Mathematics
Problem Of the Week
Fall
2006
POW #7
You are visiting an island where only
two types of people live.
Knights always say the truth (i.e.
all statements knights say are true) and knaves always lie (i.e. all statements
knaves say are false).
(1) You ask person A
living on the island, “Are you a knight?”
A replies, “If I am a knight, then I will eat my hat.”
Prove that person A has
to eat his hat.
(2) Three inhabitants B,
C, and D of the island are being interviewed.
B and C make the following
statements:
B: “C is a knight.”
C: “If B is a knight, then so is D.”
Can you determine what
type of person any of B, C, and/or D are?
Justify your answer(s).
(3) It has been rumored
that there is gold on the island of knights and knaves. You ask local person E whether there is any gold
on the island.
E: “There is gold on this island if and only if I am a knight.”
(a) Can it be determined whether E is a
knight or a knave?
(b) Can it be determined whether there is
gold on the island?
Justify your answers in each case.
Solutions
should be submitted to Dr. R. Lock's mailbox in the Math office or sent via
e-mail to rlock@stlawu.edu.