# 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.