
The Hardest Logic Puzzle Ever - infinity
http://en.wikipedia.org/wiki/The_Hardest_Logic_Puzzle_Ever
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ambulatorybird
This reminds me of the following installment of a webcomic called "Partially
Clips":

<http://partiallyclips.com/2002/09/11/paradox-dragon/>

The webcomic version is easily solvable with counterfactuals (I think -- I
could be wrong).

~~~
Karzyn
I'm pretty sure that the webcomic is flawed.

In the last panel the center head says "Yes. I was lying." A head that always
tells the truth would never say this because they weren't lying before and
wouldn't lie about not lying. Conversely, a lying head would lie about lying
and say that they weren't lying. Thus, the center head must be the random one.
However, in the first panel both the right and the left heads agree that the
hero may only ask one question. Since the right and left are both either
truth-telling or a liar they cannot agree with each other.

Thus, the puzzle is flawed and probably just a joke.

~~~
epochwolf
Given that the comic has a punch line in the bottom of the last panel, the
comic is definitely a joke.

The question becomes is the author trying some subtle play or unaware of the
error? (Or are you incorrect about the comic being incorrect)

Either way, it was amusing and has value.

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tetha
I have to say, I have grown to dislike these 'logic puzzles'. I have read 'To
mock a mockingbird', and examined several of these puzzles and non of them are
actually hard (as in: challenging) once you realize: often you just need
simple things: double negation and massive case exhaustion (and especially the
latter here is just tedious and not interesting). Often, you just need the
case exhaustion.

~~~
tome
Your "case exhaustion" criticism disappears when you generalise the puzzle to
more than three gods:

[http://www.srcf.ucam.org/~te233/maths/puzzles/evenharder.htm...](http://www.srcf.ucam.org/~te233/maths/puzzles/evenharder.html)

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miloshh
I really like the trick of determining the identities of 3 gods with just two
yes/no questions. Simple, just ask questions that might make their heads
explode. :)

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Rantenki
Productivity denial of service attack for OCD sufferers.

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jpwagner
is it assumed they know each other's identities or not? (you know like the
gatekeeper puzzle: "what would he say if I asked him [question]...")

~~~
qeorge
Yes, as I read it. All of the solutions provided depend on it.

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dasht
That is a flawed puzzle. Warning, partial spoiler to the bogus solution
included here.

\------------- \-------------

The solution proposes that for the first question, I turn to god A and ask
"Hey, A, if I were to ask you 'Are you Random', would you say 'ja'?"

Either A is Random or is not - we have those two cases to consider. Let's
consider the case where A is, in fact, Random - only I don't know it yet.

The problem, as stated, says that since A is Random, for each question posed
to him, he "mentally flips a coin" and then answers as if he were Truth or as
if he were Falsity. So while we're considering the case that A is Random, we
have two sub-cases to consider: that he will answer as Truth or that he will
answer as Falsity. Let us consider the sub-case where A, although Random, will
answer as Truth.

Finally, we have sub-sub-cases more: perhaps 'ja' means "yes" or perhaps it
means "no". And so, we'll consider the sub-sub-case where 'ja' means "yes".

So A is Random, has chosen to answer my first question as Truth, and 'ya'
means yes. My question (in the bogus solution) is:

"Hey, A, if I were to ask (in your current mental state) 'Are you Random?'
would you say 'yes'?"

God A (who is Random and who will answer this one question as Truth) reasons
thusly. He thinks to himself:

"If this guy had asked me 'Are you Random', since I have decided to tell
Truth, I would answer 'yes' ('ja').' However, that was _not_ the question put
before me - a different question was put before me.

Random (that's me, A) flips the coin once per question asked. If he were to
have asked 'Are you Random' that would have entailed a coin flip separate from
the current coin flip. That other coin flip could have gone either way and so,
really, I might have answered 'yes' or 'no' ('ya' or 'da').

In other words, we made it clear to this guy that he was supposed to ask a
simple yes or no question and instead he asked a question to which the only
True answer would be 'dajadaja' [god-speak for 'maybe']."

At that point, god A would produce from his sleeve the Magic Wand of Heyting
(which was manufatured during the god-wars recorded in the book of
RejectingTheLawOfTheExcludedMiddle) and smite me for failing to ask a yes/no
question after being clearly instructed to do so.

In other words, with the proposed (bogus) answer, should A happen to be
Random, and his coin come up Truth, and should 'ja' happen to mean 'yes' --
then I will go to my grave never knowing which god is which. (Similar analysis
probably applies to other sub-sub-cases.)

Very badly framed puzzle (but one that would be hard to frame soundly without
giving away the intended answer).

~~~
jurjenh
I do believe the puzzle stated that the Random God would only flip his random
coin once for the session - ie he would lie for the 1-3 questions put to him,
or he would tell the truth - not flip the coin for each session.

This is in fact quite clearly described in the section _Random's Behaviour_
about a third down the page...

~~~
dasht
I'm not sure I agree that that's what Boolo's clarification says however, it
makes not a whit of difference to my analysis. For, if, indeed, "Hey A, are
you Random?" is _not_ one of my three questions, then it has no True answer
other than "maybe". Conversely, if that is one of my three questions, then the
bogus solution offered fails.

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Dove
Am I the only one disappointed that this link didn't go to Fermat's Last
Theorem?

~~~
jurjenh
Any of the Millenium Prize Problems could equally have been appropriate (with
the possible exception of the poincare conjecture), but most of these require
deep mathematical knowledge.

I would expect this problem to be accessible to most people with a basic grasp
of logic, hence the generic title.

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tome
I generalised this puzzle to any number of gods:

[http://www.srcf.ucam.org/~te233/maths/puzzles/evenharder.htm...](http://www.srcf.ucam.org/~te233/maths/puzzles/evenharder.html)

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nazgulnarsil
doesn't require very large inferential jumps. just formulate a hypothesis that
postulates a unique solution and test it.

i'd say the average code base works your brain harder.

~~~
miloshh
You have only 3 questions. If you formulate a hypothesis like "A is True, B is
False and C is Random", with a naive approach you will waste much more than 3
questions just to test this single hypothesis (and statistically, it will be
false, so you will need to test more).

~~~
Groxx
I think there's also the implied "or they will kill you" to encourage a non-
brute-force method. While this one in particular doesn't mention any death
threats, most problems of this kind do. I mean, why else limit yourself to
their constraints, if not to preserve life / sanity / coffee / beer?

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khafra
I believe the hardest logic test ever actually goes more like this:

There are four gods:

One always tells the truth.

One always lies.

One always chooses a random answer.

One always stabs people for trying to be clever or asking tricky questions.

