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October 24, 2026

Math Puzzle: Ring of liars

Flex your math muscles with this weekend’s brain teaser. Play now

By Jack Murtagh edited by Sarah Lewin Frasier

Five people sit in a circle in chairs, tight focus on their hands.
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  1. Imagine a world in which liars always lie and truth-tellers always tell the truth. Five people sit in a circle. Everybody says the same thing: “The two people sitting next to me are liars.” How many of them are lying?

  2. What if the first person says “Exactly one of us is a liar,” the second person says “Exactly two of us are liars,” the third says “Exactly three of us are liars,” and so on? Who is lying then?

In puzzle #1, three people are lying, and two are telling the truth. Start with two observations:

  1. Three liars can’t sit next to each other in a row. If they did, then the middle one accidentally told the truth—their neighbors are both liars.

  2. Two truth-tellers can never sit next to each other. If they did, then they both accidentally lied—it is not true that both of their neighbors are liars.

If two truth-tellers can’t sit next to each other, then there can be at most two truth-tellers in the circle of five people—any more and you can’t keep them separated. But if there are fewer than two truth-tellers, then three liars will necessarily sit in a row, which also can’t happen. So there must be exactly two truth-tellers.

In puzzle #2, only the fourth person is telling the truth. Each statement is mutually exclusive, so at most, one can be true. (There can’t be exactly two and exactly three liars, for example.) Furthermore, the statements can’t all be false. If they were, then the fifth person, who says “Exactly five of us are liars,” would be telling the truth, creating a contradiction. So there is exactly one truth-teller and therefore four liars. This is precisely what person four is saying.

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