Any Well Shuffled Card Deck Is Almost Certainly Unique

From Paul Graham we have this interesting fact:

Any time you pick up a well shuffled deck, you are almost certainly holding an arrangement of cards that has never before existed and will likely never exist again.

– Yannay Khaikin pic.twitter.com/afOpu0y7qA

— 〈 Berger | Dillon 〉 (@InertialObservr) September 18, 2019

We’re nerds and are used to dealing with large numbers so \(10^{68}\) doesn’t seem especially huge but let’s do a little back-of-the-envelope calculation. From Emacs Calc we learn that \(52! \approx 8 \times 10^{67}\), a slightly tighter estimate than that given in the tweet. I asked DuckDuckGo and it told me that the earth is about 4.5 billion years old and that as of April this year there are about 7.7 billion people living on it.

Another quick calculation with Calc shows us that if every person alive today had been shuffling cards since the birth of the earth, it would have required \(2.3 \times 10^{48}\) shuffles per person per year to generate the \(52!\) possible arrangements¹. That means that each person would have to shuffle \(7.4 \times 10^{40}\) times a second for the entire age of the earth. Put that way, the claim in the tweet is eminently believable. Still not convinced? Suppose all the people who have ever lived (\(\approx108 \times 10^{9}\)) shuffled cards for the entire age of the universe (\(\approx 13.8 \times 10^{9}\) years). Would that make an appreciable difference on how many shuffles they would have to make per second? (Spoiler: no)

UPDATE [2019-11-29 Fri 12:09]: lets → let’s