# Island of Liars

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

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

On 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 on the party, how many truth tellers did he dance 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 on the party, assuming everyone knows who is truth teller and who is liar?

• Just to clarify a few things: 1. Do the liars have to put down a specific number, or can they just write down any number at all as long as it’s not the number of people they danced with? 2. Does the king participate in the dancing? Thanks! – PiIsNot3 Mar 27 at 9:37
• 1. Everyone can type whatever he/she wants as long as it's truth or lie. 2. Doesn't matter if king danced at all. – Kradec na kysmet Mar 27 at 9:47
• Is the King lying? What does a truth-teller say about dancing with the King? – Weather Vane Mar 27 at 9:52
• Got the right answer, but even if the king was liar/truth teller it would still fit in, but no he didn't dance at all, he dislikes dancing. – Kradec na kysmet Mar 27 at 9:57

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.

• That's it, that was fast even with all the baits I put. – Kradec na kysmet Mar 27 at 9:57
• @Kradecnakysmet Thanks, I was thrown for a second by the numbers and thought there might be a large number of solutions. – hexomino Mar 27 at 9:59
• I will admit I was laughing while typing all those random numbers :D – Kradec na kysmet Mar 27 at 10:00