# The Hardest Logic/puzzle Question Of All Time!

1. Feb 7, 2005

### gravenewworld

George Boolos, a brilliant logician claimed this puzzle to be one of the hardest logic problems of all time.

You are stranded on an Island and on that island are 3 all knowing all powerful gods. One god is the god of truth, who always tells the truth and can never lie. The second god is the god of lies, he always lies and never tells the truth. The 3rd god is the god of chaos, he tells both lies and truths, however, completely randomly. The gods appear as identical twins, they all look the same. The gods also speak a language that you do not understand, except that you know that uga and booga are the responses yes and no (you however do not know which word is yes and which is no). You can only ask 3 yes or no questions to the gods in order to figure out which god is which. What 3 questions do you ask?

2. Feb 7, 2005

3. Feb 7, 2005

### dextercioby

It's damn hard. :yuck: I don't wanna burn half of my available neurons with a logics problem.
Sorry.

Daniel.

4. Feb 8, 2005

### Pseudopod

If you ask all three gods one question, does that count as asking one question or three?

5. Feb 8, 2005

### gravenewworld

3. In the end you can only have a total of 3 responses.

6. Feb 8, 2005

### RandallB

I believe this works – I’ll type in WHITE
GODS 1 2 3 are GODS of T F C
Even an all knowing god of Truth or False cannot know, or be able to reply, to what the God of Chaos might say.

Ask God 1 “if I ask god 2 ‘is God 3 god of truth’”
no reply 2 = C Y/ N; C= 1 or 2

IF 2=C
Ask God 1 "if I ask god 3 'if he is god of truth'"
Y 1= T 3=F; N 1=F 3=T -done in two-
IF C= 1 or 3
Ask God 3 “if I ask god 1 is God 2 god of truth’”
no reply 1 =C Y or N 3= C

IF C=3
Ask God 1 “if I ask God 2 ‘if He is god of truth’”
Y 1= T 2=F ; N 1=F 2=T
IF C=1
Ask God 3 “if I ask God 2 ‘if He is god of truth’”
Y 3= T 2=F ; N 3=F 2=T

Good one, and no I’ve never seen before.

RB

Last edited: Feb 8, 2005
7. Feb 8, 2005

### Bartholomew

Good idea!

You don't know whether "uga" is true and "booga" is false, or vice versa, so this doesn't work. The same goes for your last 2 questions.

8. Feb 8, 2005

### Bartholomew

Okay, I got it.

Number the gods 1, 2, and 3. Ask god 1: Would god 2 say that 2+2=4? If he answers, then remember his answer and ask god 2: would god 3 say that 2+2=4?

If god 1 doesn't answer your question then god 2 is the chaos god. You now have 2 questions to determine which god is which from gods 1 and 2. So ask god 1: would god 3 say that 2+2=4?

If god 2 doesn't answer your question then god 3 is the chaos god. You have god 1's answer. And if gods 1 and 2 both answer, then god 1 is the chaos god and you have god 2's answer.

So you've pinpointed the chaos god. Call the remaining gods A and B, so that A is the one who you have already asked the "would god B say that 2+2=4" question of.

However god A answered, it meant "no," so now you know which word is "no." Now ask god A, "would god B say you are the lying god?" If he says yes, god A is the truth telling god and god B is the liar; if he says no, god A is the liar and god B is the truth telling god.

9. Feb 8, 2005

### gravenewworld

I'll give you a hint, Boolos was able to publish a solution to this problem WITHOUT figuring out what uga and booga mean. The questions you have to ask ARE NOT EASY questions. They require complex questions to be asked.

10. Feb 8, 2005

### gravenewworld

maybe I should clarify further, each god has to answer your question, and they will only answer yes or no.

11. Feb 8, 2005

### Bartholomew

If you asked the truth god what the chaos god would say if you asked the chaos god if 2+2=4, what do YOU think the truth god would say? He can't say anything, as Randall pointed out.

Perhaps Boolos simply overlooked this.

12. Feb 9, 2005

### RandallB

OOps -- messed up on the (you however do not know which word is yes and which is no) part.

So to extend the hint, we must ask questions that can be answered.
Complex can include something like:
If you are not God T
all the statements in list 1 are true or at least one in list 2 is true
but if you not God F
all the statements in list 3 are true or at least one in list 4 is true
Where the statements in the lists are: ……

Plus a useful couple of statements would be "the last reply from C was 'yes'"
"the last reply form C was 'uga'"

Also some "If ???? is ? then statement ???? goes in list ?" might help
Even “ as to true or false, if statement ??? and statement ??? are both the same then statement ???? is in list ? and list ?.

That should give a fairly wide area of the complex to make some ugly looking yes or no questions!

Plus it's likely nessesary not to find out what uga means, because that would require wasting a question to get unneeded info, and why Boolos didn't find what the word for Yes was.

If anyone does get this please put it white.
RB

13. Feb 17, 2005

### honestrosewater

Eh, you mean asking each god the same question counts as asking 3 questions, right?
Asking conditionals counts as one question? Does the liar lie about a conditinal as a whole or about each of its propositions? That is, if I ask, "Does S imply P," and S is false and P is true, will the liar answer as if S were false and P were true or as if S were true and P were false?

Last edited: Feb 17, 2005
14. Feb 20, 2005

### RandallB

Only these one answer given to you is fasle or not. As Gods they can correctly figure out any problem.
If is was as you fear asking the false god is 4+5=9 would mean a false calculation of 8 for 4+5 would NOT = 9, and the false would have to say YES. That would be Chaos and you already have one of those.

What ever the logic - it must come down to one yes or no (I mean ooga or booga) reply. Have you stopped beating your wife? Cannot resolve to a meaning full answer.

ALSO altough he didn't say so - I think you must address the Gods seperatly so you cannot get by with addressing them as a groups saying.
1. "God of Truth are we alone?"
2. "God of Lies are we alone?"
As you'd have it figured out in just two guestions, by getting four answers as just the act of answering is a reply that gives infromation.

The answer is in another thread, so those that want to work on this can do so from here.

15. Feb 20, 2005

### honestrosewater

RandallB,
Your answer doesn't address my question. 4+5=9 is not a conditional. I've read the solution, and the solver implicitly assumes that Liar lies about compound propositions as a whole but not about their parts.
And, yes, you have to address every question to a god- as in, the first, second or third god. It wouldn't be much of a puzzle otherwise. :tongue2:

16. Feb 21, 2005

### RandallB

I'm assuming any God will truthfully and accurately work thru all conditionals just as accurately as non-conditionals to come up with a yes or no answer and then give the a)correct b)opposite or c)random reply as their answer.

17. Feb 22, 2005

### Icebreaker

Not necessarily. Let's assume that the Truth God knows what the Chaos God will say (they communicate with each other through telepathy or something), and whether it is a lie won't matter, because he is telling the truth about what the chaos god will say.

18. Feb 22, 2005

### Bartholomew

If the chaos god knew his own answer beforehand, then the chaos god's answer wouldn't be truly random.

19. Feb 23, 2005

### Your.Master

What if the random determination is made before the question is asked?

As in:

Chaos God's thoughts: "Okay, next time I answer false."

Brave Adventurer: "Okay, I'd like a Pizza, hold the anchovies."

Chaos God: "FALSE!!!"

Chaos God's thoughts: "Okay, next time, I say true."

Brave Adventurer: "What on Earth...you're very strange, do you know that?"

Chaos God: "TRUE!!!"

Chaos God's thoughts: "Okay, next time, I say true again. This is good fun."

-- except that he of course speaks in ooga and booga rather than true and false.

Then the all knowing Truth God could know the answer (telepathy, however, is out of the question: the truth god knows that answering the question could change the answer.

So you might think to yourself, what if I asked the Truth God about two questions down the road?

Well...maybe the Chaos God has already "rolled the dice" for the next two or three questions, which after all are all he needs random values for.

Of course, telepathically asking is out of the question: the Truth God would potentially change the Chaos God's answer by asking. Sort of like Quantum Mechanics.

20. Feb 23, 2005

### Bartholomew

If the chaos god decides he will say true before he says it, then his answer is determined by the fact of his decision, and not random. An outcome is perfectly random with a certain probability only if no information may be obtained about what that outcome may be other than that probability.

21. Feb 23, 2005

### Icebreaker

It's simple: it's random if you don't know how it's done.

22. Feb 23, 2005

### Bartholomew

That's a weak, relativistic definition of random (though practical). A god of chaos would use the much stronger form I have described.

23. Feb 24, 2005

### Your.Master

Although for the sake of the question I assume you're right, Bartholemew, I don't think your perfect randomness holds in this situation since we have three omniscient beings involved here, which, although completely impossible, contradict the conception of randomness in a big way. My example above is to illustrate the one and only way the random information could be found beforehand: via the "mechanism" of omniscience.

24. Mar 8, 2005

### Alkatran

Guys, the intent of the puzzle is there, don't bother worrying what happens when you ask another god a question about the god of chaos: just assume the answer will be as random as a chaos god answer.

As for the puzzle. I can solve a simpler form of it... If I can ask 3 questions to any of the gods and I know what truth/false is:

Ask the god on the left:
"If I asked you if the god on the right was the God of Chaos, what would you answer?"
The form of the question will ensure that the God of falsehood and god of truth will give a true answer.

If he says "true":
Either a) you are talking to the god of chaos and he is deceiving you or b) the one on the right is chaos. However, you KNOW that the god in the center is either truth or false and not the god of chaos.

if he says "false":
There is no way that the god on the right is the god of chaos. Either a) you are talking to the god of chaos and he happens to give the right answer or b) the god or truth/falsehood is giving you the right answer. So the god on teh right is not the god of chaos.

Alright, now ask the god that you know is not the god of chaos:
If I were to ask you if the god on the left was the god of chaos, what would you answer?

if he says "true"
You know who the god of chaos is, use your final question to figure out who is truth.
if he says "false"
You know the god on the right is the god of chaos. You'll need to ask the god on the left or center again to eliminate that last truth/falsehood answer.

25. Mar 17, 2005

### lawtonfogle

you ask the question 'Will your next word will be no" to all three then...

the god of truth can not answer because it will be a lie either way

the god of lie he will answer either yes or no because both will be lies

the god of chaos will only answer if he lieing

at least this way you figure out wich one is the god of truth

can anyone help me figure out the last two, and since the god of truth does not answer, it does not count againts the 3 responses