52!

28 May

‘Shuffle up and deal!’

Each year, those four words signal the start of The World Series of Poker (WSOP).

The WSOP started in 1970 with just seven players and one event. 53 years later, the WSOP is now a huge multi-event jamboree of poker that runs for weeks.

However, the No Limit Hold ‘Em $10k Main Event is the jewel in the crown: in 2023 it attracted 10,043 entrants, all gunning for a first prize in excess of 15 million dollars. That’s a lot of money.

That’s a lot of shuffling.

The American Contract Bridge League–the largest organization of its kind in North America–sanctions ‘over 3.5 million tables of bridge, played in more than 3,000 bridge clubs and 1,100 sectional and regional tournaments’ each year.

That’s a lot more shuffling.

Scores of other games use the 52-card deck version that originated in France and made its way to England around 300 years ago.

And yet, in all that pokering and bridging, the chance that any two shuffles–at any time or any place–have yielded the exact same combination is vanishingly small. That’s because the unique permutations possible from 52 playing cards is an eye-wateringly large number. How large?

52! large.

What is ‘52!’?

52! is the mathematical notation for 52 factorial. In mathematics, a factorial is the sum of all the multiplications smaller than the number in question. For example:

1! is 1

2! is 1x2 = 2

3! is 1x2x3 = 6

4! is 1x2x3x4 = 24

5! is 1x2x3x4x5 = 120

So far, so normal. But as we climb just marginally higher, the numbers get very large very quickly.

For example:

10! is 3,628,800.

15! is 1,307,674,368,000.

20! is 2,432,902,008,176,640,000.

And 52!–the number of unique combinations from a standard deck of playing cards–is 8,065,817,517,094,387,857,166,063,685,640,376,697,528,950,544,088,327,782,400,000,000,0000.

Quite a few, then.

Start your timers

In his excellent article that illustrates the mind-melting magnitude of 52!, Scott Czepiel invites us to ‘start a timer that will count down the number of seconds from 52! To 0.’ From there, we take up a spot on the equator and walk the 40+ million metres around the world, 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 (707.6 million cubic kilometers of water) each time you circle the globe.

Continue until the ocean is empty. When it is, 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.

1 Astronomical Unit, the distance from the Earth to the Sun, is defined as 149,597,870.691 kilometers. So, take the stack of papers down and do it all over again. One thousand times more. Unfortunately, that still won’t do it. There are still more than 5.385e67 seconds remaining. You’re just about a third of the way done.’

Some other outrageous numbers

Obviously, 52! isn’t the largest number in the playground. Remember There’s Something About Mary? If we take a leaf out of the hitchhiker’s book–the twitchy genius who came up with 7-minute abs instead of 8-minute abs–we could just go in the opposite direction and say 53! Or, if we’re feeling fruity, 54!...

We could even temporarily abandon the exclamation marks and check out some vertical arrows. Knuth’s up-arrow notation–named after mathematician Donald Knuth–is one way of representing massive numbers.

One arrow represents exponentiation. Exponentiation is just repeated multiplication. For example 10↑2 is 102. 102 is 10x10=100. 10 is the base number, 2 is the exponent. Here, the exponent represents the number of zeroes needed to express the base number in full.

Tetration

Three arrows represents tetration. Tetration is just repeated exponentiation. Unlike exponentiation, where the exponent lives to the right of the base number, tetration flips things horizontally and sticks it to the left. For example 4↑↑3 is 43. 43 is equal to 3 to the power of three, four times.

Tetration works from the top down. In this instance 33 is 27. 327 is 7,625,597,484,987. 37,625,597,484,987 is a number with over three trillion digits.

Pentation, as you may have guessed, is repeated tetration. Knuth’s arrows come in handy for expressing the kinds of numbers generated by tetration and pentation. Perhaps the most famous large number to use Knuth’s arrows is Graham’s Number, named after the mathemetician, Ronald Graham.

It is one of the largest numbers ever used in a serious mathematical proof; a number so big that a digital representation of it is made impossible thanks to the puny restrictions of the known observable universe. The general idea, though, is that the number of Knuth’s arrows grows cumulatively, and results in this huge, but still finite, number.

Middle World

So, if 52! isn’t as big as 53!–never mind Graham’s Number and a whole bunch of numbers even bigger than Graham’s–then why even bother writing about it?

Well, the answer lies in Middle World. No, it’s not a Tolkien spinoff; it’s the area humans inhabit - between the very small and the very large.

‘Our brains have evolved to help us survive within the order of magnitude of size and speed which our bodies operate at,’ said evolutionary biologist Richard Dawkins during a Ted Talk in 2006.

‘We never evolved to navigate the world of atoms. Moving to the other end of the scale, our ancestors never had to navigate through the cosmos at speeds close to the speed of light.’

Dawkins called this area that we inhabit, Middle World. As denizens of Middle World, Dawkins argues it is ‘intuitively easy’ to imagine Middle World objects like rabbits moving at middling velocity colliding with rocks.

In this sense, the humble deck of cards is a distinctly Middle World object. We can open the deck, shuffle the cards, deal them out. There is no cognitive load; none of these things–rabbits, rocks, playing cards–break our brains.

Problems arise when the Middle World collides with the mega galactic. Given the time and the inclination, the permutations from a deck of playing cards could unspool from our hand out into the cosmos at a scale Dawkins says we are unequipped to handle. The combinatorics of 52!, therefore, is a decidedly un-Middle World phenomenon.

So, while Graham’s Number and the up arrows of Knuth might dwarf 52!, the friction with Middle World is absent.

‘Science does violence to common sense’ - Richard Dawkins

If you want to demonstrate Dawkins’ point, find someone unfamiliar with 52!. Ask them the likelihood of two identical deals, and gauge their response. By framing it in this manner, perceptive participants might realise they’re being set up and get the hint it might not be a simple yes. But still, their Middle World brains won’t be able to comprehend the answer, even after you’ve shown them Czep’s article and had them stand on the equator, too.

When the author of Ecclesiastes complained ‘there is no new thing under the sun’, he was quite mistaken. All he needed was a time machine, a deck of cards, and a simple imperative: