Richard is a strange liar. He lies on six days of the week, but on the seventh day, he always tells the truth. He made the following statements on three successive days:
Day 1: ''I lie on Monday and Tuesday.''
Day 2: ''Today, it's Thursday, Saturday, or Sunday.''
Day 3: ''I lie on Wednesday and Friday.''
On which day does Richard tell the truth?

We know that Richard tells the truth on only a single day of the week. If the statement on day 1 is untrue, this means that he tells the truth on Monday or Tuesday. If the statement on day 3 is untrue, this means that he tells the truth on Wednesday or Friday. Since Richard tells the truth on only one day, these statements cannot both be untrue. Therefore, exactly one of these statements must be true, and the statement on day 2 must be untrue.

Assume that the statement on day 1 is true. Then the statement on day 3 must be untrue, from which follows that Richard tells the truth on Wednesday or Friday. Consequently, day 1 is a Wednesday or a Friday. Therefore, day 2 is a Thursday or a Saturday. However, this would imply that the statement on day 2 is true, which is impossible. From this, we can conclude that the statement on day 1 must be untrue.

This means that Richard told the truth on day 3 and that this day is a Monday or a Tuesday. Therefore, day 2 is a Sunday or a Monday. Because the statement on day 2 must be untrue, we can conclude that day 2 is a Monday.

Consequently, day 3 is a Tuesday. Therefore, the day on which Richard tells the truth is Tuesday.