Every person on the island is either a knight or a knave. Knaves always lie and knights always tell the truth. 2019 people get together on the island. Each person makes one statement taking turns. The first person says"We are all knights" Every other person at their turn says "at least 2/3 of the people who made their statement before I lied."How many knights are among 2019 people?
Suppose person 1 is telling the truth.
Then person #2 says that at least 2/3 of person 1 has told a lie which I will take to mean he says person 1 lied.
But we know that person 1 told the truth so this means that person #2 lied.
But person #1 truthfully said that none of them lied so we have a contradiction.
Thus person 1 must have lied.
If you work through the responses and the count of liars you see that 2/3 of the people are liars i.e. knaves.
\(\dfrac 2 3 \cdot 2019 = 673\)