# The hardest logical puzzle according to George Boolos

PhiloLearner
According to the late philosopher and logician George Boolos, who was a professor at MIT, the hardest logical puzzle is one that was invented by the logician and puzzle-master Raymond Smullyan, and modified slightly by the computer scientist John McCarthy. Here is the puzzle :

Three gods A , B , and C are called, in some order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter.

Your task is to determine the identities of A , B , and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer in their own language, in which the words for yes and no are “da” and “ja”, in some order. You do not know which word means which.

mapper
In computer programming we use this all the time. IIF statements. Ask questions If god a = true when god b = condition etc..

lueffy
Uhm, i believe I've seen this puzzle before in this brain teasers forum...........

PhiloLearner
lueffy said:
Uhm, i believe I've seen this puzzle before in this brain teasers forum...........

Similar subject titles but different contents

Moo Of Doom
PhiloLearner said:
Similar subject titles but different contents

No, I am certain I have read this exact problem. In fact, I remember it to have had a whole separate thread for the answer. The only difference between it and this one is the yes and no words were changed slightly.

wisredz
My first question would surely be are you a god? :) If I am lucky I learn two things :)

Moo Of Doom
How can you learn anything if it depends on luck? If the god says "Da" what does that tell you? If A=True Then Da=yes. If A=False Then Da=no. If A=Random Then Da=yes or no. Not much information there...

We definitely had this question posted before. The answer has to do with some technicality of "iff".

PhiloLearner

http://hcs.harvard.edu/~hrp/issues/1996/Boolos.pdf

Mr. dude
well da means yes in Russian and Ja means yes in German so we have a couple funked up gods here.

delton
WOW this is tough... but since I know it's solvable mayby I'll try to take a crack at it when I'm bored....

Your first move is to find a god who you can be certain is not Random, and
hence is either True or False. (so he claims he finds the god (which is not random) with one question..LoL!)

To do so, turn to A and ask Question 1: Does da mean yes iff,you are True iff,B is Random?

(Assume that Da means no,A is true god and b is random. A's answer would be Da..

Now Assume that Da means no, A is False God and B is random.
Da means no but While A is false god he will think of the first statement;

- Da means yes? as True
- You are True God? as True
- B is random? as False

He will end up with a False, therefore will answer with a True which is "Ja" )

If A is True or False and you get the answer da(According to what i showed you this is megabull****), then as we have seen,
B is Random(Really? Why is that?), and therefore C is either True or False; but if A is True or False and you get the answer ja, then B is not Random, therefore B is either True or False.But what if A is Random?

(How the **** do you make logical conclusions? you have no idea if "da" is yes or no; Let alone if A is random)

If A is Random, then neither B nor C is Random! (Really? No ****!... Now tell me how did that help you?)

So if A is Random and you get the answer da, C is not Random neither is B,
but that's irrelevant(You said that before and i said "Really? No ****!...), and therefore C is either True or False; and if A is Random and you get the answer ja, B is not random neither is C, irrelevantly, and therefore B is either True or False.

Thus, no matter whether A is True, False, or Random, if you get the answer da to Question 1, C is either True or False, and if you get the answer ja, B is either True or False!

(sorry but this answer is total bull****, only confusing people it is no answer..

Add: Above you can see A gave the answer "Ja" while B was Random..As i said this answer is just to confuse people it is bull****))

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whozum
Thus, no matter whether A is True, False, or Random, if you get the answer da to Question 1, C is either True or False, and if you get the answer ja, B is either True or False!

Shocking conclusion, theyre either true.... or false. :surprised

gravenewworld
The answer is already in the forums. I posted the Q and A a while ago...and yes the solution is very difficult.

gurkhawarhorse
:tongue: it goes like this.
ask A="hey, A if i ask B a question ="do u speak truth ?" what will B say.
ask B again="hey , B if i ask C a question ="do u speak truth ?" what will C say.
ask C again="hey , C if i ask C a question ="do u speak truth ?" what will C say.

now u will get the god answer. whoever says no either is random or false speaker but not truthful.
the truthful god will say yes to the question.
false one says no.random one either says yes or no.

but u do not know what do those gods mean do u; language problem? so we have a trick. i can assure u that two gods answer same words and one is the odd one. so u get a pair.one thing is that the god who speaks lie and the one who is truthful do not have the same answer. ie they don't couple up.
see, whoever is the odd one out in the answer ie. whose answer does not match with others either says truth or lies.

got it? so u get the god who is random. now again compare their answers but exclude the random god. now u have either false god caught or truthful then catch the next one. :tongue2:
gurkha-war-horse

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LarrrSDonald
Hmm, that was confusing enough.. I explained this problem once to someone or the other, so even though it may be beat into the ground I'll go ahead and post it on the off chance that someone cares - at least the person I wrote it for seemed entirely satisfied with it.

I'll call the gods T, F and R and their postitions A, B and C.

First, we need to deal with the language barrier. We do this by tacking on an "iff da means yes", "iff" meaning "if and only if" i.e. true if both are true or both are false, false otherwise. Thus, when asked "X iff da means yes", T would answer:

X is true:
Da means yes: da (both true, so da as in yes)
Da means no: da (one true, one false, hence da as in no. Meanings switched though, see)
X is false:
Da means yes: ja (one true one false, da as in yes)
Da means no: ja (both false, ja as in yes)

As you notice, if the god answers da, same as in the iff statement, then that means yes, ja means no. Da and ja could be switched in the entire above agrument, of course. You can now consider this abstract, a black box if you will, you no longer need to worry about the language barrier. Just tack on "iff da means yes" and compare your answer per above.

Second, we have F, who lies. We get past this the same way by adding "iff you are T". For "X iff you are T":

X is true:
T says: true (both true)
F says: true (one is true (X) and one is false (being T), thus the honest answer is false. However, F lies, so true)
X is false:
T says: false (one false one true)
F says: false (both are false, but again, he lies and says false)

With the language barrier they will obviously report their "true" and "false" via that, but you cleaverly don't care since you're already past that.

Thus, asking T or F "X iff you are T iff da means yes" will give you a da on true and a ja on false, regardless of what da/ja means and regardless of it it's T or F talking. Again, you can now quit worrying about this. If you can't, reread it until you can before moving on - because you won't want to worry about it later.

Ask A, per above format, "Is B R iff da means yes iff you are T?". If A is T or F, they will give you a solid answer, da if B is R, ja if C is R. Either way, you have now found a non-R god, which you can procceed with. If A is in fact R, he will randomly and sneakily finger B or C as R. It doesn't really matter, however, because *neither* of them are (on account of the whole A being R business) so which ever one he manages to trick you into will in fact be fine. So long as you don't draw any conclusions about A (we still have no clue who he is) or anything else, you're fine.

For your second question, ask the god that you have now decided isn't R something you know, dropping the "iff you are T" but still using the "iff da means yes", for instance "Do I exist iff da means yes?". The language barrier killer is still in effect, so you'll get da for yes, ja for no regardless of their actual meaning, but the T/F filter is gone. Thus, Da means you are speaking to T, Ja means you're speaking to F.

Last question, ask about one of the other two "Is A R iff da menas yes?" for instance. You can tack on the "iff you are T" if you wish (why should us humans do all the work) or just reverse the answer if your speaking to F. Either way, you now know the identity of A, and by elimination the other of B and C (depending on whom you've already managed to secure earlier).

In case the question "Is B R iff da means yes iff you are T?" bothers you, I will go ahead and truth table it. I'll call the answer W and "da means yes" Y. R gets extra lines because he can give two answers.

Y A B C W Implicated as not-R
t T F R ja B
t T R F da C
t F T R ja B
t F R T da C

t R T F da B
t R T F ja C
t R F T da B
t R F T ja C

f T F R ja B
f T R F da C
f F T R ja B
f F R T da C

f R T F da B
f R T F ja C
f R F T da B
f R F T ja C

Those are the only 16 possible routes (6 god posistions w/ R giving two possible answers in the two cases where he speaks, two options for da/ja), nothing else can possibly happen. As you can see, in no case is someone implicated as not being R without, in fact, not being R. It's mostly downhill from there, just check which god you're speaking to, then find out about the other two.

[EDIT] Realized I should probably white text it since it's kind of a spoiler if you'd like to solve it. Changed the truth table to Courier, forgot that the first time around.

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gurkhawarhorse
LarrrSDonald said:
Hmm, that was confusing enough..

NOO!!! mommy
my aint confusing... :grumpy: :uhh: :grumpy:

gurkha-war-horse

LarrrSDonald
<shurgs> You're welcome to chalk it up to stupidity on my part if you wish, I don't mind. You do have a point, it should be possible to avoid (or rather perhaps burry) the "iff you are True" thing in "If I were to as B about X, what would he say?". I considered the da/ja pair thing (if you were to have two da and one ja, the ja has to come from not-random), but with only three questions I can't see it being viable, you have to work it into the previous statements to conserve. I have no immediate desire to work out a version that hides the iff, but I certainly feels like it'd be possible.

PhilosophyofPhysics
I am so illogical.

Xargoth
Lars ;

X is true:
Da means yes: da (both true, so da as in yes)
Da means no: da (one true, one false, hence da as in no. Meanings switched though, see)
X is false:
Da means yes: ja (one true one false, da as in yes)
Da means no: ja (both false, ja as in yes)

Please include X is random, because X can be the random one totally ruining your claim..

lawtonfogle
I answered this one a while ago. It is not that hard. Basic princible, two of the gods can each be asked one question they cannot answer. Now the two questions are different, so you need to be careful how you ask them. It is possible, just differently per which one you ask.

LarrrSDonald
If X is random, the answer is random but this is delt with later. You're right in that the two first sections only deal with T and F, however the random case is delt with by using a god to choose between the other two thus choosing one of T and F at random if the questioned god is R. This has been discussed at length elsewhere, I only posted this because I had it pre-written anyhow. I took it as obvious that "iff da is yes" and "iff you are T" only work for T and F.

GOsuchessplayer
Does anyone know

Does anyone know wether they can respond "da or ja" in stead of just saying either or such as "da".??

GOsuchessplayer
My answe

Assuming that if the gods are asked a question that they can't answer with one word then the following answer is a solution..

Ask god A doesn't really matter which.. What would B have said if he asked C if "Da" meant yes would he have responded "Da"??

Then you must ask another god B or C the same question but with letters different obviously.

This will auto eliminate your random since The truth and the false must either A. Not answer or B. Respond "da or ja". Where as the random will say either "da" or "ja"

Then you will ask one of the 2 remaining gods this question If you were the truth teller & i was to ask you are you telling the truth about da meaning yes would you respond da?

This will automatically cause the truth teller to say "Da" NO MATTER WHAT
And the Liar will say "Ja" NO MATTER WHAT. thus you have eliminated either your truth or your false and the puzzle is solved. :)

jhe1984
can they be any yes-no questions?

Could I ask something that I know to be true - like am I holding up five fingers?

I think the hardest part about brain teasers is understanding the rules.

{EDIT} I've read over the answer method and think I follow, but here's what I don't like. Isn't using the IFF clause bending the rules pretty far? I mean it's pretty close to a two part question (or a question that is so conditional that it is not really fair). Why could I not simply ask "Answer DA iff I am holding up two fingers and da means yes." Well then that's not pretty difficult from there. I understand it works - it just seems a little removed from the spirit of a riddle. If you are familiar with the procedures, it is relatively simple; on the other hand, if you're not (like most of us), then the question seems unanswerable.

What say yall?

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