UKMT - Senior Mathematical Challenge - 2023 - 11

11

Wenlu, Xander, Yasser and Zoe make the following statements:

Wenlu: “Xander is lying.”

Xander: “ Yasser is lying.”

Yasser: “Zoe is telling the truth.”

Zoe: “Wenlu is telling the truth.”

What are the possible numbers of people telling the truth?

A. 1 or 2 B. 1 or 3 C. 2 D. 2 or 3 E. 3

Answer

B. 1 or 3

SOLUTIOn

Suppose Wenlu is telling the truth. Then Xander is lying. Therefore Yasser is telling the truth.

Hence Zoe is telling the truth, and this agrees with the fact that Wenlu is telling the truth.

Hence it is possible that Wenlu is telling the truth. We have seen that in this case Wenlu, Yasser and Zoe are telling the truth.

Suppose Wenlu is lying. Then Xander is telling the truth. Therefore Yasser is lying.

Hence Zoeis lying, and this agrees with the fact that Wenlu is lying.Hence it is possible that Wenlu is lying.

We have seen that in this case only Xander is telling the truth. Therefore the number of people telling the truth is either 1 or 3.

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