1. Rocky the dog is alive.
2. Rocky the dog is dead.
3. Exactly one of statements 6 and 9 is true.
4. Exactly one of statements 2 and 6 is false.
5. Statements 4, 5 and 10 are all false.
6. Exactly one of statements 1 and 10 is false.
7. Exactly 5 statements are true.
8. Exactly one of statements 3 and 10 is false.
9. Exactly one of statements 6 and 10 is true.
10. Exactly one of statements 1 and 2 is false.
11. Statements 1, 8 and 11 are all false.

Which of the above statements are definitely true? Is Rocky the dog alive or dead?

*** Based on an original problem by Louise Mayor.

Let's walk through this one:

Leaving statements 1 and 2 aside for the moment.

Statement 10 must be true, unless we’re talking about Schrödinger's Dog.

Assume statement 3 is false.

So, this means either both statements 6 and 9 are true or both statements 6 and 9 are false.

If 6 and 9 are true:  Then statement 1 must be false (Rocky is then dead) [from statement 6]

and to make statement 9 true, 6 must be false.  This makes statement 6 both true and false.

So, for statement 3 to be false statements 6 and 9 must both be true.

This means that (from statement 6) Statement 1 is false (and Rocky is dead) and statement 6 is false (from statement 9).   This makes statement 6 both true and false.  

Thus statement 3 must be true.

There are two possibilities to make statement 3 true:

Statement 6 is true and statement 9 is false.

Or statement 6 is false and statement 9 is true.

If we assume statement 6 is true and statement 9 is false.

Then (from statement 6) statement 1 must be false (as statement 10 is true) and Rocky must be dead.
And (from statement 9) either 6 is true and 10 is false or 6 is false and 10 is true.

We know 10 is true so 6 must be false.   This makes statement 6 both true and false.

Thus we know that statement 3 is true and it means that statement 6 is false and 9 is true.

So far we have as true:  statements 1, 3, 9, and 10.

And as false:  statements 2, and 6.

From this statement 4 must be false as both statements 2 and 6 are false.

Statement 5 is false as we know statement 10 is true.

Statement 8 is false as we now know statements 3 and 10 are both true.

Statement 11 is false because if it were true it would make itself false.

This leaves statement 7, which fulfils the condition if it is false or if it is true, but since it fulfils the condition by being true then it is true.

So, statements 1, 3, 7, 9, and 10 are all true making Rocky the dog alive.

Posted by Astilius on 2014-11-28 11:00:58