Cool page on the so-called "Sleeping Beauty" problem in probablity. You'd do well to just skip down to the ones about the clones, because for whatever reason those are a lot easier to understand.

For what it's worth, I think the thirders are committing a logical error, for exactly the reason expressed by this variant:

A coin is flipped 10 times in a row.
If (and only if) it comes up tails each time, then you are cloned a million times.

When you are wakened, after the possible cloning, you (and clones of you) are asked "Were they all tails?"
If you answer correctly, you live happily.
If you answer incorrectly, you are tortured and killed.

The paradox is that thirders would answer yes to this question.

Flipping a coin and having it come up heads doesn't retroactively change the probability of heads coming up to 1.

UPDATE: This one says it even better.

A fair coin is to be tossed without your seeing it, and you will be asked for your credence that it has landed tails. If it lands heads, you will be asked once; tails, you will be asked twice, but the second time you will be compelled to give the same answer as the first time. You know all this in advance.

The answer is obviously 1/2.

This is effectively the same as SB's scenario.

Does anyone support the thirder position here?