If she was told she would be woken up 3^n times if n is even, 0 times if n is odd, then it seems obvious enough that when asked upon being woken up what she thought the probability that n is even, she would rationally and correctly say 100%. And that this would make sense. So Why wouldn't it make sense if the answer is some number other than 100%?
What she would do differently is bet on things she cared about based on the odds. Like "would you rather your relatives are given $5 if the number of coin flips is odd or $3 if the number of coinflips are even?" The answer for a rational beauty would depend on the probability that the number of coin flips is even.
I got into a heated debate a couple days ago with some of my (math grad student) colleagues about the Sleeping Beauty Problem. Out of this discussion came the following thought experiment:
Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: She will be put to sleep. During the experiment, Beauty will be wakened, interviewed, and put back to sleep with an amnesia-inducing anti-aging drug that makes her forget that awakening. A fair coin will be tossed until it comes up heads to determine which experimental procedure to undertake: if the coin takes n flips to come up heads, Beauty will be wakened and interviewed exactly 3^n times. Any time Sleeping Beauty is wakened and interviewed, she is asked, "What is your subjective probability now that the coin was flipped an even number of times?"
I will defer my analysis to the comments.