Optimum number of shuffles to randomize cards?

Factual Questions
21

I have actually read the original paper. It is a journal called something like Applied Statostics. One of the authors is magician turned mathematician called Persi Diaconis and I have forgotten the other one. In the paper he discusses at some length a construction called the shuffle idempotent of which I am the discoverer, but as far as I can determine, makes no use of it. The name of the paper is something like “Chasing the riffle shuffle to its lair”.

But that is an aside. The main result is that after seven random (as defined in the paper, but both the size of each pack and the riffle itself are random) shuffles, the order of the deck is within 1% of being random. This assumes you start from a perfectly sorted deck. There is certainly no claim that an 8th shuffle will leave it less random and I assume that more shuffles simply cause it to approach, but never reach, perfect randomness. Fewer shuffles will leave a less random deck. So it is a purely pragmatic answer and the number doubtless does increase, probably slowly, with the size of the deck.

22

I can confirm this result, not mentioning it earlier lest Chronos think I was working too hard. :cool:

If “consistent shuffle” is defined to allow any 52-sized permutation, then there are some shuffles which must be perfomed 180,180 times to restore the deck. This is the 52nd term of Landau’s g(n); you can find it as A000793 in The On-Line Encyclopedia of Integer Sequences, although the list there seems to stop at the 51st term, just shy of the interesting term! :smiley: