Island of Liars

Question

A king goes to an island, where there are 5823 people who always tell the truth and 8723 who always lie.

The king is a special guest for the party celebrating the year 2053.

At the party, liars and truth tellers dance in couples (everyone can dance with everyone). Some people leave the dance, others join, and some don't even participate in the dances as they are not dance lovers.

After the party, the king asked everyone who had danced at the party how many truth tellers they danced with, and put the numbers into his diary. Surprisingly, when he checked his diary later on he found out he had all the numbers from 0 to 956 typed in uniquely.

How many truth tellers danced at the party, assuming everyone knows who is truth teller and who is liar?

Answer 1

I think that the answer is

Just $1$ truthteller danced.

Reasoning

Suppose more than one truthteller dances, say $N$. Now consider the graph of dancers just involving truthtellers, where an edge represents "danced with". Within this graph, either the lowest degree of a node is $0$, where then the highest possible is $N-2$, or the lowest degree is $1$, where the highest possible is $N-1$. In either case, by the Pigeonhole Principle, there will be at least two truthtellers who report the same number to the king. Of course, if there are zero truthtellers who danced then one of the liars would have to have told the truth - saying $0$.

Proof that this works

Suppose there is $1$ truthteller, he will report $0$. There just needs to be one liar who doesn't dance with the truthteller who reports $1$. Then everyone else is verifiably lying.

Answer 2

Note one very important thing that has been mentioned, "he found out he had all the numbers from 0 to 956 typed in uniquely". This means that no two people said the same number. Also, since the king got 957 total replies then it means that exactly 957 people danced."

Now, let's look at the various possibilities.

Case 1: There were 0 truthtellers among these 957 people. This means that all the 957 were liars. But if this was the case, then the person saying 0 is saying the truth, which is a contradiction. So, all 957 people cannot be liars because then, the person who said 0 is telling the truth.

Case 2: There was one truthteller. This means that he would be the person who would have said 1. And all the others who danced were liars and lied. It is easy to see that this is a valid possibility. There can indeed be one truthteller and 956 liars and everything checks out.

So, we have found one valid solution namely, there was one truthteller and 956 liars who danced at the party. Could there be other valid possibilities? Let's explore further.

Case 3: There were 2 truthtellers. If this was the case then both these truthtellers would have said 2. But , there was only one person who said 2. So, this is not a possibility.

In fact, number of truthtellers cannot be >1 because if there were n truthtellers ( where n >1) then there would have been n people whose answer would have been "n". For instance,

if there were 3 truthtellers then exactly 3 people would have said 3 and the rest, who were all liars, would have said some other number.

If there were 4 truthtellers then exactly 4 people would have said 4 and the rest, who were all liars, would have said some other number, etc.

But everybody says a unique number and no two people say the same number.

So, in a nutshell, the only possibility is that there was one truthteller and 956 liars who danced at the party.