Where do people find this many friends?
friends are seated around a closed circle. They can each be a liar or a truth-teller. Every one of them states, "Exactly one of the two people sitting next to me tells the truth" (although this statement can either be true or false, depending on their own identity).
How many liars are seated at the table?
The answer is 2018.
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One possibility for any number of friends is that they are all lying. But suppose there exists at least one truth-teller. Then they must be flanked by a liar and another truth-teller. This other truth-teller, since they already have the first truth-teller on one side of them, must have a liar on their other side. This liar, for the given statement to indeed be a lie, must then have another truth-teller on their other side. This process continues with every truth-teller being flanked by a liar and a truth-teller, and even liar being flanked by two truth-tellers. Since there would then be one liar for every two truth-tellers, the number of friends would need to be a multiple of 3 in order for this circle to 'close'. But as 2018 is not a multiple of 3, a circle with any truth-tellers would not close, and thus the only possibility is that all 2 0 1 8 of the friends are lairs.