1978 AMC 12 Problem 22

Question

The following four statements, and only these, are found on a card:

On this card exactly one statement is false.
On this card exactly two statements are false.
On this card exactly three statements are false.
On this card exactly four statements are false.

(Assume each statement on the card is either true or false.) Among them the number of false statements is exactly

0

1

2

3

4

Answer: D

Concepts:logical deduction truth-tellers and liars

Difficulty rating: 1860

Small hint:

Assume the actual number of false statements is

m

Big hint:

For each possible

m,

count how many of the four displayed statements would then be true

Solution:

If the actual number

m

of false statements is one of

1,

2,

3,

4,

exactly one displayed statement—the one naming

m

—is true. Therefore exactly three statements are false, forcing

m=3.

This is consistent: the third statement is true and the other three are false.

Therefore, the correct answer is D.