The Mad Emperor Strikes Back

[Dintiradan]

Well, it's happened again; your ragtag regiment of rebels have been captured by the forces of the Mad Emperor. Many of the younger members of your troop don't know about him, and who can blame them? Your last meeting with the Mad Emperor was so long ago (the last topic purge on General might also have something to do with it). The Mad Emperor is known as much for his cruelty as his insanity, brutally torturing all who oppose him. However, he does possess a sporadic sense of mercy. From time to time, he will let captives go free, provided they can solve fiendishly difficult puzzles of logic.

 

True to form, the Mad Emperor is in one of his generous moods tonight (perhaps being captured on Christmas Day explains this leniency). He says that you and your comrades may walk free, if you can solve a trivial task of identification. At least, you think that's what he said; you were somewhat distracted by the growing puddle of spittle at his feet.

 

He takes you to his throne room, where you meet three silent crones. You have heard of these women -- the Mad Emperor's oracles. The oracles are reputed to know everything, and to always speak the truth. There is a hitch, however: all three are mute, and each one has her own peculiar form of sign language when advising the Mad Emperor. One hag raises her left hand to indicate that the answer to a certain question is "True", and raises her right hand when it is "False". Another raises her right hand to indicate "True", and her left to indicate "False". The final crone is supposedly as mad as the Emperor himself. She answers all questions truthfully, but after each question is asked, she randomly chooses whether to raise her left hand for "True" and her right for "False", or vice versa. If only her insanity outwardly manifested itself; all three hags are identical in mannerisms and form. One final peculiarity: an oracle will only answer a question if the only possible answers are "True" or "False", and will only answer if it is addressed to her, and her alone. They refuse to answer questions addressed to more than one oracle.

 

The Mad Emperor informs you that you may ask three questions of his advisors. After that, you must correctly identify which oracle is which. Answer correctly, and you and your entire regiment will be released. Answer incorrectly... well, best not to dwell on that.

 

You stare at the three crones quietly, thinking about your first question. The only sound breaking the silence is the Mad Emperor gnawing on his throne's armrest. What do you do?

[Enraged Slith]

So you can't ask different oracles the same question or address all of them at once?

[Erebus the Black]

retracted

[nikki.]

I expected more Paint drawings, given the title of this thread.

[Mistb0rn]

Hmm, tricky. I keep thinking I have it solved, then realize that it wouldn't work.

[ĐªгŦĦ Єяŋϊε]

i think it has to do with asking them if a different oracle is lying and using that as a baseline

[Dantius]

Since I know the answer to this riddle, I will only voice my disappointment that you didn't opt to use Sheogorath instead.

 

 

[Goldengirl]

Well, as a basis for questioning, we know a few things to be true in this scenario, so we can discover who answers in which way.

 

1. Each hag is mute.

2. The emperor is mad.

3. It is Christmas.

4. We are prisoners.

 

etc. How to ask these in only a few short questions so as to know which adviser is which, though, is something I haven't yet discovered.

 

EDIT:

 

Originally Posted By: Darth Ernie

i think it has to do with asking them if a different oracle is lying and using that as a baseline

 

Definitely not. They all tell the truth.

[Cairo Jim]

Well, if you ask the oracles what you know the answer is going to be, you could pick out quite easily.

[Ceiling Durkheim]

No. It's more complicated than that, because of the one that answers randomly. Assume we use your implied solution, and ask each one a question of known truth value, e.g. in a decimal number system, does 2+2=4? Either two will answer with the right hand (right and random) and one with the left, or vice versa. Using this method, there is no way to discern which is consistently right/left and which is random.

 

That said, I don't know what the actual solution is. I think it's something in which the answers to your questions are conditional on how the other crones would answer, but I'm not sure what questions of that sort will generate a correct solution.

[Lazarus.]

Ask the following question to two of the oracles: "Is it true that you raise your right hand in response to true questions?"

 

The two sane oracles will both raise their right hand in response to this question. The insane will not answer, since for her there is no true/false answer for this question. If you pose this question to two oracles you will know which one is the insane one. Now pose any question that you know to be true to one of the remaining sane oracles. "Is it true that today is Christmas?" You know she's sane and that the answer is true, so you can identify her.

 

Do I go free?

[Enraged Slith]

Got it, I think.

 

Click to reveal..

Ask the oracle on the right if the oracle in the middle will always raise their right hand when answering true.

 

Ask the oracle on the left if the oracle in the middle will always raise their left hand when answering true.

 

While the answers to these questions can both be false, only one can be true.

 

If the oracles raise the same hand then one of the answers must be true. Both answers cannot be false in this instance, as this would require the oracles to use opposite hands. If two rights were raised, then the middle oracle will raise their left when indicating truth, and vice versa. We simply ask the middle oracle about the nature of one of the other oracles to solve the puzzle.

 

If the oracles raise different hands, it means that either both questions are false (the middle oracle is the switcher) or that one is true and the other is false (one of the other two is the switcher). We can correctly identify which is the case by identifying which hands were raised.

 

When asking about the right hand, if the oracle raised her own right hand, and vice versa with the left, both responses are "false" and the oracle in the middle must be the switcher as neither response could be "true" (an oracle raising their right hand to indicate the truth of the statement "the middle oracle raises their right hand in response to truth" cannot happen). This yields our answer.

 

If the hands are the opposite of the question (right hand for questioning about the left), then the oracle in the middle is a constant. We ask her a question of obvious nature (i.e 2+2 = 4) and note which hand she raises. Whichever of the first two oracles raised the same hand is the switcher, yielding our answer.

This took me way too long to solve.

 

EDIT: Clarified a little more. This isn't pretty, but it should work.

[Nioca]

I thought on this for a while, and finally decided on an answer:

Click to reveal.. (Nioca's Answer)

No Solution. No matter what three questions are asked, there is no 100% reliable way to correctly identify all three oracles. The odds can be improved, yes, but it'll still come down to hoping that the random oracle didn't undermine your results.

Then again, I suck at logic puzzles, so I'm probably screwed either way.

[Enraged Slith]

Originally Posted By: Lazarus.

Ask the following question to two of the oracles: "Is it true that you raise your right hand in response to true questions?"

The two sane oracles will both raise their right hand in response to this question. The insane will not answer, since for her there is no true/false answer for this question. If you pose this question to two oracles you will know which one is the insane one. Now pose any question that you know to be true to one of the remaining sane oracles. "Is it true that today is Christmas?" You know she's sane and that the answer is true, so you can identify her.

Do I go free?

This is clever, but I think that the insane one's response would technically be "false, I change up my hands randomly."

If all three raised their right hands, you wouldn't be any better off than you were at the start.

[ĐªгŦĦ Єяŋϊε]

all three cant raise their right hands.

 

if i ask all three what is 2+2?

i will get one right and two left or one left and two right.

so whichever one is alone is the sane one.

then...

[Cairo Jim]

Originally Posted By: Enraged Slith

Originally Posted By: Lazarus.

Ask the following question to two of the oracles: "Is it true that you raise your right hand in response to true questions?"

The two sane oracles will both raise their right hand in response to this question. The insane will not answer, since for her there is no true/false answer for this question. If you pose this question to two oracles you will know which one is the insane one. Now pose any question that you know to be true to one of the remaining sane oracles. "Is it true that today is Christmas?" You know she's sane and that the answer is true, so you can identify her.

Do I go free?

This is clever, but I think that the insane one's response would technically be "false, I change up my hands randomly."

If all three raised their right hands, you wouldn't be any better off than you were at the start.

I reckon that solution sounds a bit too complicated for little or no result. Besides, the insane one doesn't do it at random, she alternates between questions. One question she all answer true with her left, next question with her right, then left again. There has to be a way to capitalise on the definate alternation

[Ceiling Durkheim]

Quote:

I reckon that solution sounds a bit too complicated for little or no result. Besides, the insane one doesn't do it at random, she alternates between questions. One question she all answer true with her left, next question with her right, then left again. There has to be a way to capitalise on the definate alternation

Wrong. What Dintiradan said was:

Quote:

She answers all questions truthfully, but after each question is asked, she randomly chooses whether to raise her left hand for "True" and her right for "False", or vice versa.

Not, "alternates," "randomly chooses." There is no reason, given the way the puzzle is written, that she couldn't respond the same way twice in a row.

[Enraged Slith]

Originally Posted By: Darth Ernie

all three cant raise their right hands.

if i ask all three what is 2+2?
i will get one right and two left or one left and two right.
so whichever one is alone is the sane one.
then...

All three can raise their right hands, because the answer to "Is it true that you raise your right hand in response to true questions?" depends on the oracle.

Oracle 1 raises her right hand because it is true.

Oracle 2 raises her right hand because it is false, as she raises her left for those questions.

Oracle 3 raises her right hand because it is false, and she randomly determined that her right hand would indicate falsehood.

[Cairo Jim]

Originally Posted By: Enraged Slith

Originally Posted By: Darth Ernie

all three cant raise their right hands.

if i ask all three what is 2+2?
i will get one right and two left or one left and two right.
so whichever one is alone is the sane one.
then...

All three can raise their right hands, because the answer to "Is it true that you raise your right hand in response to true questions?" depends on the oracle.

Oracle 1 raises her right hand because it is true.

Oracle 2 raises her right hand because it is false, as she raises her left for those questions.

Oracle 3 raises her right hand because it is false, and she randomly determined that her right hand would indicate falsehood.

Not so. 2+2 has a definate asnwer, not something silly like which one raises which hand, so thusfore what Darth Ernie was correct in the first place

[Zummi]

Edit: NVM.

[Enraged Slith]

Originally Posted By: Cairo Jim

Originally Posted By: Enraged Slith

Originally Posted By: Darth Ernie

all three cant raise their right hands.

if i ask all three what is 2+2?
i will get one right and two left or one left and two right.
so whichever one is alone is the sane one.
then...

All three can raise their right hands, because the answer to "Is it true that you raise your right hand in response to true questions?" depends on the oracle.

Oracle 1 raises her right hand because it is true.

Oracle 2 raises her right hand because it is false, as she raises her left for those questions.

Oracle 3 raises her right hand because it is false, and she randomly determined that her right hand would indicate falsehood.

Not so. 2+2 has a definate asnwer, not something silly like which one raises which hand, so thusfore what Darth Ernie was correct in the first place

I thought he was responding to my earlier statement. My mistake.

[Cairo Jim]

I just had a thought, but thinking over it, it seems less likely it'd work. But it could still be possible. By making 2 obliviously obvious true statements, such as "2+2=4" and one blatantly false statement like "2+2=5". The thing is, I was thinking of as asking one of the crones a true and false question.

[Ceiling Durkheim]

Quote:

all three cant raise their right hands.

if i ask all three what is 2+2?
i will get one right and two left or one left and two right.
so whichever one is alone is the sane one.
then...

Even if we assume you're right, that leaves us with certainty as to one sane one, and no certain distinction between the other sane and the mad. Also, all three questions used up. This solution does not work.

[Enraged Slith]

Originally Posted By: Cairo Jim

I just had a thought, but thinking over it, it seems less likely it'd work. But it could still be possible. By making 2 obliviously obvious true statements, such as "2+2=4" and one blatantly false statement like "2+2=5". The thing is, I was thinking of as asking one of the crones a true and false question.

You cannot isolate the oracle that randomly changes her hands, so asking a single oracle two questions offers no more certainty than one. The best available option, aside from solving the riddle, is simply yielding a truth from all of them and guessing between the two that raise the same hand.

It's time to flex those heuristics.

[Cairo Jim]

Yes, I realised that. Tis why I mentioned that it seemed less likely to work.

[Erebus the Black]

Originally Posted By: Darth Ernie

all three cant raise their right hands.

if i ask all three what is 2+2?
i will get one right and two left or one left and two right.
so whichever one is alone is the sane one.

then the entire regiment is brought to the execution block and is immediately executed as you had only three questions

[Dintiradan]

I'm glad people are having fun with this, and also pleased that it's a bit of a challenge. The last two times I made a topic like this, people got the answer right away. I don't know if I'll be able to raise the bar for the next time, though.

 

Originally Posted By: Enraged Slith

So you can't ask different oracles the same question or address all of them at once?

You can ask the same question multiple times, but every asking counts against the limit. You can't cheat by saying "All three of you, answer this question," as a way to get three answers for one question.

 

Originally Posted By: Dantius

Since I know the answer to this riddle, I will only voice my disappointment that you didn't opt to use Sheogorath instead.

Ehhh, I used the Mad Emperor the last time I did something like this. Also, I haven't played any of the Elder Scrolls games. I used to play western CRPGs, but then I took an arrow to the knee.

 

Originally Posted By: Lazarus.

Ask the following question to two of the oracles: "Is it true that you raise your right hand in response to true questions?"

 

The two sane oracles will both raise their right hand in response to this question. The insane will not answer, since for her there is no true/false answer for this question. If you pose this question to two oracles you will know which one is the insane one. Now pose any question that you know to be true to one of the remaining sane oracles. "Is it true that today is Christmas?" You know she's sane and that the answer is true, so you can identify her.

 

Do I go free?

Haha, totally didn't see that coming. The whole bit I added about the oracles not answering questions that can be answered with true/false was just to stop people from asking paradoxical questions. But I never considered that you can gain extra information by asking questions that are ambiguous to some oracles but not to others.

 

Okay, Lazarus gets to go free for sheer panache, but now the Mad Emperor is making everyone else ask unambiguous questions. :-P

 

Originally Posted By: Enraged Slith

Got it, I think.

 

Click to reveal..

Ask the oracle on the right if the oracle in the middle will always raise their right hand when answering true.

 

Ask the oracle on the left if the oracle in the middle will always raise their left hand when answering true.

 

While the answers to these questions can both be false, only one can be true.

 

If the oracles raise the same hand then one of the answers must be true. Both answers cannot be false in this instance, as this would require the oracles to use opposite hands. If two rights were raised, then the middle oracle will raise their left when indicating truth, and vice versa. We simply ask the middle oracle about the nature of one of the other oracles to solve the puzzle.

 

If the oracles raise different hands, it means that either both questions are false (the middle oracle is the switcher) or that one is true and the other is false (one of the other two is the switcher). We can correctly identify which is the case by identifying which hands were raised.

 

When asking about the right hand, if the oracle raised her own right hand, and vice versa with the left, both responses are "false" and the oracle in the middle must be the switcher as neither response could be "true" (an oracle raising their right hand to indicate the truth of the statement "the middle oracle raises their right hand in response to truth" cannot happen). This yields our answer.

 

If the hands are the opposite of the question (right hand for questioning about the left), then the oracle in the middle is a constant. We ask her a question of obvious nature (i.e 2+2 = 4) and note which hand she raises. Whichever of the first two oracles raised the same hand is the switcher, yielding our answer.

This took me way too long to solve.

 

EDIT: Clarified a little more. This isn't pretty, but it should work.

I don't think that works. I find that looking at a truth table helps. There are six possible scenarios:

Code:

1: AX BR CL2: AX BL CR3: AR BX CL4: AR BL CX5: AL BX CR6: AL BR CX

(Here A, B, and C refer to the three oracles, X refers to that oracle being random, and R/L refer to that oracle raising R/L for "True" and the opposite hand for "False".)

 

Now let's ask the first two questions, first asking C, and then asking A. 'X' indicates a random answer.

Code:

1: AX BR CL    C: L (true)     A: X (false)2: AX BL CR    C: L (false)    A: X (true)3: AR BX CL    C: R (false)    A: L (false)4: AR BL CX    C: X (false)    A: R (true)5: AL BX CR    C: L (false)    A: R (false)6: AL BR CX    C: X (true)     A: R (false)

- "If two rights were raised, then the middle oracle will raise their left when indicating truth, and vice versa.": Nope, look at scenario #6.

- "If the hands are the opposite of the question (right hand for questioning about the left), then the oracle in the middle is a constant." Nope, look at scenario #5.

 

You are on the right track, though. You've got to find someone who gives you non-random answers as soon as possible.

 

Originally Posted By: Nioca

I thought on this for a while, and finally decided on an answer:

Click to reveal.. (Nioca's Answer)

No Solution. No matter what three questions are asked, there is no 100% reliable way to correctly identify all three oracles. The odds can be improved, yes, but it'll still come down to hoping that the random oracle didn't undermine your results.

Then again, I suck at logic puzzles, so I'm probably screwed either way.

Oh, there is a solution -- the Mad Emperor only gives puzzles that can be solved with a perfect chance of success. Difficult, yes, but impossible, never. He's not like those semi-barbaric kings who give their captives even odds at best. Whenever the Mad Emperor conquers one of their kingdoms, he invariably feeds the ruler to a horde of ravenous maidens.

 

Say what you will of the Mad Emperor, he's got a smashing sense of humour.

[Enraged Slith]

DAAAAAAAAAAAAAAAAAMMIT

 

this is what happens when you try to do everything in your head with a fart brain

[Dantius]

(This is the solution I came up with when I was first asked the problem by someone else. It gets you out 5/6ths of the time, but it might be possible to improve that, I just could't think of the right final question to ask. Maybe someone here could improve on it)

 

Click to reveal..

There are actually, by my reckoning, three possible answers that a crone can give- true, false, or refuse to answer. For instance, if I asked "What color is the sky?" then they would remain silent, because that's not a yes/no question to them. So all I need to do is find a question that some crones would see as true/false, and some would see as unanswerable. The one I came up with was:

Click to reveal..

If a response of "no" is equivalent to the numerical value 0, and a response of "yes" equivalent to 1, then what is the probability that you will raise the same hand as the Random crone to any given question I ask you?

The two crones that answer in a deterministic manner will be unable to respond, because the answer for them will be 1/2, which is not an option. However, the Random crone will ALWAYS answer the same as herself, so she will be able to raise a hand. Therefore, the crone who raises her hand AT ALL in response to that question is Random

 

If the random crone is first, then ansk a trivially true question the the second crone, if they raise their left, the order is {X,L,R}, otherwise {X,R,L}.

 

If the random crone is second, the ask the third a trivially true question, left raised gives the order of {R,X,L} else {L,X,R}.

 

If the first two both refuse to reply(only a 1/3 chance of that), then you could guess {G,not G && not X,X} and be right 50% of the time, or you could be cleverer than I and come up with a question that winnows out the identity of one of the prior two crones, since you know that the third is Random.

 

Feel free to build off that.

[A less presumptuous name.]

That's a pretty clever question. I wonder how the Mad Emperor feels about you converting language into a binary system. I guess that's for Dintiradan to decide.

[Erebus the Black]

Originally Posted By: Dintiradan

I don't know if I'll be able to raise the bar for the next time, though.

This is after all a variant of "The Hardest Logic Puzzle Ever", so there's only going down from here

[Student of Trinity]

I believe this is a reliable solution.

 

 

Click to reveal..

Here is a question that reliably checks crone sanity. Sane crones will raise their left arms in response to it, while the insane crone raises her right arm, no matter which arm she is choosing to represent truth.

 

Click to reveal..

If I suggested you were insane, would you raise your right arm?

 

This question mentions another question, "Are you insane?" If the crone is insane, then the answer to the mentioned question is True. So then if she is insane and choosing right for true, the answer to the full question is True, and so the crone will raise her right arm. But if she is insane and choosing left for true, then the answer to the full question is False, so she'll raise her right arm in this case, too.

 

On the other hand, if the crone is sane, then the answer to the mentioned question is False. So if the crone is sane and raises right for truth, then the answer to the full question is False, hence she will raise her left arm. If she is sane and raises left for truth, then the answer to the full question is True, so she will raise her left arm.

 

The nested question thus produces a right arm raise from the insane crone, and a left arm raise from either sane crone.

 

Here is another question that reliably checks whether a crone always raises the right hand for truth. Only the sane crone who raises her right arm for truth will raise her right arm in response to it; the other two will both raise the left arm, again regardless of which arm the insane crone is using for truth.

 

Click to reveal..

If I suggested that you always raised your right arm to denote True, would you raise your right arm?

 

By a similar logic to the sanity check question, this one produces a right arm raise only from a sane crone who raises the right arm for True. The insane crone is presumed to know that she doesn't always raise either arm for truth. The sane crone who always raises right for truth obviously raises her right arm to this question. For the other crones, the mentioned question is False, so for a right-arm-true insane crone the overall question is also False, so she'll raise left. For a left-arm-true crone, the overall question is true, so she'll raise her left arm.

 

So the plan is this:

 

Ask the sanity check question to one of the crones. If she comes out sane, then ask her the right-hand-truth-check question to find out whether she raises the right hand for true, then ask one of the other crones the sanity check question, and with a little bit of logic, you're done. If the first crone comes out insane, just ask one of the other crones the right-hand-truth-check question, and again a little logic sorts them all out. So you can always solve the problem with three questions, and 1/3 of the time you can do it in only two.

 

I believe this solution is optimal.

Click to reveal..

This solution requires an average of 2-2/3 questions to determine the crones. With six distinct cases, there are log(6)/log(2) = 2.58 bits of information to determine. I guess it's not so easy to prove that there isn't some tricky question-order-dependent procedure that might do better than 2.67 question bits on average, since with question order included there are more than six cases to consider. But there's no way to get the right answer in one question, and a way that needed three questions in only 3 cases out of six, instead of 4 cases out of six like mine, would be 2.5 bits per solution on average, and that is too few to be possible.

 

This solution does make one certain assumption concerning the way in which the insane crone's insanity manifests itself. But I think it's a reasonable assumption, and anyway it hasn't been clearly ruled out.

 

Click to reveal..

I assume that if the insane crone is asked a nested question, that involves the answer she would hypothetically give to another mentioned question, then she does not separately randomize her arm-raising policy for the mentioned and actual questions. That is, the whole thing still only counts as one question, and she selects one arm-raising policy for the whole question, including its nested mentioned questions, and responds to it on that basis.

[Dantius]

(SOT, I don't think you can ask multiple questions to the same crone. If you want to use three responses, you have to ask each crone in turn. So you're pretty much in the same boat I am: there's a 1/3 chance that, after two questions, all you'll wind up knowing is that the last crone you can ask a question to is insane,and it's tricky to derive a meaningful answer from insanity)

[Goldengirl]

Originally Posted By: Dantius

(SOT, I don't think you can ask multiple questions to the same crone. If you want to use three responses, you have to ask each crone in turn. So you're pretty much in the same boat I am: there's a 1/3 chance that, after two questions, all you'll wind up knowing is that the last crone you can ask a question to is insane,and it's tricky to derive a meaningful answer from insanity)

There's nothing that seems to indicate that in the rules, so SOT's solution should work.

[Dikiyoba]

Originally Posted By: Dantius

(SOT, I don't think you can ask multiple questions to the same crone. If you want to use three responses, you have to ask each crone in turn. So you're pretty much in the same boat I am: there's a 1/3 chance that, after two questions, all you'll wind up knowing is that the last crone you can ask a question to is insane,and it's tricky to derive a meaningful answer from insanity)

If we were only allowed to ask each crone one question, Dintiradan would have wrote "you may ask each advisor one question" instead of "you may ask three questions of his advisors." I think "they refuse to answer questions addressed to more than one oracle" refers just to asking multiple crones a question at the same time.

Also, whatever the solution, if we are lucky enough to know the identity of the crones in two questions, then Dikiyoba suggests we ask a third question to try to get some information useful to the rebellion without tipping the Mad Emperor off (we really don't want him to renege on his deal and torture us anyway). Dikiyoba's initial idea is "Is it true the future is mutable?" but is open to other suggestions.

[ex post slarto]

SoT, I don't think your assumption is tenable. Dinti specifically said that she chooses which arm to use randomly "after each question." When she is arriving at her answer to your question, she has to evaluate the theoretical case of being asked a question, and she knows that her arm choice will vary, so she isn't going to be able to answer the question.

 

I'm also skeptical about asking questions that one crone won't be able to answer because they aren't true-or-false for her due to her arm-choice algorithm. I read the rules as stating that "true" and "false" must be the only POSSIBLE answers to that question, for a question to be legit; not just the only answers that that particular crone could give. At any rate, if you can ask questions that break the crones' AIs the puzzle becomes very easy, so I can't imagine that was intended.

 

---

 

Here's what I think is more promising. If you ask an objectively true (or false) question, you actually DO get information. Specifically, if you ask an obviously true question ("do you always tell the truth?"), a LH raise means you are talking to either the insane crone or the LH-true crone. And a RH raise means you are talking to either the insane crone or the RH-true crone.

 

So, here's my attempt at survival:

 

 

1) Ask Crone #1: "Is Crone #3 the insane crone?"

 

2) Ask Crone #2: "Is Crone #3 the insane crone?"

 

If Crone #3 is the insane crone, you will get two different hand raises.

 

Thus, if both crones raise their LH, you know that crone #3 is not insane. That also means that either crone #1 or crone #2 must be the RH-true crone, so crone #3 is the LH-true crone. Then you can ask her if crone #1 is insane and be done with it.

 

And thus, if both crones raise their RH, you know that crone #3 is not insane. That also means that either crone #1 or crone #2 must be the LH-true crone, so crone #3 is the RH-true crone. Then you can ask her if crone #1 is insane and be done with it.

 

The problem is, what if you do get two different hand raises? Then one of two things happened. Either you got two "trues" and crone #3 is insane, or you got two "falses" and crone #3 is sane. I tried a bunch of things but at this point, and I'm stuck. But I'm hopeful that there is a solution in this direction.

[Erebus the Black]

Because it seems you guys have figured out what's the procedure for finding which crone is random, I'll give you a hint in the form of a puzzle that might help you move forward:

 

Click to reveal..

Suppose that, somehow, you have learned that you are speaking not to Random, but to one of the other crones, you don't know which, and now you need her to tell you if Dushanbe is in Kirghizia. What one yes-no question can you ask her from the answer to which you can determine whether or not Dushanbe is in Kirghizia (not who she is)?

 

p.s. Remember that in the full riddle you also need to figure out which crone is LHT and which is RHT. Or which witch is which

[Student of Trinity]

I think my interpretation of the insanity is tenable. The crones know everything, so the insane one must know how she will choose to answer any possible question. It's just that nobody other than the omniscient crones knows this. In any case, my interpretation of the insanity is that the insane crone has a switch that is reset randomly every time she is asked a question, but that then has a definite value, known to her, until it is reset. I think this is consistent with the stated problem.

 

I also initially assumed that you had to ask each crone one question, and was stuck with some bad cases, but then I re-read the problem and realized that this condition was never stated.

[Lilith]

Originally Posted By: Student of Trinity

I think my interpretation of the insanity is tenable. The crones know everything, so the insane one must know how she will choose to answer any possible question. It's just that nobody other than the omniscient crones knows this. In any case, my interpretation of the insanity is that the insane crone has a switch that is reset randomly every time she is asked a question, but that then has a definite value, known to her, until it is reset. I think this is consistent with the stated problem.

Even if the crone doesn't innately work that way, I don't see why amending the question to "If I asked you whether you would raise your right hand if I asked you whether you were insane, and you answered that question using the same hands to denote 'true' and 'false' as you do for this question, would you raise your right hand?" would break any rules. It's a complicated question, but it's still one with a clear, single yes/no answer.

[Student of Trinity]

Good idea. Sure the insane crone is insane, but she's still gotta be omniscient, so insane can't mean stupid. She'll cope.

[Cairo Jim]

Again I have thought about it long and hard, and then realised, how is this a "logic" puzzle, where one of the main factors of the puzzle is illogical?

[Erebus the Black]

Originally Posted By: Cairo Jim

Again I have thought about it long and hard, and then realised, how is this a "logic" puzzle, where one of the main factors of the puzzle is illogical?

because the "illogical" part can be neutralized and ignored with the right logical (and complicated) question.

[Dintiradan]

Apologies if the original post is unclear. As Erasmus points out, this Mad Emperor problem (and the previous one) is an obfuscated version of a famous logic puzzle. In my defence, a lot of the ambiguities and subtleties in my version are present in the original.

 

Originally Posted By: Erasmus

Originally Posted By: Dintiradan

I don't know if I'll be able to raise the bar for the next time, though.

This is after all a variant of "The Hardest Logic Puzzle Ever", so there's only going down from here

Well, I'm not too worried. I've got nearly a thousand posts to come up with a better one. :-p

 

Originally Posted By: Cairo Jim

Again I have thought about it long and hard, and then realised, how is this a "logic" puzzle, where one of the main factors of the puzzle is illogical?

Hmmm, maybe the next Mad Emperor problem will feature error-correcting codes.

 

Originally Posted By: Dikiyoba

Originally Posted By: Dantius

(SOT, I don't think you can ask multiple questions to the same crone. If you want to use three responses, you have to ask each crone in turn. So you're pretty much in the same boat I am: there's a 1/3 chance that, after two questions, all you'll wind up knowing is that the last crone you can ask a question to is insane,and it's tricky to derive a meaningful answer from insanity)

If we were only allowed to ask each crone one question, Dintiradan would have wrote "you may ask each advisor one question" instead of "you may ask three questions of his advisors." I think "they refuse to answer questions addressed to more than one oracle" refers just to asking multiple crones a question at the same time.

Dikiyoba is correct, you may ask a crone more than one question, it's just that you can only talk to one crone at a time, and you are limited to three questions.

 

Originally Posted By: Lilith

Originally Posted By: Student of Trinity

I think my interpretation of the insanity is tenable. The crones know everything, so the insane one must know how she will choose to answer any possible question. It's just that nobody other than the omniscient crones knows this. In any case, my interpretation of the insanity is that the insane crone has a switch that is reset randomly every time she is asked a question, but that then has a definite value, known to her, until it is reset. I think this is consistent with the stated problem.

Even if the crone doesn't innately work that way, I don't see why amending the question to "If I asked you whether you would raise your right hand if I asked you whether you were insane, and you answered that question using the same hands to denote 'true' and 'false' as you do for this question, would you raise your right hand?" would break any rules. It's a complicated question, but it's still one with a clear, single yes/no answer.

The original phrasing for the question was probably more clear. It had the oracle choose the mental state before the question is asked. One of the solutions has questions of the form "If I asked you "..." in your current mental state, would you ...?"

 

Student of Trinity: that solution seems to work. I'll trust your analysis that it's the optimal solution -- I'm one of those lazy people who only worry about the worst case analysis. Solutions that use the "embedded question" trick seem to be the clearest (note that the "embedded question" trick is used in the classic "Knight and Knave" puzzle). There's another solution that has questions of the form "Are an odd number of the following statements true: ...", but I find it kinda ugly.

 

Anyway, congratulations to Lazarus and Student of Trinity for correctly solving the challenge. Now, I know a lot of you are disappointed with the two current solutions, because you feel they rely on loopholes. Good news! The Mad Emperor feels the same way! He's decided to make some changes for the next batch of unlucky rebels who end up in his throne room...

 

Challenge Variation #1

 

First, the Mad Emperor has instructed the oracles to only answer questions that all oracles are capable of answering. That is to say, if one oracle is incapable of answering a question with "true" or "false", then no oracle will answer that question.

 

Second, the insanity of the final oracle has progressed. Now, she just randomly raises her right or left hand whenever asked a legal question.

 

By the way, I've rephrased this variant using the more classic "Knights and Knaves" terminology, and without the awkward "Mad Emperor" frame story. I've made a few modifications, but the solutions to the two problems should be equivalent:

Click to reveal..

Your task is to determine the identities of three oracles standing before you, using yes/no questions. Any oracle may be asked any number of questions, so long as each question is directed at a particular oracle, and a maximum of three questions are asked.

• One oracle is a Knight, who always answers correctly.
• One oracle is a Knave, who always answers incorrectly.
• One oracle is a Jester, who's answering scheme is unknown (the Jester could always answer correctly, always answer incorrectly, or answer at random).

The oracles all use the same form of sign language to respond to questions. The right hand is raised to indicate "yes", and the left hand is raised to indicate "no".

[Student of Trinity]

Okay, I think this works for the revised problem. It never uses less than three questions, but it doesn't need any assumptions about the Jester and it's quite simple.

Click to reveal..

Ask an oracle, "If I were to ask you if the second oracle were the Jester, would you say yes?"

If the second oracle IS the Jester, then I'll get a Yes. If the third oracle is the Jester, then I'll get a No. If the one I've just asked is the Jester, then of course I could get either Yes or No.

 

But the point is that if I get a Yes, then I know that the THIRD oracle is NOT the Jester. And if I get a No, I know that oracle 2 is not the Jester. So either way, I have identified one non-Jester (though I don't yet know if he's the Knight or the Knave).

 

The rest is easy. Just ask the known non-Jester a standard nested question about himself, then ask if the first oracle is the Jester.

 

[Alorael at Large]

Clever, but I have one nitpick. Don't you have to ask, "If I were to ask your sane sibling if the second oracle were the Jester, would she say yes?" If you don't, the first oracle could answer yes or no depending on whether you're speaking to the knight or the knave. Then you know the answer from the first oracle you interrogate is going to be false unless it's the jester. From there, proceed according to SoT.

 

—Alorael, who can see how the mad oracle might not parse that. Substitute "one of your sane siblings" if you want; it gives the sane oracles only one option.

[Enraged Slith]

 

- "If two rights were raised, then the middle oracle will raise their left when indicating truth, and vice versa.": Nope, look at scenario #6.

- "If the hands are the opposite of the question (right hand for questioning about the left), then the oracle in the middle is a constant." Nope, look at scenario #5.

 

Click to reveal..

Scenario 6 is a small oversight, I think. While my solution can't decipher the hand the middle oracle is going to raise, it can determine that they aren't the random oracle if the other two oracles raise the same hand. The reason for this is that both statements must be false for the oracle in the middle to be the random variable, but a double false would yield opposite hands. If we yield a truth from the middle oracle, the hand they raise should tell us which of the other two oracles is random and which is constant.

 

Scenario 5:

 

As both statements cannot be true at the same time, different hands can either mean a false/false or true/false.

 

If the hands are opposite, the middle oracle is a constant. If an oracle asked about a right hand raises their left, the response is potentially true. However, if the other oracle raises their right hand in response to a question about the left, another potentially true response, it would either indicate that one of the two oracles is random or that both statements are true (impossible). In this instance, we can yield a truth from the middle oracle and use the hand they raise to decipher who the random oracle is.

 

If the hand raised matches up with the question, then the answer cannot be true unless that oracle is random. In this case the possible outcome for each question is:

 

The oracle in the middle raises their right hand for truth:

RH - False

RH - True (I am the random oracle)

 

The oracle in the middle raises their left hand for truth:

LH - False

LH - True (I am the random oracle)

 

However, when combined, neither response can ever be true. If the oracle who raises their right hand to imply that the middle oracle always raises their right hand for truth, the oracle asked about the other hand, who must be a constant, would have to imply falsehood. However, using their left hand to imply falsehood would indicate that they use their right hand to imply truth. As there is only one random oracle, this cannot ever happen. Because of this, in this situation, the oracle in the middle is always random and you wouldn't need to ask any further questions to figure out who is who.

 

Did I do good, or am I missing something obvious?

[Dintiradan]

Sorry everyone for not responding sooner.

 

Originally Posted By: Enraged Slith

If the hands are opposite, the middle oracle is a constant. If an oracle asked about a right hand raises their left, the response is potentially true. However, if the other oracle raises their right hand in response to a question about the left, another potentially true response, it would either indicate that one of the two oracles is random or that both statements are true (impossible). In this instance, we can yield a truth from the middle oracle and use the hand they raise to decipher who the random oracle is.

For reference, here's the truth table again:

Code:

1: AX BR CL    C: L (true)     A: X (false)2: AX BL CR    C: L (false)    A: X (true)3: AR BX CL    C: R (false)    A: L (false)4: AR BL CX    C: X (false)    A: R (true)5: AL BX CR    C: L (false)    A: R (false)6: AL BR CX    C: X (true)     A: R (false)

Regardless of how the results are interpreted, it's impossible to distinguish scenarios #1, #2, #4, #5, and #6 from each other if C answers L and A answers R. Think of it this way: the worst case for your method is when a random C always answers L, and a random A always answers R. There's no possible way to distinguish between all five scenarios after your third question. For any method to work, you have to make sure that after any two questions, you only have to distinguish between two scenarios.

 

Student of Trinity has a solution for Challenge Variation #1. I leave you with Challenge Variation #2 (which is actually equivalent to the puzzle I was basing all of this on). It's got one extra layer of complication, but modifying the previous solution will work.

 

Click to reveal..

Your task is to determine the identities of three oracles standing before you, using yes/no questions. Any oracle may be asked any number of questions, so long as each question is directed at a particular oracle, and a maximum of three questions are asked.

• One oracle is a Knight, who always answers correctly.
• One oracle is a Knave, who always answers incorrectly.
• One oracle is a Jester, who's answering scheme is unknown (the Jester could always answer correctly, always answer incorrectly, or answer at random).

The oracles all use the same form of sign language to respond to questions. However, it is unknown whether oracles raise their right hands to indicate "yes" and their left hands to indicate "no", or vice versa.