hmm, when you call the startNewPuzzle function and use:
it completely kills the initialized SWF it seems like. Like natty said, you could put the call to embed the SWFObject into a separate div and utilize like a flipper.
Looking at the API (I haven't messed with SWF either in a while) it seems like it will only re-initialize on a refresh. What happens if you re-register the object after you kill it in your startNewPuzzle function?
I'll try that soon right now there seems to have been a problem with my website I'm not quite sure what happened. my code didn't update from the other day and it's broken at the moment.
EDIT: I missed a comma when trying your idea so it messed up my initialize function lol
Thanks for your help!