Is It Possible For A Deck Of Cards To Be Shuffled Accidentally Into Perfect Order?

Are you a self-proclaimed poker god or do you enjoy shuffling a deck of cards? Have you ever wondered is there a way to shuffle a deck of cards into perfect order accidentally?

Firstly we must make two assumptions: one that there is a singular entity that we can call a “perfect order”, and secondly that the “shuffling” is perfectly random.

We will tackle “perfect order” first. What is “perfect order”? If we assume it’s the order the cards were in when you first open the box, or that it has a single definition (such that the suits and values have to be of a fixed ordering), then there is only 1 out of 52 ways to achieve a “perfect order”.

But what if we don’t care about the order of the suits? Maybe SPADES, HEARTS, DIAMONDS, CLUBS is just as valid as HEARTS, DIAMONDS, CLUBS, SPADES. You can reduce the permutations by a factor of 24 if the suit order doesn’t matter. And what about the ace? Is it before the two or after the king? If it can be either, you can reduce the probability by another factor of 2. Combine both together, and you’ve reduced the permutations by a factor of 48. That reduces the permutations from 1 in ~8 * 1067 down to ~1.6 * 1066. That will still take you forever, but it is a slight improvement.

A better way to do it

But what if we told you we could guarantee you can do it in only eight shuffles?

“Shuffling” is usually thought of conceptually as “randomization”, and while that’s the intent, the mechanical principals of how cards are actually shuffled aren’t purely random events. Take the standard riffle shuffle for example. You split the deck into two nearly even halves, and then alternately drop a few cards from each pile into a new pile. This isn’t a purely random series of events: the bottom card of the pile that has the first drop will always be on the bottom of the shuffled deck, for example.

Indeed, if you’re so very skilled in shuffling that you always split the deck into two perfect 26-card sub-decks, and always start the shuffle with a card drop from the same sub-deck, AND always perfectly interleave the cards one at a time, you can “shuffle” the deck in a completely predictable way (this actually has a name, and is known as an out shuffle). And if you do that, after only 8 shuffles you’ll have your original deck layout again!

The number of possible shuffles of a standard deck of cards (52 cards) is 52 * 51 * 50 * … * 1, or otherwise written as 52!. This number is approximately 8 * 1067 (an 8 followed by 67 0’s).

So how long does it takes to achieve this?

To get an idea of how unimaginably large that number is, let’s assume that we can create a billion different deck-orderings every second. That’s 3.6 trillion per hour or 86.4 trillion per day. At this rate, it would take about 1054 days to exhaust all possible orderings. Or about 3 * 1050 years. For reference, our universe is about 1.5 * 1010 years old.

The chance of shuffling a deck of cards into one specific order is so incredibly small, that it is effectively impossible. You stand a better chance at trying to win the jackpot in the lottery multiple times in a row.

How many times do I have to shuffle?

Imagine you shuffle a deck of cards once per second, every second. You shuffle 86400 times per day.

You start on the equator, facing due east. Every 24 hours (86400 shuffles), you take one step (one metre) forward. You keep shuffling, second after second, each day moving one more metre. After about 110 thousand years, you will have walked in a complete circle around the Earth (We know you can’t walk on water. Just ignore that part).

When you have completed one walk around the Earth, take one cup (250mL) of water out of the Pacific Ocean. Then, start all over again, shuffling, once per second, every second, taking a step every 24 hours. When you get around the Earth a second time (another 110000 years), take another cup of water out of the Pacific Ocean.

Eventually (after approximately 313 quadrillion years, or so, about 22 billion times longer than the age of the universe), the Pacific Ocean will be dry. At that point, fill up the Pacific Ocean with water all over again, and place down one sheet of paper. Then, begin the process all over again, second by second, every 24 hours walking another metre, every lap around the Earth another cup of water, every time the Pacific Ocean runs dry, refilling it and then laying down another sheet of paper.

Tall enough to reach the moon…

Eventually, your stack of sheets of papers will be tall enough to reach the Moon. I think it goes without saying that, at this point, the numbers become very difficult to comprehend, but it would take a very very very very very long time to do this enough to get a stack of paper high enough to reach the Moon. Once you get a stack of papers high enough to reach the moon, throw it all away and begin the whole process again, shuffle by shuffle, metre by metre, cup of water by cup of water, sheet of paper by sheet of paper.

Once you have successfully reached the Moon one billion times, congratulations! You are now 0.00000000000001% of the way to shuffling 8 * 1067 times!

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