Solution
Nine Interlocking Claims
Show the problem again
Nine statements are written in a list. Each is true or false according to what it says. 1. Statements 5 and 6 are both true. 2. Exactly 2 of these nine statements are true. 3. Statements 2 and 9 are both false. 4. At least two of statements 3, 7 and 8 are true. 5. Statements 7 and 9 are both true. 6. If statement 9 is true, then statement 7 is false. 7. Statements 2 and 8 are both true. 8. Exactly one of statements 1 and 6 is true. 9. Statements 4 and 6 are both false. How many of the nine statements are true?
Worked solution
Suppose statement 5 were true. Then statements 7 and 9 are true, which makes statement 6 false (its condition holds but its conclusion fails). Statement 1 needs statement 6, so it is false too. Statement 8 says exactly one of 1 and 6 is true, but both are false, so statement 8 is false. But statement 7 is true and requires statement 8 to be true, a contradiction. So statement 5 is false, and with it statement 1. Now statement 8 requires exactly one of 1 and 6 to be true; since 1 is false, 6 is true, and so 8 is true. Statement 9 requires statement 6 to be false, so 9 is false. Statement 3 therefore says only that statement 2 is false, and statement 7 simplifies to "statement 2 is true" (since 8 is true). If statement 2 were true, statement 7 would be true and statement 3 false, and statement 4 would hold (8 and 7 true), giving the true set 2, 4, 6, 7, 8: five statements, contradicting "exactly 2". So statement 2 is false, hence 7 is false and 3 is true. Statement 4 then holds (3 and 8 are true), so the true statements are 3, 4, 6, 8, and statement 2 is correctly false because the count is 4. The answer is 4.
Source: Original puzzle written for AxiomIQ; the format of statements referring to each other is a staple of recreational logic.