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August 17th, 2005
04:12 am
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Riddle: The Paradox Dragon
Wow, I've been on an unusual mathematical kick lately. Must be the aftereffects of all the good news I'll mention in my next post, or something.

First amthrax came up with the river-crossing riddle I posted here two days ago; I not only solved it, but came up with a proof for why the given solution is optimal.

Now, by chance I've stumbled upon the PartiallyClips strip (LJ: partiallyclips) named "Paradox Dragon". And the logic circuits of my brain kicked in.

Right off the bat, the third panel contains a fairly obvious paradox. (Warm-up puzzle: What?) I'm going to assume that it was put in there intentionally, to give the strip its title; or maybe it simply drifted in as part of the punchline. However, if you discard the third panel entirely ... the first two panels are a self-consistent logic puzzle. I've reprinted the dialogue below, modified to clarify it for problem-solving purposes:

NARRATOR: On the hundredth day of my quest, I faced the Paradox Dragon. I knew that one of its three heads always lied, one always told the truth, and the other head would alternate [back and forth between telling the truth and lying].
FIRST HEAD: You may ask us one question, then you must guess which head is which.
SECOND HEAD: He's lying. You get three questions.
THIRD HEAD: Oh no, it's definitely one question.

NARRATOR: Not willing to risk counting on three questions, I asked [the dragon] ... what the second head would say if I were to ask it if the third head had been lying when it agreed with the first head that I could only ask one question.
FIRST HEAD: (*non-answer*) Oh ... okay, hold on a minute. ...
SECOND HEAD: He'd say, "Yes, the third head was lying."
THIRD HEAD: ... [The second head is] lying [in the statement he just gave].

To my delight, I discovered that the information given was sufficient to triumph over the Paradox Dragon! Anyone else out there care to reproduce my work and solve Rob Balder's riddle?

(n.b.: Comments for river crossing riddles now unscreened. Comments here, as with my other riddle posts, are screened to give everyone time to play along at home.)


Edit, 1-11-2005: Going back through old entries to tag everything up, realized that I never unscreened anyone's solution. Oops! Fixing that now -- keep in mind all answers below were the product of independent work.

Current Mood: tiredinsomniac
Current Music: Elastica, "Connection"

(15 comments | Leave a comment)

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Date:August 17th, 2005 12:19 pm (UTC)
Leaving a comment here with my solution before I jet off to bed:

Warm-up puzzle: The second head's statement that "I was lying." Neither a truth-teller nor a liar would be capable of that statement. However, we already know the second head is not a waffler from panel one (see full explanation below). Thus it cannot be any of the three, a paradox.

The first head is the waffler, the second is the truth-teller, and the third is the liar. Outline of proof in comment to come.
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Date:August 17th, 2005 01:12 pm (UTC)
I'm not sure I got the riddle right... The alternating head will be switching between telling the truth and lying *every time*, right? No two successive questions answered the same way? If so...

The second and the third head always disagree. This means the alternating head cannot be one of those two. Hence, head 1 is the alternating one.

To sort out 2 and 3, let's first assume that 2 always lies and 3 always tells the truth.

If so, head 3 would say, when asked if it had been lying, that it hadn't.

Hence, head 2 would claim that it WAS lying.

Hence, head 2's actual statement, that head 2 would claim head 3 was lying, is the truth. Which contracts both hypotheses regarding heads 2 and 3.

Unless head 2 got confused, which I wouldn't blame it for, honestly.

However, if we assume that 2 always tells the truth and 3 always lies, it all works. Head 2 would indeed claim that it would say head 3 was lying. Head 3 would claim that this truth was a lie.

Hence, head 1 alternates, head 2 tells the truth, head 3 lies.

For a much simpler solution, though, try asking two other questions and see if the dragon answers at all. :)
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Date:August 17th, 2005 02:16 pm (UTC)
Attempted solution:

  • First and Third both agree in Panel 1. Therefore, one of them must be consistent (either truth-teller or liar), and the other the alternator.*
  • Second makes two statements in Panel 1. Since we already know Second's not the alternator**, this fact is of little relevance, other than guaranteeing that the statements are either both true or both false.***
  • In Panel 2, Second says that Second (the "he") would claim that Third was lying when Third agreed with First (whew!). Let's take it backwards.
    • Third agreed with First. We knew that.
    • If Third had been lying when Third agreed with First, both of them must have been lying.
    • Second said this. Since we know Second is consistent (i.e. not the alternator****), he would be guaranteed to say this irregardless, since he said they were lying before. Therefore, Second would claim that Third was lying when Third agreed with First, and this is guaranteed.
    • Second says this. Therefore, Second is the truthteller.

  • Since Second is the truth-teller, Third is lying in both Panel 1 and Panel 2. Therefore, Third is the liar.

Thus, the final solution: First is the alternator (and opens by lying), Second is the truth-teller, and Third is the liar.

* This is not exactly true. Careful reading of First's opening statement shows that it is a compound claim, whereas Third's statement is not. This will lead to an alternate solution, which will be enumerated after the footnotes.

** This is not true either. Since, in the third panel, Second admits to having lied, Second must be the alternator. (This works even if the admission was a lie, because it means that the earlier statement was true. Either way, changing from true to false, therefore alternator.) Hence the second solution, below.

*** Liars can say compound claims (e.g. "I am red and he is blue") even if only one of the claims is false (e.g. "I" and "he" are both blue).

**** I already told you this wasn't true, of course.

Second Solution:

In order to make this solution properly, I will retranscribe the conversation, recreating the Narrator's dialogue as best as I can.

First: You may ask us one question, then you must guess which head is which.
Second: He's lying. You get three questions.
Third: Oh no. It's definitely one question.
Narrator: First Head, what would the Second Head say if I were to ask it if the Third Head had been lying when it agreed with the First Head?
First: Oh. Okay, hold on a minute. What was the, um... the third part of that question? About the agreeing?
Second: He'd say, "Yes, the Third Head was Lying."
Third: He didn't ask you. Besides, you're lying.
Second: Yes. I was lying.
Third: No you weren't.
First: Shut up! I'm working on the answer!
Second: No you aren't.

The first head is the liar, the second the alternator, and the third the truth-teller. I'll explain, it's a bit contrived complicated.

Panel One. First's lies because the narrator never has to guess which head is which (they all die of migraines). The Second tells the truth first – the first head is lying – but then lies about the number of questions (which was the true part of the compound sentence). Third points out Second's lie.

Second Panel. First pretends to be working on the answer, but since First is a logic puzzle entity, and therefore an instantaneous puzzle solver, his entire dithering is a lie. Second (back on Truth mode) correctly answers the question – he had been on the same parity as before, and therefore would have been guaranteed to say that. Third points out that it wasn't Second's turn to answer, and that Second has now switched to Lying mode (thereby changing the answer).

Third Panel. Second lies by claiming he had been lying. Third points out that he hadn't. First continues his farce of Not Answering The Narrator's Question. Second, knowing this, and being now in Truth-Teller mode, points this out.

[User Picture]
Date:August 17th, 2005 06:50 pm (UTC)
B cannot be alternating, because when two heads agree and a third disagrees, the sole dissentor cannot be alternating between true and false answers because that would place the two heads in harmony with each other on opposite truth values. Therefore, B is one of the absolutes, and A or C is the non-deterministic. What cannot yet be determined from the first paragraph is which way "non-deterministic" went.

Paragraph 2: Since B is absolute, then C was lying on the first answer and either you have three questions or B tells the truth. (If B is telling the truth, it's saying how it would answer the question, and therefore C was lying. If B is lying, it would have answered a direct question with "C is not lying", and that would be itself incorrect. This is the standard double-negative gambit for getting information out of an absolute accurate/inaccurate source of unknown veracity.) Therefore, C was lying in question 1, as was A. Therefore, B tells the truth in all instances (because knowing that statement C1 is inaccurate confirms that B1 is accurate, and B is known to be absolute.) The final statement of C confirms C as the absolute inaccurate source, as it gave a wrong answer to two consecutive questions- firstly the question count, and secondly the claim that B was inaccurate, causing A to be the alternating source, B to be absolutely true, and C to be absolutely false. Much more importantly, you aren't required to guess which head is which because A made this statement while not telling the truth.

...The worrisome corrolary is that you're going to get eaten no matter what. Oops.

In the comic strip, the first head actually did have something to say, accusing the third of lying that time, which was a true statement. Note that it made a false statement the first time. However, you can omit this, as you ahve, because the information is simply redundant.

As an amusing note, the third panel is clearly inconsistent from the information given in only the first panel and one of its dialogue points. B cannot be alternating, and is either the correct or incorrect source. Such an absolute source will never claim to lie, because it always lies and it would be a true statement or it never lies and it would always be a false statement. B's "I lied" cannot be valid in any circumstance given the first panel.
Date:August 17th, 2005 07:13 pm (UTC)
T3h melony one again. Head A alternates, head B lies all the time, and head C tells the truth... I think.
And which head had the migraine, anyway?
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Date:August 17th, 2005 08:59 pm (UTC)
There are only 12 possible situations - there are 6 truth permutations, and the mixed head is either telling the truth or lying during his first response.

However, the first and third head agree initially, therefore the second dragon cannot be the mixed teller.

Since the second and third dragon consistently disagree, the third dragon can't be mixed. Therefore the first dragon is mixed.

This leaves us with two possibilities:
The first dragon initialy lied, and the second dragon tells the truth
The first dragon initialy told the truth, and the second dragon lies.

Now, the second head says that, if [the second head] was asked whether the third head lies it would answer with yes. However, it would answer the interior question with yes regardless of whether it was the the liar or truth teller, so it must have be the truth teller.

Therefore we have:
The first head is mixed
The second head tells the truth
The third head lies

P.S. The author's single question cannot have been sufficient to identify the heads, unless he was trusting the dragon to honestly tell him when he was out of questions (and, in that case asking the first dragon, "Do I really only get one question" would have been sufficient, and easier.)
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Date:August 17th, 2005 11:53 pm (UTC)
I would say that the second dragonhead was the lying one

Not too sure 'bout the others, but I'd take a stab at guessing the third head always tells the truth? and the first head is alternating..

though I might have those two mixed up
Date:August 18th, 2005 02:10 am (UTC)
Thanks for the comment about the bridge problem; I was able to solve it. This problem, however, has some problems. The first is that the second head's second statement is triply ambiguous. He could mean "The first head would say that I would say that the third head is lying", which is what I take it to mean in the paragraph below. Here he's acting as if the first head will answer the question. He could also mean, "I would say that the third head is lying"; here he's acting as if the question is addressed to him, and he's just sardonically responding in the third person because it wasn't. And finally, the literal reading is, "The first head would say that the third head was lying", which would mean that the second head misunderstood the question, and no conclusions can be drawn from that answer.

Anyway, assuming that the first case is true (since it holds most closely to the question), there are two solutions. The first set of responses establishes that the second head either only lies or only tells the truth, since the other two agree, and it also establishes whether the alternating head starts out on a truth. In one solution, the first head alternates, starting with a falsehood; the second head tells the truth; and the third head always tells a lie. So the third head merely confirms what the second head is saying, and the second head is saying (truthfully) that the first head (now due to speak the truth) would say that he, the second head (a truth-speaker), would say that the third head lied (which is true).

In the other case, the first head alternates, starting with the truth; the second head always lies; and the third head always tells the truth. In this case, the second head's second sentence is false, and the third head merely confirms it. So the second head says that the first head would say that the second head would say that the third head was lying. Now the third head was telling the truth. So the second head, if asked whether the third head was lying in the first statement, would lie and say that he was, indeed, lying. The first head is due for a falsehood, so if he had answered the question, he would have said that the second head would have said that the third head was telling the truth. But the second head is once again obliged to lie, so he would say that the first head would say that the second head would say that the third head was lying, which is indeed what happens. Thus there are two solutions. *grins* Good thing for that migraine.
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Date:August 18th, 2005 02:29 am (UTC)
Third panel first:

2 says "3 was lying" and "My previous statement was a lie," i.e. "3 was telling the truth."
3 says "2 was lying" and "2 was telling the truth."

Both these pairs self-contradict. Therefore neither 2 nor 3 can be always truthful or always lying. They must both be alternating truth and lies, which contradicts the initial conditions.

Now for the main event. Watch me flounder!

For Head 2, the second question is a double positive if he's telling the truth and a double negative if he's telling a lie. Therefore head 2's answer to the second question has to be true no matter what.

(Insert Flounder: (I tell truth(about the truth I'd tell)) means he reports a true statement. (I lie(about the lie I'd tell)), also means he reports a true statement.)

If my logic is correct, then when Head 2 says "I'd say Head 3 was lying," Head 3 must have lied on the first question. That means 2 told the truth on the first question.

If 2 told the truth once, then 2 must either always tell the truth or must alternate truth and lies-- if he told the truth once, he doesn't always lie, so that possibility is right out.

3 lied the first time. If 3 tells the truth now when he says "2 is lying," then 3 is alternating between truth and lies. But 2 must ALSO be alternating between truth and lies. The conditions of the riddle are that only one head alternates between truth and lies. Since both can't, neither can. 3 must always be lying, 2 must always tell the truth.

Which means that 1 lied the first time (since he agreed with 3, who was lying) and is telling the truth the second time when he says he's confused. Well, I sympathize, I am too.
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Date:August 18th, 2005 06:16 am (UTC)
The truth-teller and the liar must always disagree, so since heads 1 and 3 agree in panel 1, one of them must be the alternator. Therefore, head 2 cannot be the alternator.

Also in panel 1, heads 2 and 3 disagree. Head 2 does not alternate, so he would have claimed that head 3 was lying. He says so, therefore head 2 is the truth-teller. And since head 3 is still disagreeing with him, head 3 is the liar. Which leaves head 1 to be the alternator.

The paradox in the third panel is that both heads 2 and 3 are disagreeing with their own previous statements, which contradicts the given fact that only one of the heads is an alternator.
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Date:August 18th, 2005 09:25 pm (UTC)
The first head is mixed, the second tells the the truth, and the third lies.
(And the narrator couldn't have figured out the heads just from the answer to his second question thus the puzzle isn't actually self-consistent.)

The first and third head agree on the first question, so they're either both telling the truth, or lying - either way, one of them must be the mixed dragon. The second and third consistently disagree, so neither can be the mixed dragon.

In that case, it's clear that the second head would say that the third was a liar, so the fact that the second head says so makes the second dragon the truth teller.

That leaves the third head as the liar.
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Date:August 19th, 2005 10:48 am (UTC)

Excellent. (How can one possibly leave a riddle unsolved?)

The first head is alternating, the second is telling the truth, and the third is lying.

There are twelve possibilities: all permutations of truthful (T), lying (L), and alternating, where alternating can be either initially telling the truth ([T]) or initially lying ([L]). I'll represent a possibility as a triple like so: T L [L].

Starting with the heads' statements prior the narrator's question, any possibility where the first and third heads conflict can be discarded because this would contradict their agreement on the number of questions. This leaves four possibilities:

  1. L T [L]
  2. [L] T L
  3. [T] L T
  4. T L [T]

Now, consider the second head's response. There are two cases. If the second head is telling the truth (possibilities 1 and 2), the second head's response is consistent with the third head lying, so both 1 and 2 remain possible. If the second head is lying (possibilities 3 and 4), then that head would have said that the third head was telling the truth. This would have been a lie, so these two possibilities are only consistent with the third head lying. This leads to a contradiction for both 3 and 4, so both can be discarded. Luckily, we don't have to consider the possibility of the second head alternating. The possibilities are now:

  1. L T [L]
  2. [L] T L

*ding*ding*ding* Finally, we come to the third head's response. First, consider possibility 1. The third head has now alternated to telling the truth, so the second head is lying. This is a contradiction, so possibility 1 is out. Now, consider possibility 2. The third head is lying, so the second head must be telling the truth. Thus, possibility 2 is consistent and is the only remaining possibility.

Date:August 19th, 2005 11:16 pm (UTC)
The first and third heads agreed with each other in the first panel, so one of them has to be the alternator. Therefore the second head is either the truth-teller or the liar. If he's the liar, then the third head told the truth in the first and second panels, making the first head the alternator. If he's the truth-teller, then the third head lied in both panels, making the first head the alternator. Since the first head is the alternator in both cases, the first head must be the alternator.

If the first head told the truth in the first panel then the second head is the liar and the third head is the truth-teller. So in the second panel the second head would say that the third head lied, the first head would lie and say that the second head would say that the third head told the truth, and the second head, answering what the first head would say, lies and says that the third head is lying. This would be a lie and the third head would truthfully say that the second head is lying. This matches the dialogue in the second panel.

If the first head told a lie in the first panel then the second head is the truth-teller and the third head is the liar. So in the second panel the second head would say that the third head lied, the first head would say that second head would say that the third head lied, and the second head would say that first head would say that the second head would say that the third head lied. The third head would then be lying when he said the second head was lying. This too matches the dialogue.

Therefore I don't know which head is the truth-teller and which is the liar.

"Yesterday upon the stair
I met a man who wasn't there."
Date:August 29th, 2005 05:15 pm (UTC)
This would be a great deal easier if the questor had asked a sensible question in the second panel, such as "Are you the Paradox Dragon?". Then whichever head agreed with the second head would be the alternator, and of the remaining two heads the truth-teller would answer yes and the liar would answer no. Simple. But it would have killed the joke.
"Yesterday upon the stair
I men a man who wasn't there."
Date:September 5th, 2005 07:29 am (UTC)
more puzzles

"Yesterday upon the stair
I met a man who wasn't there."
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