Seven people sit in a circle, and each makes the same accusation about the person to their right. Sometimes the shape of a problem matters more than the words.
The chain of lies (7 in a circle)
Riddle statement
Seven people are seated in a circle. Each says exactly:
> “The person to my right is a liar.”
Each person is one of two types:
- a truth-teller, who always tells the truth;
- a liar, who always lies.
Can such an arrangement exist?
Show solution
Solution
Answer: No. Such an arrangement is impossible.
If a person is a truth-teller, the statement “the person to my right is a liar” is true, so their right-hand neighbor is a liar.
If a person is a liar, the statement is false. Their right-hand neighbor is therefore not a liar, but a truth-teller.
In either case, every person must be of the opposite type from the person to their right. The types are forced to alternate around the circle:
Start with any person. If they are a truth-teller, the seven positions are forced to be
But the seventh person and the first person are also neighbors. They would both be truth-tellers, even though neighboring people must be of opposite types.
Starting with a liar produces the same problem:
Again, the seventh and first people have the same type.
An alternating pattern of two types can close around a circle only when the number of positions is even.
Conclusion: there is no valid arrangement for seven people.