'Impossible' Logic Puzzle - How Many Liars Are At The Party?

A party has 100 people who are either liars are truth tellers. Liars always lie and truth tellers always say the truth. After the party you ask each person, "How many truth tellers did you shake hands with?" Each person gives a different answer from 0 to 99 (the whole numbers 0, 1, ..., 98, 99). How many liars were at the party? This seems like it's impossible to figure out, but it's not! Watch the video for the surprising solution to the logical puzzle.

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[TheUniqueImpact]

It must've been a nightmare to host this party if 99% of attendees said they couldn't make it, then they all showed up.

[mesplin]

3 logicians walk into a bar. the bartender asks "Do you all want beer?"
The first says "I don't know"
The second says "I don't know".
The third says "Yes!"

[Firehead]

A crucial unstated assumption in this puzzle is that everyone in the party already know each other's identity (lier or truth teller). I thought they revealed their identities to each other while shaking hands, telling a truth or a lie. That got meta ready quickly.

[otter.phillips]

"You callin me a liar?"
"Well I ain't callin you a truther!"

[SethalaTheGamer]

For me, this problem became much simpler when I worked on a slightly different problem, a party with 4 people where each of them gave a different number of handshakes (0, 1, 2, or 3) with truth tellers. Once I had a small enough group to actually work with, it was easy to figure out that 0 must be a truth teller and everyone else must be lying, and then the same logic works perfectly fine with the full puzzle.

[pollaeng]

Isn't that scary to be in a party of all Liars? My sympathy to Person0

[bgbbft]

I started with "what if nobody shook hands?" and saw that only Person 0 would be telling the truth.
Then I went "meh; that's probably right"

[andrewchou]

"Hey this is pressure locker" -youtube auto generated subtitles

[megarotom]

It also means that person 0 and person 1 didn't shake hands otherwise person 1 told the truth

[L]

I incorrectly assumed that everyone shook hands with everyone else. Thus, it is impossible. If Person 0 is a truth teller, then it would necessitate Person 1 as a truth teller, which would then necessitate Person 0 as a liar. It’s a paradox. So, I think this solution’s explanation would have been better by explicitly stating that it is necessary for Person 0 and Person 1 to have never shook hands ... and there is no assumption that everyone shakes hands. Therefore, Person 1’s statement about shaking 1 truther’s hand can be a lie even though there is exactly 1 truther at the party.

[OnEiNsAnEmOtHeRfUcKa]

What if they just... didn't shake hands with everyone at the party?
That's never exactly made clear.

[theoldfinalchapters]

Alternate answer: Everyone is telling the truth, but everyone shook Person 0's hand when he was asleep.

[dkmt_king]

That would make person 1 a truth teller though! This doesn't solve the problem, it just makes it more difficult!

[KnakuanaRka]

I found a simpler/more intuitive way to figure this out:
They can’t all be liars, since then the one who said 0 would not be lying, so there must be at least 1. What if there was more than 1?
If there was 2, they must have said 0 and 1 (can’t say 2 or more since each only had 1 other truth teller to shake hands with); however, the 1 could only have shaken hands with the 0, who didn’t shake hands with the 1, so this is impossible.
Similarly for any other number greater than 1 (for 4, they must have said 0-3, and 3 must have shaken hands with all the others, including 0, which is impossible).
So the only truth teller is 0; everyone else is a liar.

[BrokeNSings]

I thought of it differently, though now, looking back, i assumed(perhaps unrealistically) that everyone shook hands with each other, no one missing anyone.

Anyway, there can only be 1 truth teller, because each person has a different answer. The truth is a constant, therefore if there were more than 1 truth teller, we'd have repeat answers.

[bgbbft]

Note that we are _assuming_ that everyone at the party knows whether everyone else is a liar or truth teller. It could be the case that they think they are telling the truth, but are mistaken; or think they are lying but accidently be correct.

[tuerda]

Unstated assumption: Handshakes are mutual. If A shakes B's hand B then B also shakes A's hand. This sounds obvious, but you never know, maybe A used her foot to shake B's hand.

[Lucy-ngcw]

But that means 1 is telling the truth too?

[DennisMathgod]

Wait a minute... in your scenario, person 0 is a truth teller and all others are liars. The problem is, person 1 said he shook hands with 1 truth teller, which would be a true statement. Hence, person 1 can't be a liar. This makes your scenario contradictory.

[jakubdutkiewicz]

We can also deduce, that person 0 didn't shake hands with person 1. If person 1 shook hands with person 0 and person 1 told that he has shook hands with one truth teller, he would be telling the truth, which is contradictory. Hence person 1 mustn't shook hands with person 0. I wonder if we could induce that property.