Where do people find this many friends?

Logic Level 2

2018 2018 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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2 solutions

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 2018 \boxed{2018} of the friends are lairs.

Very good solution

Stephen Mellor - 3 years, 3 months ago
Blcraft Gaming
Mar 5, 2018

If all 2018 people are liars and every one of them says that “exactly one of the two people sitting next to me tells the truth”, then they would all be lying, which is what they should be doing. Therefore, since only one answer is correct, the answer must be 2018.

Can you prove that this is the only possible answer (and not just because it is a feasible solution and Brilliant only accepts one answer!)? Hint: you have found a solution with no truth-tellers. If there were to be a different solution then there would have to be at least one truth teller. Starting with one truth-teller, can you state why it doesn't work with 2018 people and include that in your solution?

Stephen Mellor - 3 years, 3 months ago

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