Who's telling the truth?
In this puzzle we have $32$ people. Some of these always tell the truth and the rest always lie. Our task here is to find out the liar/s. For these, every person writes down each of these sentences:
Exactly one of these $32$ people is a liar
Exactly two of these $32$ people are liars
…
Exactly thirty two of these $32$ people are liars
Who is/are the liar/s? Why?
Solution
First, it is not possible that two (or more) people are telling the truth, because in that case we have that exactly $i$ and $j$ people are liars, which is not possible. Second, is not possible that everyone is a liar, because in that case everybody is telling the truth and then there are $i$ liars which is not possible. Finally, the only person telling the truth is the one saying "Exactly thirty one of these $32$ people are lying", which are the rest of the people.
Manuel Cáceres
Postdoctoral Researcher
My research interests include algorithmic bioinformatics, graph algorithms, string algorithms, algorithmic bioinformatics, compressed data structures, safe & complete algorithms and parameterized algorithms.