A puzzle from the 1998 Moscow Mathematical Olympiad, via Peter Winkler’s excellent Mathematical Puzzles, 2021:
An anthropologist is surrounded by a circle of natives. Each native either always lies or always tells the truth. The anthropologist asks each native whether the native to his right is a liar or a truth teller. From their answers, she’s able to deduce the fraction of the circle who are liars. What is the fraction?

SelectClick for Answer 
The key is to notice that if all the liars were changed to truthtellers and vice versa, none of the answers would change. So if the fraction x can be determined, it must have the property that x = 1 – x, and x must be 1/2.
