## Twin Liars

Kerry and Kelly are twins. One of them lies on Mondays, Tuesdays, and Wednesdays and tells the truth on the other days of the week. The other lies on Thursdays, Fridays, and Saturdays and tells the truth on the other days of the week. On which day of the week did they have this conversation?
Kerry: I lie on Saturdays.
Kelly: I will lie tomorrow.
Kerry: I lie on Sundays.
Source: NCTM Mathematics Teacher, September 2006

Solution
There are two types of liars, the Mon-Tue-Wed liar and the Thu-Fri-Sat liar. We need to determine who is who? Clearly, Kerry is lying when he said that “I lie on Sundays” since both of them tell the truth on Sundays. Therefore, his statement “I lie on Saturdays” is false; he tells the truth on Saturdays and so Kerry is the Mon-Tue-Wed liar and Kelly is the Thu-Fri-Sat liar.

Since we know Kerry is lying, the conversation can only take place on Monday, Tuesday, or Wednesday.

Suppose it was Monday. Since Kelly is telling the truth on Mondays, the statement “I will lie tomorrow” (Tuesday) is true, which characterizes her as a Tue. liar. But that is not possible, because she is a Thu-Fri-Sat liar.

Suppose it was Tuesday. Since Kelly is telling the truth on Tuesdays, the statement “I will lie tomorrow” (Wednesday) is true, which characterizes her as a Wed. liar. But that is not possible, because she is a Thu-Fri-Sat liar.

Suppose it was Wednesday. Since Kelly is telling the truth on Wednesdays, the statement “I will lie tomorrow” (Thursday) is true, which characterizes her as a Thu. liar. There is no contradiction here, she is indeed a Thu-Fri-Sat liar.