
The Hardest Logic Puzzle: A step-by-step guide to true, false, and random - dnetesn
http://nautil.us/issue/30/identity/how-to-solve-the-hardest-logic-puzzle-ever
======
throwaway000002
This is an aside, but somewhat similar in spirit.

In high school, ages ago, I remember discussing the following problem and
never being satisfied with the solution. If someone could clarify the exact
wording and source of the problem I'd appreciate it.

What I loosely remember is as follows. A teacher tells her students on a
Friday that they are to have a test next week. However, she adds, they won't
know the day before that they're going to have a test on the following day.

The students, being somewhat savvy, deduce that if it transpires that they
don't get tested by the end of the following Thursday then they'd have to be
tested on Friday, which would mean that they know this beforehand, so the test
cannot possibly be on Friday. And so, inductively, cannot be on Thursday, and
so on.

This "logic" is very unsatisfying, so I need to find out what the actual form
and solution of the problem is.

This has bugged me for decades.

~~~
thrwwy77911234
[https://en.m.wikipedia.org/wiki/Unexpected_hanging_paradox](https://en.m.wikipedia.org/wiki/Unexpected_hanging_paradox)

~~~
PeterWhittaker
And read this, too:
[http://www.ams.org/notices/201011/rtx101101454p.pdf](http://www.ams.org/notices/201011/rtx101101454p.pdf)

~~~
throwaway000002
Wow Peter, thanks for that. I like their analysis.

It reminded me of a great article I read years ago that dealt with the issue
of self-referential paradox. In fact, at the time I tried applying the
techniques therein towards the surprise examination, however I didn't get
anywhere.

It's a paper by Noson Yanofsky entitled "A Universal Approach to Self-
Referential Paradoxes, Incompleteness and Fixed Points"
([http://arxiv.org/abs/math/0305282](http://arxiv.org/abs/math/0305282))

I think you'll enjoy reading it. I read the paper on a long train ride loved
working out how given "Ann believes that Bob believes that Ann believes that
Bob has a false belief about Ann" asking "does Ann believe that Bob has a
false belief about Ann?" results in a paradox. This is supposedly called
Brandenburger’s Epistemic Paradox.

Anyway, thanks again for your reference.

~~~
PeterWhittaker
Thanks, downloaded. I have to read the paper I quoted a few more times before
I grok it fully. Perhaps it will be clearer after reading the one you cite,
which appears most intriguing.

Thanks again!

EDIT: Just started reading. Mind already blown by the clarity of the
"limitation, not paradox" idea. Slowing down for a longer, deeper, read.

------
n0us
The trick resides in how you can ask questions like "if and only if" which in
my opinion are really multiple questions masked as one. For example you can
formulate iff as "(not q or p) and (not p or q)" where the "not" refers only
to the term directly to its right. This is far from a simple assertion like
"is it true that if the earth is round then I am a knight?" because you are
also asking "is it true that if I am a knight then the earth is round" and
thus get the chance to ask two questions in one.

It's certainly still a tricky puzzle but you would need to understand the
rules on what kinds of questions you are allowed to ask before starting

~~~
shasta
Asking compound questions isn't the same as asking multiple questions because
you can only ever get one bit of information out of each question.

~~~
n0us
Not exactly because you can combine as many questions as you like and get an
answer about whether one of them or all of them is correct, furthermore if you
can ask question in first order logic, like "for all x is y true?" as opposed
to just "if x then y?" The wording would be come awkward very quickly but you
can gain an almost unlimited amount of information from compound questions. I
wouldn't be surprised if there were a solution to this problem involving only
two questions or even one, though at the moment I don't have the time to work
through it so don't take my word for it.

In binary terms, let's say you have one bit, 0 or 1 that represents a single
propositional statement, "x or y" or "x and y" but with compound questions you
can get as many bits as you want "(x or y) and (x and y) and (p and q)" so you
can only ever get the truth value of the entire compound statement but you get
far more than a single truth value

~~~
shasta
If you take an information theory point of view, then a single question could
provide more than a single bit of information, but it never will on average
(for some suitable meaning of "average"). If I'm speaking to 1 of 1024
possible people and I ask "Are this particular woman?" then 1 time out of 1024
I will learn 10 bits of information, but 1023 out of 1024 times I will learn
approximately zero bits of information.

------
Rangi42
Sam Hughes implemented this puzzle in Javascript to demonstrate how the
solution works, and why some other proposed ones don't:
[http://qntm.org/gods](http://qntm.org/gods)

~~~
aidenn0
Not impossible but makes it slightly harder to do IFF and "is exactly one of
these true"

I had formulated my yes/no questions when I worked through this as "Is exactly
one of these true: Ja means yes, X" Which makes Da yes and Ja no.

------
885895
>Suppose that you can’t remember whether Pluto is a dwarf planet, and you need
to find out by asking someone nearby—but you don’t know whether that person is
a knight or a knave. What’s the one yes-no question you can ask to figure out
whether Pluto is a dwarf planet?

>"Are you a knight if and only if Pluto is a dwarf planet?"

>If the person’s a knight and Pluto is a dwarf planet, then you get the answer
“yes,” since both statements on each side of if and only if are true, and
knights always speak truly.

I don't get how this follows. Surely, the person being a knight or a knave is
independent from whether or not Pluto is a dwarf planet so a real knight would
say "no" either way since the assumption is bad and a knave would thus say
"yes". Thereby, you have not determined whether or not Pluto is a dwarf planet
but instead whether the person is a knight or a knave.

Please explain my error.

~~~
anon4
It's easier if you see it as logical operations, rather than language. A iff B
has the same meaning as A == B, but with the added twist that !B negates the
answer in the example (knaves always lie). So let's take A = Pluto is a dwarf
planet; B = you're a knight, then "are you a knight if and only if Pluto is a
dwarf planet" has the formula (B && (A == B)) || !(!B && (A == B)), which
after a bit of simplifying you can see is the same as A == True || !(A ==
False), or A == True.

Very often logic formulas, when translated in English, sound absolutely
bonkers, not least because the words if, and, or have pretty loose meanings in
ordinary language.

------
FLengyel
I heard that Norman Steenrod was pacing back and forth in Fine Hall thinking
about this puzzle, until he solved it. This was before it was attributed to
Boolos, according to my sources. The author might have been Mark Kac, or it
might have been communicated to Steenrod through Mark Kac, I was told.

------
IIAOPSW
I thought you had to ask exactly one question to each god. This solution uses
god B twice.

~~~
burkaman
"Each question must be put to exactly one god" means you can't ask one
question to all of them and get three answers. One question, one answer.

------
jsprogrammer
The article presents a good technique for handling situations, but I can't
help but feel the solution breaks a rule. You are supposed to only be able to
ask a single question to each god, but a biconditional question is really two
different questions. You can see this simply from the structure of the English
question (as well as in the conversion to NAND logic):

> “Does ‘da’ mean ‘yes’ if _and_ only if you are True _and_ if _and_ only if B
> is Random?”

~~~
masterzora
While you can construct the answer to "Does 'da' mean 'yes' if and only if you
are True?" from "Does 'da' mean 'yes' if you are True" and "Does 'da' mean
'yes' only if you are True", you cannot construct the answers to "Does 'da'
mean 'yes' if you are True" and "Does 'da' mean 'yes' only if you are True"
from "Does 'da' mean 'yes' if and only if you are True?" alone.

Note, in particular, that at the end of the three questions in the solution
you still have no way of discerning whether 'da' does, in fact, mean 'yes'.

~~~
jsprogrammer
I'm not arguing that there is a solution that meets all of the constraints of
the problem, just that this particular solution does not seem to meet the
constraints described.

~~~
masterzora
I'm sorry if I was unclear but I did understand that and was responding to
that particular notion. I was attempting to demonstrate that the "and" does
not imply the question is now two (or more) questions. Perhaps I can be
clearer:

My primary point is that there is a difference between asking both the
question "A" and the question "B" vs. asking the question "A and B". It is
true that I can waste two questions asking both "A" and "B" to obtain the
answer to "A and B" so in that sense I can see why one might feel "A and B" is
two questions rather than one.

But really, the answer to "A and B" isn't really about the answers to "A" and
"B" as much as it is about the _relationship_ between them. If you ask "A and
B", an answer of "yes" will tell you they are both true but an answer of "no"
will not distinguish between whether A is false or B is false or both.

Alternatively, consider questions "A" and "B" again. A is either true or
false. B is also either true or false. You don't know the truth value of
either. How many questions does it take to determine the truth value of both
questions, assuming A and B are independent and the value of one doesn't
influence the value of the other? Well, you could ask "A and B" and if you get
"yes" you're done but that rides on you being lucky. In fact, there is no way
that you can differentiate all four possibilities with a single yes/no
response; you need at least two. This isn't really a _proof_ that "A and B" is
a single question, of course, but intuitively if "A and B" were, as you say,
"really two different questions" one would expect to be able to construct an
"A and B" that does differentiate four different possibilities.

Or, from another angle, per your prior post would you say "are you and the god
to your left both not Random?" is actually two different questions? How about
(ignoring for the moment that this is an entirely useless question in this
puzzle) "are you not Random and the god to your left not Random and the god to
your right not Random?" Or "Are none of the three of you Random?" I can see no
cause to say "Are none of the three of you Random?" (or, say, "does it rain
here every day?") is any more than one question nor any way to differentiate
this construct from the version using "and". I also see no reason why asking
"are you not Random and is the god to your left not Random?" would be any more
or fewer questions than "are you not Random and does 'da' mean 'yes'" or why
that would be any different from an "if and only if".

Of course, there is the final, if somewhat less satisfying, point to make: the
framing as a "question" is more for convenience and wider understanding but
the common intention for puzzles like this (especially with Smullyan credited
for the puzzle) is generally for you to choose a predicate and ask a god to
evaluate it for you, with the god possibly running the output through a not
gate before it gets to you.

~~~
jsprogrammer
You make fair points. Our disagreement seems to only be about the
interpretation of the rules.

>In fact, there is no way that you can differentiate all four possibilities
with a single yes/no response; you need at least two.

This is why I would say that the question is actually two questions. Two
separate truth values must first be produced, then combined with some operator
to make a third (and possibly combined again) value, which is the answer given
by the god.

If a god must parse your question down into individual propositions and then
answer them in some order to resolve a larger statement, it might stop after
the first proposition. I believe it's a valid interpretation of the rules
anyway.

------
dkarapetyan
The article mentions law of excluded middle which doesn't hold in constructive
logic so I wonder what exactly happens in the constructive setting.

~~~
masterzora
Well, the part where it directly mentions excluded middle is the part where it
cites Boolos commenting on just that.

~~~
dkarapetyan
That's not what I mean. What is the analog of the hardest puzzle in the
constructive setting.

