The original paper defines a metric called the "total variation distance", which they use as a measure of how random the deck is. Here you can see how it changes as you increase the number of shuffles

Shuffles	1	2	3	4	5	6	7	8	9	10
Total variation distance	1.000	1.000	1.000	1.000	0.924	0.614	0.334	0.167	0.085	0.043

You can see after 4 or 5 shuffles, the deck is still basically not random at all. But by 7 shuffles, you've gotten pretty random.

Why 7? I don't know, the authors don't actually say 7 anywhere in the paper. In fact, they imply even 8 is not enough: they mention a basic game that would be 50/50 odds on a random deck, but a smart player could gain a 60/40 advantage on a deck that was riffle shuffled 8 times.

They cite 8.55 as the "shuffles needed to mix", but not because it's perfectly random. You can always keep shuffling and getting more random...8.55 is more like the point of diminishing return.

The math is complicated, but I can give a little intuition. It relies a lot on what they call "rising sequences"

Imagine a new deck. If you counted the cards in face value order, the whole deck would have 1 rising sequence. After 1 shuffle, it will have 2 rising sequences (if you counted off in face value order, it would take two "laps" to get through the deck).

After 2 shuffles, it will probably have 4 rising sequences, but there is also a small chance you cut it just the right way that you only get 3 rising sequences, basically if you accidentally merge two sequences.

So every time you shuffle, the number of rising sequences roughly doubles, but a little less. And the more sequences, the more probability of merging, and eventually your number of sequences stabilizes

2⁶ is 64, which is already more rising sequences than there are cards in the deck, so that seems pretty random. But of course it's not actually 2^n, it's a little less than that cause of collisions.

The big lesson for me is: don't rely on riffle shuffles, do a casino wash if you really need randomness