it is annoying. And the author is a bit of jerk (Read it and you'll see
what I mean.)

~~~
spacehome
I think he did me a favor.

------
JadeNB
What bothers me about mjd's explanation of "Why doesn't every logic puzzle
fall afoul of this problem" is:

> > Portia explained to the suitor that of the three statements, at most one
> was true.

> Notice that the problem condition gives the suitor a certification about the
> truth of the labels, on which he may rely.

(the first sentence is quoted, approvingly, from a puzzle by Gardner, and the
second explains why it saves that puzzle). Sure, there is a certificate
_provided in the problem_ ; but why may we trust Portia's certification, any
more than we may trust the labels on the boxes? It seems that one needs either

\- an annoying intra-textual infinite regress: "Portia said this, and
Balthazar said that Portia was telling the truth, and Stephano said that
Balthazar was telling the truth, …" (which still doesn't really address the
trust problem, just moves it infinitely far); or

\- a _meta_ -textual reassurance: "Given that at most one of the three
statements is true, which of the caskets should the suitor choose?"

(EDIT: Oops, mdpopescu
([https://news.ycombinator.com/item?id=9837257](https://news.ycombinator.com/item?id=9837257))
made this point much more succinctly several hours ago.)

~~~
Ended
I think you are right, and the solution is a meta-textual reassurance,
specifically of the form: " _if_ one of the statements is true, _then_ which
casket should the suitor choose?"

~~~
JadeNB
I agree, although note that Gardner actually sneaks in the subtly different
statement "at _most_ one of the statements is true." Of course, even here one
has to be careful: do we require that the problem say "if at most one of the
statements is true, _and the previous description of the problem is correct_ ,
then …"? If not, what allows us to trust the previous description?

One can re-phrase every logic puzzle ever as "Assuming that `P` is true, what
can you deduce about `Q`?"; or one may assume that a logic puzzle of the form
"`P` is true. What can you deduce about `Q`?" is really shorthand for the
former. I don't begrudge mjd for pointing out this shorthand, but I do think
that his description of the 'solution' (to the meta-problem, not the actual
puzzle he poses) is facile at best, and creates bigger problems at worst.

------
shultays
So, how could there be enough info to solve any problem if we can't trust the
information at all? I mean bot boxes could be open, with the treasure being in
green box at all but it could be all dream! The treasure was in red box all
the time. Kinda stupid.

~~~
slavik81
The problem inherently indicated that the labels were untrustworthy. It was
impossible for them both to be true, because one explicitly stated that only
one was true. I see no reason to trust the red label more than the green
label, so at that point, it should be clear that neither can be trusted.

It's interesting in that people seem to trust meta-statements like the red
label's more than they trust basic statements like the green label's.

~~~
chronial
> It's interesting in that people seem to trust meta-statements like the red
> label's more than they trust basic statements like the green label's.

That are the rules of puzzles.

~~~
function_seven
Only when such statements are made by the narrator, not the puzzle artifacts
themselves. The whole point of these sorts of puzzles is that the labels can't
all be relied upon, yet some people think the red label can be, for some
reason.

~~~
JadeNB
The problem is that mjd himself, when discussing "Why don't all logic puzzles
fall afoul of this problem", says that the statement

> Portia explained to the suitor that of the three statements, at most one was
> true.

in a Gardner puzzle is evidence enough that we may trust that "of the three
statements, at most one was true". Here, Portia is a character in the puzzle;
why should we trust her any more than we should labels on boxes or (in the
set-up of the puzzle) statements on caskets? If we have to distrust
_everything_ about the problem, then it seems to me that it runs into exactly
the problem shultays
([https://news.ycombinator.com/item?id=9837381](https://news.ycombinator.com/item?id=9837381))
mentions.

(EDIT: I think that mdpopescu
([https://news.ycombinator.com/item?id=9837257](https://news.ycombinator.com/item?id=9837257))
said it better.)

------
bottled_poe
The actual scenario he presents with the physical boxes is not consistent, as
the truth value of the label on the red box cannot be decided. i.e. Its value
is the opposite of its value. This is a troll question.

~~~
phaemon
Of course it's consistent: it actually exists.

There is no paradox. It's clearly obvious that you can choose either box to
put the treasure in. It's also clearly obvious that sticking labels on the
boxes doesn't alter that, _no matter what they say_.

You just got fooled by the puzzle. It's not a big deal.

~~~
bottled_poe
Sorry for assuming the problem would adhere to standard expectations. By
breaking the typical rules of a logic puzzle, the interesting aspects are
replaced with groan-worthy ones.

~~~
TuringTest
That's true only if your interest is limited to the mechanical rules of logic
derivation. For those of us interested in the relations between symbols and
meaning, the puzzle is fascinating.

> Sorry for assuming the problem would adhere to standard expectations. That
> was the solution to the puzzle - realizing that this was an invalid
> assumption, given the formulation.

------
AgentME
tl;dr: The problem didn't actually state that the labels mean anything!

~~~
mdpopescu
Yes. Normally, problems give you information so that you can do something with
it. Otherwise, even if the author says later on:

 _Notice that the problem condition gives the suitor a certification about the
truth of the labels_

the problem author can always cop out and say "yes, but the certification
itself was meaningless!".

If I take the labels to their face value, the treasure cannot be in the green
box because the red label doesn't have a truth value. (If it's true, then
there are _two_ true labels, therefore it must be false. But if it's false,
there is only one true label, so the red one is true.)

~~~
myhf
It's more interesting than that. If you take the labels at their face value,
evaluating the red label either causes itself to be correct (flip-flop), or
causes itself to be incorrect (#REF!), based on the state of the world.

    
    
        treasure in green | treasure in red | green label   | red label
        ------------------+-----------------+---------------+-----------
        no                | no              | false         | flip-flop
        no                | yes             | false         | flip-flop
        yes               | no              | true          | #REF!
        yes               | yes             | misquantified | #REF!

~~~
linhchi
i'm so confused. the red label doesnt convey any information yet it rules out
the ability that the green label is true.

so how can the green label still be true?

or is the author saying that the symbols on labels dont mean anything? if he
says that, it's tautological. what can we get from discussion if we dont
conventionally put meaning in symbols? are what we typing now in all these
comments also jibberish??

~~~
anatoly
The contents of the green box make the labels paradoxical; but it's OK because
paradoxes exist.

Consider a paradox such as "This sentence is false". There's a problem with
deciding if it's true or false. Countless articles and books have been written
about it, and there exist different ways of dealing with this "difficulty".
However, nothing prevents you from actually writing these letters on a piece
of paper, and nothing prevented me from typing them a few seconds ago. The
puzzle we're talking about makes the same point in a more amusing fashion.

~~~
linhchi
hm, isnt that obvious? if it's just "we do whatever we want regardless of
whatever", then we dont need to go that far: elaborating a nonsense puzzle to
prove nonsense.

------
ajanuary
With logic problems the assumption is that logical information is always
relevant to the solution, no matter what the source of the information is.
It's interesting to have that assumption challenged though.

[https://www.youtube.com/watch?v=emiMj8cCL5E](https://www.youtube.com/watch?v=emiMj8cCL5E)
and the extra footage is another demonstration on how the solution can change
if you don't start with the normal "logic puzzle" assumptions.

~~~
blahblah3
the difference here is that the label on the green box is explicitly
"challenged" by the label of the red box. the step is to realize that both
labels should be challenged (i.e if the label on the green box doesn't have to
be true, the label on the red box doesn't have to be true)

~~~
ajanuary
I'm not convinced the fact one label "challenges" the other should change how
you attack the problem. I think you have to be able to apply the same
reasoning to the following problem:

    
    
        James says he is shorter than Alex.
        Jane says she is shorter than James.
        Alex says he is taller than James.
        Can you figure out who is the shortest?
    

Going by the authors premise that only things in the puzzle statement can be
taken to be truthful, you can't figure out who is the shortest because they
could all be lying. It's formulated exactly the same as the problem in the
post, it just doesn't have a potential contradiction to lead you down the path
to realising you're not meant to be assuming the logical statements are
involved in the solution.

~~~
TuringTest
If one of the possible answers is "There's not enough information to decide",
that's a good hint that something weird may be going on and you should double-
check your assumption of "everybody is telling the truth".

------
prof_hobart
I think the problem I'm having with it is that it's presented as a logic
puzzle, but I'm not sure I'd classify it that way.

It seems more of a comprehension test to me, as the trick is both to see that
the questions is a "Can you figure out the answer?" rather than "What is the
answer?" and to realise that at no point does he say that either label is
correct.

~~~
peteretep
If you've ever done any GMAT-type questions, this is an absolutely classic
example of a data-sufficiency question.

------
brador
There is the implication, since the question was asked, that enough
information is presented to get to a solution. Else, why would the question be
asked, unless to fool. In which case, he did a fancy equivalent of "it' a
prank it's a prank".

~~~
PhantomGremlin
_There is the implication ... that enough information is presented to get to a
solution_

In general I agree with you.

HOWEVER, in this case, one of the choices is explicitly "There is not enough
information to determine the answer". Given that choice is explicitly
presented, it's not a prank. IMO.

~~~
JadeNB
> HOWEVER, in this case, one of the choices is explicitly "There is not enough
> information to determine the answer". Given that choice is explicitly
> presented, it's not a prank. IMO.

That's an interesting perspective, but can it really be that the correct
answer to the question depends on the options one is given? That is, would the
same puzzle, with that choice deleted, have a different answer?

------
blahblah3
haha indeed an "annoying" trick. we're so conditioned to reading phrases like
"exactly one of the labels is true" as meta-information that we forget it's a
"box label".

