In my puzzle column earlier today I set you the following two self-referential logic list problems

**ONE**

Which of the following statements is/are true?

1. Exactly one of the statements in this list is false.

2. Exactly two of the statements in this list are false.

3. Exactly three of the statements in this list are false.

4. Exactly four of the statements in this list are false.

5. Exactly five of the statements in this list are false.

6. Exactly six of the statements in this list are false.

7. Exactly seven of the statements in this list are false.

8. Exactly eight of the statements in this list are false.

9. Exactly nine of the statements in this list are false.

10. Exactly ten of the statements in this list are false.

**Solution**

The easiest way to make sense of this problem is to restate each line as a logically equivalent statement about the truth, rather than the falsity, of the other statements. For example, look at statement 1. If exactly one statement in this list is false, this is the same as saying that exactly nine are true. Statement 2: if exactly two statements are false, this is equivalent to saying that exactly eight are true, and so on.

Seen from this perspective, it is clear that only one statement can be true (since if more than one were true, there would be a contradiction). If only one is true, then nine are false and the only statement that is true is 9.

**TWO**

Which answer in the list is the correct answer to *this* question?

1. All of the below.

2. None of the below.

3. All of the above.

4. One of the above.

5. None of the above.

6. None of the above.

**Solution**

There are different ways to solve this. One way is to look for contradictions and eliminate statements as you go:

If 1 is true, then 2 must be true, but this contradicts 1 so 1 must be false.

And if 1 is false, then 3 must also be false.

If 2 is true, then 4, 5 and 6 must be false. But if 4 is false, it is not the case that only one of 1,2 and 3 is true. This contradicts the premise, which is that 2 is true (and we know 1 and 3 are false). So 2 is also false.

Since 1,2 and 3 are false, so is 4.

If 6 is true, 5 must also be true, but if 5 is true 6 is false. So 6 is false.

5 is true, and is therefore the correct answer.

I hope you enjoyed the puzzle and I’ll be back in two weeks.

*I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.*

*I’m the author of several books of popular maths, including the puzzle books Can You Solve My Problems? and Puzzle Ninja.*