The Maths and Magic of Shuffling Cards

I’m sure that everyone that is reading this has, at some point in their life, shuffled a deck of cards. Such a simple, almost meaningless action. Yet, mathematically, shuffling a deck of cards is incredibly interesting. It’s a fascinating mathematical concept that connects to probability, randomness, and even patterns.

When you shuffle a deck of 52 cards thoroughly (so not just a simple cut of a new deck), there are many ways they can end up arranged. In fact, the number of possible arrangements is 52 factorial, written as 52!, which is a gigantic number: 52 × 51 × 50 × … × 1. Using expanded notation, it equals 8.0658×1067 Or, if you actually want to write that out:

80658175170943878571660636856403766975289505440883277824000000000000

This number is so huge that if you thoroughly shuffled a deck of cards every second since the earth was formed, you would never repeat the same order twice! To really understand how gobsmackingly big this number is, here’s another way to look at it: (a famous explanation taken from an essay by Scott Czepiel):

Set a clock timer to count down from 8.0658×1067 seconds.

Now go to the equator and wait a billion years to take one step. Wait another billion years before taking a second step. Do this until you’ve reached all the way around the earth (about 40 million steps) and now take one drop of water out of the Pacific Ocean and set it aside. Repeat this entire process until the entire Pacific Ocean is drained (about 700 million cubic kilometres of water) and then lay a piece of paper on the ground. Repeat this process of draining the ocean again to lay another piece of paper on top of the first one. Keep going until that stack of papers reaches the sun (about 150 billion kilometres). Make 1000 more stacks of paper to the Sun and then you’ll only be a third of the way through the timer!

This absolutely boggles the mind. So the next time you shuffle a deck of cards, realise that the order that those cards are in has never happened before.

Parent problem of the week

Here’s one for Star Wars fans.

On Endor, OC means a bundle of 8 sticks, OCTA means a bundle of 8 OCs, OCTIL means a bundle of 8 OCTAs, and OCTILLA means a bundle of 8 OCTILs. One day, a band of Ewoks gather OCTILLA sticks. How many did they gather?

Good luck!

Mr Preorius

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