I think the answers have been correct, but may not have answered OP’s intended question. He doesn’t want to use Fisher-Yates; he wants to physically riffle a physical deck using two thumbs and several fingers. (Can you do that with Fisher-Yates?) He doesn’t want to model shuffles mathematically; he wants to perform them physically, finger foibles and all. He doesn’t want to shuffle thousands of decks and perform elaborate statistical tests; he just wants to do a handful of shuffles (perhaps 3 or even fewer each by him and his friend) and calculate a simple but useful “randomness score.”

My advice to OP is [ul][li] Sort the cards completely before each shuffle (say Spades: Ace, Deuce, … through King; then Hearts the same way; then Clubs; then Diamonds)[/li][li] Shuffle (perhaps allowing a fixed number of seconds for the shuffling since almost any method will shuffle better if you repeat it longer)[/li][li] Compute a simple “randomness score.”[/li][/ul]

Of course there are no usable *perfect* “randomness scores,” but I think there are some usable for your purpose. One way (mentioned in the papers cited to gauge effectiveness of riffle shuffles, though it *might* be a reasonable test for some other shuffling patterns as well) is *to count rising sequences*. Counting the rising sequences may be a bit error-prone and time-consuming, but may not take much longer than the *Sort cards completely* step and, if done in a certain way, leaves the deck in that sorted order, ready for the next trial, upon completion!

Here’s how to count “rising sequences”:

Starting at the same end of the deck that had card #1 (Ace of Spades) before the shuffle, search until you find that #1 card (and then pull it out and set it aside, forming the sorted deck for the next shuffle) and continue forward from the location of #1 until you reach #2, set it aside and continue to #3 and so on. When you get to the end of the deck, start over at the beginning and increment a count.

The number of times you go back to the beginning of the deck in this procedure is your “count of rising sequences.” The count will be 1 if you did no shuffling at all, 2 after a single riffle, and will increase up to a point as the shuffling improves. (The count will reach 52 if you manage to shuffle until the cards are exactly reversed from their starting position!)

This is just one simple randomness test and may unfairly favor one shuffling method over another. Depending on how important your bet is, you might want some alternate tests.

Crudely, the larger the number of rising sequences the better but since the largest number (52) represents perfect *un*shuffling, it might be better to let the winner be whoever approaches the average rising sequence count among all orderings. (What is that “average count”? Let’s leave that as an exercise for now. )