# True or false

From Math Puzzle Wiki

Assign a truth value to each of the following statements.

- Statement 5 is false.
- Statement 1 is false.
- Statement 4 is true.
- Only one of these statements is false.
- Statements 2 and 3 are both true or both false.

**Solution**

First identify statements with equal vs opposite truth values:

- 1 has opposite truth value to 5 (from statement 1)
- 2 has opposite truth value to 1 (from statement 2), hence equal truth value to 5
- 3 has equal truth value to 4 (from statement 3)
- 5 true only if 2 has equal truth value to 3 (from statement 5)

If 4 false, then:

- 3 false (equal value to 4)
- 2 has opposite value to 5 (5 true only if 2 false, given 3 false)

But 2 has equal truth value to 5. So 4 true.

If 4 true, then:

- 3 true (equal value to 4)
- 2 and 5 true (otherwise they'd both be false, contradicting 4)
- 1 false (opposite value from 2, 5)

Hence 1 is false and all other statements are true.