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Here's an interesting variation on the classical liar puzzles:

Trutham-and-Ly is a small town in the north of Logistan. It is divided into two districts, Trutham and Ly, separated by a small stream. Truthamers always tell only the truth, and Lyers only tell lies. While walking over a bridge crossing the stream, you meet the two local policemen, Bob and Bill. You know that both of them are from Trutham-and-Ly.

Bill says: "Lyers never lie."

Bob says: "Bill is from the same district as I am."

Which district is Bob from?

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    $\begingroup$ Are you sure Bob's statement is say-able ? It's sure that Bill is a Ly-er, I find some vague paradox about Bob's statement. $\endgroup$
    – user27395
    Oct 6, 2016 at 15:59
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    $\begingroup$ Either it's a paradox, or this needs the lateral-thinking tag. $\endgroup$
    – qwertyu63
    Oct 6, 2016 at 16:08
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    $\begingroup$ Not really, they covered this case in logic class. You may just have thought it was trivial. (and/or I'm wrong and we'll see another clarification) $\endgroup$
    – mr23ceec
    Oct 6, 2016 at 16:10
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    $\begingroup$ @mr23ceec: Who is "you" ? Who is "they" ? $\endgroup$
    – user27395
    Oct 6, 2016 at 16:18
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    $\begingroup$ @Arbitrary Kangaroo (you're joking, right?) "You" is the hypothetical character symbolizing the reader of these lines. "They" is the secret society that includes everyone who ever taught logic to students, but is known to no one else (and members of it will deny being in it.) [and the lowercase you in "you're" refers to Arbitrary Kangaroo specifically] [[and "may" is shorthand for "may or may not, your choice"]] $\endgroup$
    – mr23ceec
    Oct 6, 2016 at 16:22

5 Answers 5

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Is it this?

All Lyers are mute and illitirate (more precisely dysgraphic). Bob (and Bill) are from Trutham.

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    $\begingroup$ But then they would surely lie six feet underground… $\endgroup$
    – Arkku
    Oct 6, 2016 at 16:11
  • $\begingroup$ The point is 'Lyers' is an empty set. $\endgroup$
    – mr23ceec
    Oct 6, 2016 at 16:13
  • $\begingroup$ Yes, I understood the point, but I'm hoping this is not the intended answer. (In particular, I would then argue that the statement "Lyers never lie" is not restricted to currently living Lyers, or to the future, i.e., past Lyers that have lied would still be part of the set.) $\endgroup$
    – Arkku
    Oct 6, 2016 at 16:16
  • $\begingroup$ @mr23ceec Is that the set that needs to be empty? $\endgroup$
    – Anon
    Oct 6, 2016 at 16:20
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    $\begingroup$ If this answer was the intent of the question, then the question given is flawed. "Lyers only tell lies" is disingenuous subterfuge; them being mute means they in fact DON'T tell lies, as they don't tell anything. If the question had stated "Truthamers always tell only the truth, while Lyers never tell the truth", it would have been correct, and provided to the careful reader the subtle hint needed to solve. $\endgroup$
    – Rubio ♦
    Oct 6, 2016 at 19:59
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We know that

Bill is a lyer since the statement he says is contradicted by the facts of the question

For Bob

if he was a lyer

He would be from the same district as Bill but would not say so. Thus we know he is not a lyer

if he was a Truthamers

He would be from a different district as Bill and would say so. Thus we know he is not a Truthamers.

So I conclude that

McFry is a lyer

Or

lyers sleep standing up (they don't lie down) and Bill is a Truthamers.

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    $\begingroup$ Excellent conclusion :) $\endgroup$
    – user27395
    Oct 6, 2016 at 16:05
  • $\begingroup$ This is a disappointing and generic answer that could equally well be applied to any other liars puzzle. It's little more than "lol I give up". $\endgroup$
    – Anon
    Oct 6, 2016 at 18:20
  • $\begingroup$ @McFry and the accepted answer and the question are also equally disappointing since Bill said "Lyers never lie" which includes any form of forming a lie (nodding, writing, pointing to pictures, etc) even if someone is mute, illiterate with dysgraphia they likely can still communicate an answer and thus lie. Which makes Bill's statement false. $\endgroup$
    – gtwebb
    Oct 6, 2016 at 18:27
  • $\begingroup$ @gtwebb The correct answer is "Bob is from Trutham and lyers have never made a single statement", not "lyers have reduced statement-making abilities". I accepted that answer because it was the closest to the intended one, I'm not particularly happy with the formulation myself. $\endgroup$
    – Anon
    Oct 6, 2016 at 18:33
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Well, taking the obvious

Lyers never lie -> False -> Bill is a Lyer;
Bill is from the same district as me -> True -> Bob is a Lyer who spoke truth, contradiction
Bill is from the same district as me -> False -> Bob is a Truthamer who lied, contadiction
Bob speaks Paradox

This implies that something else is required.

Easiest is that Bob is from a second town, named Paradise

Alternately

Lyers never lie, they sleep sitting, they never fall down drunk, it's really quite marvelous. They do tell lies, but that's not necessarily what the question is asking.

This (fanciful) interpretation leads us to the following

Lyers never lie -> True -> Bill is a Truthamer
Bill is from the same district as me -> True or False -> Bob can be from either Trutham or Ly without contradiction

And now to determine Bob's district

We think to the bridge, if they were both from Trutham, or both from Ly, this seems an odd spot to converse on the job; a donut shop would be far superior. Clearly the Bridge is the point where their districts, and thus duties, overlap; and so Bob is from Ly.

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It's a paradox. If we assume the rule "Lyers only tell lies" is true, than Bill is a Lyer. But that means Bob is in essence saying "I am a liar". Neither truth teller or liars can say that. Therefore, Bob must be from out of town or something, because he can't be from either district.

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  • $\begingroup$ Whoops, sorry. I forgot to specify that Bob and Bill are indeed both from Trutham-and-Ly. $\endgroup$
    – Anon
    Oct 6, 2016 at 16:03
  • $\begingroup$ Oh yeah, that's the correct answer. Bob's just come for a chit chat with his friend Bill, form Normal-O-Vania, a neighbourhood of Trutham and Ly :) $\endgroup$
    – user27395
    Oct 6, 2016 at 16:04
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Bob is ...

talking about some other Bill, not the policeman he is with.

So Bob is from ...

Ly, maybe. Or Trutham, maybe. All depends on where the Bill he is actually talking about comes from.

Or, a different possibility is that

Bill comes from Trutham-and-Ly, but lives not in either district but rather on the bridge, or perhaps in a house-boat on the stream. He can lie if he wants, but is not required to.
The question asks which district Bob lives in, so assuming he in fact lives in one or the other, he would thus be lying about living in the same district as Bill.

Ergo Bob is a Lyer.

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