Okay, The Sleeping Beauty Problem

Here's the problem.  Just read it for a minute and don't try to think about it until you get to the end:

We plan to put Beauty to sleep by chemical means, and then we’ll flip a fair coin. If the coin lands Heads, we will awaken Beauty on Monday afternoon and interview her. If it lands Tails, we will awaken her Monday afternoon, interview her, put her back to sleep, and then awaken her again on Tuesday afternoon and interview her again. The (each?) interview is to consist of the one question : what is your credence now for the proposition that our coin landed Heads? When awakened (and during the interview) Beauty will not be able to tell which day it is, nor will she remember whether she has been awakened before. She knows about the above details of our experiment. What credence should she state in answer to our question?

Okay, so if you have to choose the probability of the coin landing heads, what is it?  Well, it is one out of two, right?

Then, for whatever reasons, we start to think about the weird false bottom that we were once already asked the question, at least, there's a chance that we were once already asked the question.  So, there's a chance that this is the second time, i.e. that it is now tuesday and we're getting asked the question.  Seen in this light, the probability that heads comes up is based on tails having come up the prior day, which changes the chance that it will come up today, ad infinitum.