2000 PMWC Problems/Problem I12
Problem
During the rest hour, one of five students (, , , , and ) dropped a glass of water. The following are the responses of the children when the teacher questioned them:
- : It was or who dropped it.
- : Neither nor I did it.
- : Both and are lying.
- : Only one of or is telling the truth.
- : is not speaking the truth.
The class teacher knows that three of them NEVER lie while the other two ALWAYS lie. Who dropped the glass?
Solution
Suppose that is telling the truth. Then , and one of and are lying. But only two students are lying, so there is a contradiction. Hence is lying. Then there are two possible scenarios: both and are lying, or both and are telling the truth. Suppose that both and are lying. Then there are three students (, and ) who are lying, a contradiction. So both and are telling the truth. Using their responses, we get that dropped the glass.