# Liar, liar, shorts on fire

Football manager Stephen is a compulsive liar, he just can't help himself. He lives in a fantasy world of his own. He's always telling people about fantastic players he's about to sign up, about the phenomenal crowds that come to watch his club and how he was a great player himself in his youth - playing for some of the top sides in the country. His compulsion is however rather unusual. On six days of the week he always lies but on one 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 is Thursday, Saturday or Sunday"
• Day 3: "I lie on Wednesday and Friday"

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

## The solution

We know that Stephen 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 Stephen tells the truth on only one day, these statements cannot both be untrue. So, 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 Stephen tells the truth on Wednesday or Friday. So, 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 Stephen told the truth on day 3 and that this day is a Monday or a Tuesday. So 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. So day 3 is a Tuesday. Therefore, the day on which Stephen tells the truth is Tuesday.