How Many Different Ways Can You Shuffle A Deck Of Cards?

In my video demonstration for the Shuffle Prophecy, I casually point out “there are more ways to shuffle a deck of cards than there are atoms in the universe” I then correct myself to say more atoms on earth, not the universe. I recently received a note from Doug Dyment, retired math professor at University of Waterloo, to correct my correction.

Doug
Doug Dyment
Thought I should comment on the math in your instructional video. Your original claim is correct; the text correction you provide is unnecessary and misleading.
 
The number of possibilities of a shuffled deck is 52 factorial (written “52!”), which is about 8 × 10^67, or more precisely:
 
80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000
 
This is vastly larger than the number of atoms on earth, or even in our solar system. It is approximately equal to the number of atoms in our (Milky Way) galaxy of some 200–400 stars and all their accompanying planets (around 2 × 10^67 atoms).

It’s a number that really too large to comprehend. However, Doug goes further than that to offer an even larger number to consider…

But this is for a deck with all the cards facing the same way. If you allow for each card to be facing in either direction, the total is vastly greater than that: specifically, it is (52! × 2^52), which is about 4 × 10^83, or more precisely:
 
363,252,127,644,247,044,041,398,160,152,368,436,824,058,904,969,853,261,166,529,262,548,680,704,000,000,000,000
 
This is at least ten times greater than the number of atoms in the known/observable universe (estimated at somewhere between 10^78 and 10^82).

Interesting, yes, but is it entertaining?

Personally, I love this sort of stuff. Physics, astronomy, and math on the universal scale fascinate me. I can appreciate, however, that not all boats may be floating right now.

Many card tricks based on math can be painfully dull, but must they be? Is that inherent in the trick, or is it the choices of the presenter to let it fall flat? If you’ve seen Giovanni turn the old 10 Pennies math puzzle into a theatrical masterpiece, you may be more optimistic.

Many magician understand the difference between a boring trick and an entertaining trick is the story. So, what if you take a boring number and turn it into an interesting story?

This story comes from data scientist Scott Czepiel. I’ve adapted the wording a tiny bit.

How large, really, is 52 Factorial?

This number is beyond astronomically large. So, just how large is it? Let’s try to wrap our puny human brains around the magnitude of this number with a fun little theoretical exercise.

Start a timer that will count down the number of seconds from 52! to 0. We’re going to see how much fun we can have before the timer counts down all the way.

Start by picking your favorite spot on the equator. You’re going to walk around the world along the equator, but take a very leisurely pace of one step every billion years. 

After you complete your round the world trip, remove one drop of water from the Pacific Ocean.

Now do the same thing again: walk around the world at one billion years per step, removing one drop of water from the Pacific Ocean each time you circle the globe.

Continue until the ocean is empty.

Once it’s empty, take one sheet of paper and place it flat on the ground.

Now, fill the ocean back up and start the entire process all over again, adding a sheet of paper to the stack each time you’ve emptied the ocean.

Do this until the stack of paper reaches from the Earth to the Sun.

(Take a glance at the timer, you will see that the three left-most digits haven’t even changed. You still have 8.063e67 more seconds to go.)

So, repeat the entire process. One step every billion years, one water drop every time around, one sheet of paper ever ocean. Build a second stack to the Sun.

Now build 1000 more stacks.

Good news! You’re just about a third of the way done!

To pass the remaining time, start shuffling your deck of cards. Every billion years deal yourself a 5-card poker hand.

Each time you get a royal flush, buy yourself a lottery ticket.

If that ticket wins the jackpot, throw a grain of sand into the Grand Canyon.

Keep dealing, and when you’ve filled up the entire canyon with sand, remove one ounce of rock from Mt. Everest.

Empty out the sand and start over again. Play some poker, buy lotto tickets, ,drop grains of sand, and chisel some rock. When you’ve removed all 357 trillion pounds of Mt. Everest, look at the timer, you still have 5.364e67 seconds remaining.

Do that whole mountain levelling thing 255 more times. You would still be looking at 3.024e64 seconds.

The timer would finally reach zero sometime during your 256th attempt.

But, let’s be realistic here. In truth you wouldn’t make it more than five steps around the earth before the Sun becomes a Red Giant and boils off the oceans. You’d still be shuffling while all the stars in the universe slowly flickered out into a vast cosmic nothingness.

Anyhow, who wants to see a card trick?!?

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