Problem:
On Halloween children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let denote the number of truth-tellers, the number of alternators who tell the truth on odd-numbered questions, the number of alternators who tell the truth on evennumbered questions, and the number of liars. These four variables must satisfy the following equations.
Subtract the third equation from the second to find that . Because they received candy only in response to the first question, they received 7 pieces of candy.
Note: It may be further deduced from the system of equations that , and .
The problems on this page are the property of the MAA's American Mathematics Competitions