Edward and the seven imps' Game

Question

Excerpt of “Manipulating the Fabric of Space”

“Miscasting translocation spells can have many nasty consequences. The caster’s body could be warped or torn apart, his soul could be separated from his body - turning into an insubstantial wisp, a clone with part of the caster’s personality could appear, the caster could be transported into the endless and chaotic abyss or - worst of all... the caster could accidentally summon a few daemonia molestus, more commonly known as common imps.”

After hours of practice Edward was finally feeling sure enough to attempt a simple dislocation spell, targeting a single cup on an empty table. It only took him a moment to speak the words and cast the spell - and the cup actually disappeared!

Edward had the biggest grin on his face. “Hahaha, see? Told you that translocations are no problem for someone with my talen--”

Suddenly, he felt two weights land on top of his head before an annoying voice yelled into his right ear. “Hey!”

Edward almost fell from his chair. He tried to grab whatever was sitting on his head and pull it down, only for them to dig their claws into both of his ears. “W-what… Who the heck are you?! Get off my head!”

The two small creatures chuckled, before the one holding Edward’s right ear jumped down on his right shoulder, while the other remained in place, still clutching Edward's left ear.

The small creature on Edward’s right shoulder then started explaining. “We’re imps! You have summoned us - and we came to play a little game with you. I’m Acredo and that’s my brother Bugia. Every sentence I say is the truth, while every sentence my brother says is a lie. As soon as you want to start the game you just tell us, and we’ll switch positions at random. You can ask us as many yes or no questions as you want, to detect who of us is my always lying brother, Bugia. But I should warn you… you only get one chance to guess who of us it is before the game is over. Every time you ask one of us a question that can’t be answered with just yes or no, like you just did, another one of our brothers appears after the imp you asked finished answering your question. I will explain you the further rules once you accept to play our game”, he said.

As just mentioned, Edward could feel another weight land ontop of his head, quickly grabbing his free left ear with it’s sharp claws - before an equally annoying voice yelled “Hey!” into it.

Edward winced. “Stop yelling, I’m not deaf!”, he yelled. He crossed his arms and angrily stared down on his empty table. “And why should I bother to play your game?”, he asked the imp on his shoulder in an annoyed tone. “I could always just get rid of you the ‘old fashioned’ way...”

All three imps chuckled and the imp on Edward’s right shoulder answered again. “Because we’ll stick around until your last days if you lose or don’t play! And if you kill an imp, two more show up! By the way, all of my brothers appearing because of your illegal questions(1) are completly unreliable. Each time you ask them a question, they'll decide between behaving like an imp who always tells the truth or an imp who always lies, randomly, and then answer the question”, he said.

( (1) Illegal Questions: All questions that cannot be answered with yes or no.)

Edward could feel a fourth weight landing on his left shoulder. “Hey brothers!”

The sorcerer makes another futile attempt to get the imps off his shoulders and head but isn’t able to remove even one of them, only hurting himself in the process as they keep a firm hold on him.

Willy, Edward’s familiar, who was sitting just next to the imp on Edwards right shoulder wiggles his antennae. “How many brothers do you have?”, he asked.

“Oh, we are a big family. Well, at least compared to humanoid families. I have 257 brothers, currently. And at this rate, your master will be the host of quite a few of them! Hehehe… well, he already is by now.”, the imp on Edward’s right shoulder answered.

Another imp appeared, this time grabbing the back of Edward’s head. “Hey! Nice to meet you ~”

Edward’s face stated to turn red from anger.

“Nice to meet you too!”, Willy answered joyfully, before chatting with the imp on Edwards right shoulder again. “You know, I have many brothers too! And we all look quite similar as well. How do you imps always know who is who?”

Edward suddenly slammed his fist against the table in front of him. “WILLY, JUST SHUT UP!!!”, he screamed.

“Oh, we don’t know who is who, actually!”, the imp on Edwards right shoulder answered.

Edward could feel a leathery tail wrapping around his throat, another imp appearing - hanging upside down against his chest. “Heya!”

In resignation, Edward just rests his head on his palms.”Ugh… FINE!! I’ll play! Explain the rules and let’s get this over with.”

The imp on Edwards right shoulder nodded. “Very well! The rules are simple. As I mentioned, you have to find my always lying brother, Bugia, by asking us as many yes or no questions as you want. We’ll switch positions randomly before the game starts - but not physically. We can simply switch our consciousness between bodies at will, so you won’t be able to notice it. Oh, and I almost forgot to mention it, but every time you ask one of us a question, that imp randomly chooses another imp who behaves differently and switches consciousness with it after answering the question. So, an imp who always lies won’t switch with another imp who always lies and instead choose an imp with a different behaviour, like for example one that always tells the truth. But as you already know, there is only one imp who always lies between us - my brother Bugia. Also, just to clarify: The switch happens instantly after the question was answered. Keep that in mind when you want to point out who Bugia is, since you’re always going to do that after the switch happened. And yes, you can win this game - without asking dozens of questions, too! Alright, lets begin! We’ll all shuffle our consciousness now - good luck!”

After thinking about it for a while, Edward grit his teeth. “This is impossible! The way the rules are laid out, it’s not possible to ever guess who Bugia is with 100% certainty!”, he mumbled.

Willy started to wiggle with his antennae again. “Oh… Oh! Master! I think I know! I figured it out! Those imps really are tricky! I can solve the riddle with at most two questi--”

“Shut up, Willy! You already caused enough problems - it’s your fault there are so many of them in the first place!”, Edward interrupted. “Uggh… Why does this kind of stuff always happen to me?”, Edward sighed.

“It’s probably bad karma!”, one imp answered - before Edward could feel another imp land ontop of his head. “Hey there!”

“THAT WAS A RHETORIC QUESTION!!”

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Edward’s Notes

• There are now 7 Imps. One of them always tells the truth, one of them always lies and 5 of them give random answers (I'll refer to them as unreliable).
  • Each time I ask one of the unreliable imps a question, they'll decide between behaving like an imp who always tells the truth or an imp who always lies, randomly, and answer the question.
• I can ask any imp a yes or no question. If I ask an “illegal” question that can’t be answered with yes or no a new imp will spawn that gives random answers.
• There was no mention about a limit of how many times I’m allowed to ask yes or no questions.
• Apparently, they cannot tell each other apart. So they probably don’t know who’s a liar and who isn’t (except for themselves, obviously).
• After answering my question, the asked imp will instantly switch position with another random imp who has a different behaviour. Examples:
  • An imp who always lies, will switch with an imp who always tells the truth or an unreliable imp.
  • An imp who always tells the truth will switch with an imp who always lies or an unreliable imp.
  • A unreliable imp (who answers randomly) will switch with an imp who always lies or one who always tells the truth.
• They never clarified if an imp knows with who he’s going to switch. Since they said it’s random, the imps probably don’t know their future position themselves.
• The imps switch consciousness between bodies, so it’s impossible for me to know who switched with who just by looking at them.
• My goal is to identify Bugia, the imp who always lies.
  • Pointing out who Bugia is will always happen after the imps switched consciousness. This means I’ll have to make sure in some way, that I know who is going to switch in.
• At least two informations seem to contradict themselves… and how am I supposed to find Bugia if I can only guess after they switch consciousness? This makes no sense! Maybe I missunderstood something?

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What questions could Willy have in mind, to solve this seemingly impossible game with at most 2 questions? And would it still work, now that there are seven imps - one more than when Willy mentioned having a solution?

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Hint 1

There seems to be a very subtle contradiction in Edward's notes. Maybe that's a good point to start?

Hint 2

If the puzzle seems impossible to you as well, maybe you should analyze what the imps are saying a bit closer. It could be that they try to confuse you with "context".

Hint 3

Trying to figure out what questions to ask the seven imps is the last step! You have some work ahead of you before doing that.

Answer 1

Contrary to my previous answer, I will assume that imps start switching consciousness right after the first question. This assumption is supported by the highlighted word "every" when Acredo explains the rules. Let's look at the answers given by the imps:

Answer 1: "We’re imps! You have summoned us - and we came to play a little game with you. I’m Acredo and that’s my brother Bugia. Every sentence I say is the truth, while every sentence my brother says is a lie. As soon as you want to start the game you just tell us, and we’ll switch positions at random. You can ask us as many yes or no questions as you want, to detect who of us is my always lying brother, Bugia. But I should warn you… you only get one chance to guess who of us it is before the game is over. Every time you ask one of us a question that can’t be answered with just yes or no, like you just did, another one of our brothers appears after the imp you asked finished answering your question. I will explain you the further rules once you accept to play our game"

As the first sentence states the obvious this answer seems to be true. This means that Acredo and Bugia now switch places and every sentence of the next answer will be a lie.

Answer 2: "Because we’ll stick around until your last days if you lose or don’t play! And if you kill an imp, two more show up! By the way, all of my brothers appearing because of your illegal questions(1) are completly unreliable. Each time you ask them a question, they'll decide between behaving like an imp who always tells the truth or an imp who always lies, randomly, and then answer the question"

We know every sentence of this answer is a lie. Which means that every imp appearing is actually reliable, telling always the truth or always lying. Together with the assumption that the imps switch after each question, the next imp must be a truth-teller.

Answer 3: "Oh, we are a big family. Well, at least compared to humanoid families. I have 257 brothers, currently. And at this rate, your master will be the host of quite a few of them! Hehehe… well, he already is by now."

Nothing interesting in this answer, but continuing with our assumption, the next answer must be a lie.

Answer 4: "Oh, we don’t know who is who, actually!"

As already said, this is a lie. Not as useful as it might look though, because of the next quote which explains the rules and must be true again.

The rules: "Very well! The rules are simple. As I mentioned, you have to find my always lying brother, Bugia, by asking us as many yes or no questions as you want. We’ll switch positions randomly before the game starts - but not physically. We can simply switch our consciousness between bodies at will, so you won’t be able to notice it. Oh, and I almost forgot to mention it, but every time you ask one of us a question, that imp randomly chooses another imp who behaves differently and switches consciousness with it after answering the question. So, an imp who always lies won’t switch with another imp who always lies and instead choose an imp with a different behaviour, like for example one that always tells the truth. But as you already know, there is only one imp who always lies between us - my brother Bugia. Also, just to clarify: The switch happens instantly after the question was answered. Keep that in mind when you want to point out who Bugia is, since you’re always going to do that after the switch happened. And yes, you can win this game - without asking dozens of questions, too! Alright, lets begin! We’ll all shuffle our consciousness now - good luck!"

Much information here, and we know each sentence must be true. Lets look at the highlighted parts:

every time you ask one of us a question, that imp randomly chooses another imp who behaves differently

This is our assumption from the beginning.

So, an imp who always lies won’t switch with another imp who always lies and instead choose an imp with a different behaviour, like for example one that always tells the truth.

Here the truth-teller tries to avoid to say anything about unreliable brothers. Which is one more hint that they don't exist.

there is only one imp who always lies between us - my brother Bugia

This means all other imps tell the truth. Having this information the solution is pretty simple (actually the same as in my previous answer):

• Choose any imp and ask him a question like "Is 1 bigger than 2?".

• If he tells the truth, he has to switch consciousness with Bugia because he is the only liar, and then we know that Bugia is in the body of the imp who just answered.

• If he tells a lie we were asking Bugia, but he is gone now. Ask the same imp the same question, now he must tell the truth and switch with Bugia again. Then we know that Bugia is in the body of the imp who just answered.

Answer 2

You don't need to ask any questions to "win" the game (one way or another).

At the very start, there are only two imps (Acredo, the truth teller, and Bugia, the liar). One of them starts talking and states "we're imps", which based on Edwards reaction, we'll trust to be true, meaning the speaker must be Acredo (since every sentence he speaks is the truth).

From this point, we need to skip ahead to the explanation of the rules, specifically to the point where the imp says, "every time you ask one of us a question, that imp randomly chooses another imp who behaves differently and switches consciousness with it after answering the question".

If we assume this statement is true (by chance the imp speaking was Acredo, or a coincidentally truthful but unreliable imp), then the second imp to speak back at the start, has to have been Bugia (the liar), since he spoke only after the second question was asked, which has to have triggered this rule, and at that point there were no other imps to trade with. Therefore his statement that, "if you kill an imp, two more show up!" is a lie. So Edward can in fact win the game "the old fashioned way", as he puts it.

If we assume the rules statement is false (again by chance), then the claim that the imps can swap bodies is false. Which means Bugia, the liar, is still sitting in the same place he started - hanging onto Edward's left ear.

So to win, Edward should first identify the imp hanging onto his left ear as being Bugia, and if he's right, he wins. If he's wrong, then he should just kill them all. The end.

Answer 3

Pick any imp body to direct the questions to. The one clinging to Edward's chest is convenient. Pick a different imp body (e.g. the one on Edward's right shoulder) and call it the target. Assume that the imp explaining the rules was telling the truth about the rules. (If the rules can't be trusted, this solution doesn't work.)

This is part of the rules:

every time you ask one of us a question, that imp randomly chooses another imp who behaves differently and switches consciousness with it after answering the question.

I will take that to mean that the choice is random and occurs while the question is asked (the choice is made before answering), and that switching occurs after answering. That is, the imp knows and is able to tell you (given an appropriate question) which position it will switch to.

For any given question, any imp will consistently tell the truth or say its opposite. Questions of the sort "Would you say X is true?" always produce truthful answers from an imp that is telling the truth. For a lying imp, "Is X true?" would give the untruthful answer, but then "Would you say" would invert the truth value, giving the truthful answer.

The imps always switch with a random imp of a different "behaviour", so after every non-Bugia answerer, you have a 2/3 probability of getting Bugia, and each time Bugia answers, you are certain to get a non-Bugia imp. So given any pair of successive answerers, there is a non-zero probability that one of them is Bugia. Since the switches are random, Bugia will eventually turn up as answerer.

Now, consider just the times Bugia turns up as answerer. Bugia's "behaviour" is unique, so can switch with any other imp. Since the switches are random, Bugia will eventually choose the target to switch to.

Putting this all together, eventually Bugia will eventually appear as the answerer and will switch to the target. So keep asking the following question until you get a "Yes":

Would you say that you are Bugia and that you will switch to my right shoulder after answering this question?

Only Bugia will answer "Yes", but only if it is going to the target location. Every other imp will answer "No", and even Bugia will answer "No" if it isn't going to switch to the target. So once you get a "Yes" answer, point to the target (Edward's right shoulder) to identify Bugia.