5. As follows:
if 6 were true then 5 would be untrue and therefore 6 would be untrue. The contradiction means that 6 can’t be the answer.

And immediately therefore 1 is wrong too. And so 3 can’t be right.

So we’re left with just 2, 4 & 5 as contenders. But then the conflict between 2 and 4 rules out 4. So now we’re down to just 2 & 5, which can be consistent only if it’s 5 that’s right.

The only way to win is not to play.

5. As follows:

if 6 were true then 5 would be untrue and therefore 6 would be untrue. The contradiction means that 6 can’t be the answer.

And immediately therefore 1 is wrong too. And so 3 can’t be right.

So we’re left with just 2, 4 & 5 as contenders. But then the conflict between 2 and 4 rules out 4. So now we’re down to just 2 & 5, which can be consistent only if it’s 5 that’s right.

I’d say both are right – either 5 or don’t play.

Actually, it would be good to run a quiz where no one would answer at all, something would prevent you. Hmmmmm, have to think that one out.