1. Bob tells the truth on only one day of the week.
2. One day he said: "I lie on Mondays and Tuesdays."
3. The next day he said: "Today is either Thursday, Saturday, or Sunday."
4. The next day he said: "I lie on Wednesdays and Fridays."

On which day of the week does Bob tell the truth?

Since Bob tells the truth on only one day of the week, only one of statement 2 or statement 4 can be true, it follows that at least one of these two statements must be false. In order to make one of these statements false, the true day has to be one of the two days that it mentions, thereby falsifying the ANDed statement. This necessarily makes the other of these two statements true, and therefore must be spoken on the true day. Further, this means that the true day must be Monday, Tuesday, Wednesday or Friday, so as to falsify either statement 2 or statement 4.

If the true day is Wednesday, statement 2 is true and therefore on a Wednesday. But that makes statement 3 fall on a Thursday, which would make it also true because it claims to be stated on one of a set of days that includes Thursday. So the true day can’t be Wednesday.

If the true day is Friday, statement 2 is true and therefore on a Friday. But that makes statement 3 fall on a Saturday, which would make it also true. So the true day can’t be Friday.

If the true day is Monday, statement 4 is true and therefore on a Monday. But that makes statement 3 fall on a Sunday, which would make it also true. So the true day can’t be Monday.

But if the true day is Tuesday, statement 4 is true, and therefore on a Tuesday, putting statement 2 on a Sunday and statement 3 on a Monday, consistent with their falsity.

Therefore the True Day is Tuesday, and the three statements were made on Sunday, Monday and Tuesday.

Posted by Charlie on 2003-03-17 03:27:48