1997 PMWC Problems/Problem T5

Problem

During recess, one of five pupils wrote something nasty on the chalkboard. When questioned by the class teacher, the following ensued:

'A': It was 'B' or 'C'

'B': Neither 'E' nor I did it.

'C': You are both lying.

'D': No, either A or B is telling the truth.

'E': No, 'D', that's not true.

The class teacher knows that three of them never lie while the other two cannot be trusted. Who was the culprit?

Solution

Case 1: A is telling the truth

Thus it was either B or C. Since A is telling the truth, C is lying. If D is telling the truth, then B is lying, and E is lying, but that's impossible. Thus D is lying. Then E is telling the truth, and so is B. Thus two people are lying (C and D), and the other three are telling the truth. Since it was one of B or C, and it wasn't B, it must have been C.

Case 2: A is lying

Thus it is either D or E. If E did it, then B is lying, and C is telling the truth. But that means D is lying, a contradiction from our original assumption that there were only 2 people who were lying. Therefore it was D who did it. B would be telling the truth, so C would be lying. D would be telling the truth, but that would mean E is lying, another contradiction. So our final answer is