A perfectly valid chain of logic proves two contradictory things at once
In a barbershop with barbers Allen, Brown, and Carr, one rule holds: whenever Carr is out, Allen is out too, since one of them must always mind the shop. A customer reasons: suppose Carr is out. Then Allen must be out. But Allen never leaves without Brown going with him, so Brown must be out too. Yet if Brown and Carr are both out, Allen must be in, since someone has to be there, contradicting Allen being out. Using only valid logical steps, can Carr actually be out at all?
Reveal the answer
Lewis Carroll used this scenario to needle a real disagreement among logicians of his time about how conditional statements should be interpreted. The puzzle shows that premises which each sound individually reasonable can combine, through perfectly valid inference, into a hidden contradiction, forcing one of the premises itself to be false, in this case the assumption that Carr could ever be out, rather than any step in the reasoning being wrong.