Problem:
Agnes writes the following four statements on a blank piece of paper.
Each statement is either true or false. How many false statements did Agnes write on the paper?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let the four statements be in order, and let be the number of true statements and the number of false ones, so .
If , then all are false. But then says “at least two of these statements are false,” which would be true. Contradiction. So .
If , then all are true. But then (“at least two are false”) and (“at least one is false”) would both be false. Contradiction.
If , then . Then (“at least two are false”) is true, and (“at least one is false”) is also true, and (“at least one is true”) is true as well. That gives at least three true statements, so , a contradiction.
If , then . Then (“at least one is true”) must be true. Since , (“at least two are false”) is also true, giving at least two true statements, so , a contradiction.
The only remaining possibility is
Indeed, if three statements are true and one is false, then , , and are true, and is false, which is consistent.
Therefore, the number of false statements Agnes wrote is .
The problems on this page are the property of the MAA's American Mathematics Competitions