"What could a mathematician do with a perfect shuffle?"

Perfect? That's a word which should not be deployed lightly!

By definition a shuffle can never be "perfect" - a card shuffle should strive to mix up the cards but it can never do that job and then say "these cards are optimally shuffled"

Oh so now we have a new definition for perfect: "Praeger, however, is interested in just those permutations of the cards that you can get by using perfect shuffles, when a pack of cards is exactly split in half and the cards from each half perfectly interleaved one by one."

I have a different idea of what perfect means and it is not that. I suggest the word "precise" or "exact" when describing fancy shuffling schemes.