Fall 2008 Problem 5: Knights and knaves
You are on an island with knights (who always tell the truth) and knaves (who always lie). Unfortunately for you, you cannot tell knights and knaves apart by sight. You have the following discussion with Alphonse, Bertrand, and Cecil:
Alphonse: Both Bertrand and Cecil are knaves.
Bertrand: Cecil is a knave.
Cecil: Exactly two of us are knights.
Which of the three (if any) are knights?
Solution
Chris Patterson got the first correct answer (for three points). Dallas Krenzel had the best explanation (three points):
If A is a knight, then B and C both must be lying, which is impossible, because then B would be telling the truth in calling C a knave. If C is knight then there must be another one who is telling the truth, but both A and B call C a liar, so C and A or B can not be knights simultaneously. Therefore, neither A nor C can be knights, so both must be knaves. If this is true than B would be telling the truth about C being a knave and B would therefore be a knight. So, Alphonse is a knave, Bertrand is a knight, and Cecil is a knave.
Other correct answers (earning two points) came from: Matthew Chumley, Brent Krise, Susan Edwards, Robert Poole, Sterling Williams, Logan Parkerson, Jeffery James, and Tracy Apgar. Correct solutions from non-Centenary students came from Chris Evert, Don Dinnerville, and Nancy Simpson.