[Math] solve the puzzle how many liars

Question

Each boy in a group of $20$ boys either always tells thuth or always tells a lie. These boys are sitting around a table. Each boy says that his neighbours are liars. Prove that at least $7$ out of $20$ must be truth tellers.

Answer 1

If there are $\le 6$ truth tellers, then there exist $3$ liars sitting together by pigeon-hole principle.

Now the one in the middle of these three is telling the truth when he/she says "my neighbours are liars", a contradiction!