South African Mathematics Olympiad Third Round

Question

Bee, Cee and Vee live in Microphyllia where there are only two types of creatures – those that consistently tell the truth and those that consistently lie. The former creatures are Trudees and the latter Falsees. When I last visited Microphyllia I asked Bee: “Who of you all are Trudees?” Bee mumbled its answer so I didn't quite catch what she said. “Bee said that only one of the three of us is a Trudee”, Cee noted. Vee turned to me and said: “Don’t believe Cee, he’s not telling the truth”. Who of them are Trudees and who are Falsees?

Accepted Answer

Suppose Cee tells the truth, then Bee said that only one of the three tells the truth, so Bee can't tell the truth or otherwise there will be at least two truth tellers which would be a contradiction. Hence, if Cee tells the truth Bee lies and there must be more than one truth teller, so Vee must be telling the truth. However, Vee says Cee lies contradicting our assumption. Hence Cee can't tell the truth and lies. We don't know what Bee said, so we don't know if he tells the truth or lies. Thus Cee lies, Vee tells the truth and we don't know about Bee.