If I shuffle a deck of 52 cards and hold it out for you to pick one, I can do so with an astonishing guarantee: No other deck in history has ever existed in this exact configuration — and no other deck ever will.
That doesn’t seem right, does it? Playing cards have been around since the 9th century, the standard deck since 1516. A lot of shuffling has happened since then. Cowboys played cards. World War II soldiers played cards. Today, millions of people play poker, black jack, and bridge. How do configurations not repeat?
Well, as I learned from Tim Urban, the number of unique arrangements can be calculated with a simple formula: 52 factorial, or 52! in math terms.
A factorial is simply a multiplication of decreasing factors. 3! is 6. You take 3 * 2 * 1. 5! is 120. You take 5 * 4 * 3 * 2 * 1. Therefore, 52! is just 52 * 51 * 50…and so on. Here is the number this results in:
Yeah. There’s no decimal point in there. Let’s see if we can wrap our brains around this number.
If you were to reshuffle the deck every second while only getting new combinations as you continue, here’s how long it would take you to go through all possible options: First, you should stand close to the Pacific Ocean. San Diego, perhaps? Pick a spot near the water. Then, you start shuffling. Once a second. Shuffle. Shuffle. Shuffle.
You shuffle for a billion years. That’s right. A billion. Then, you take one step. One. Hold your horses and such. Then, you shuffle for another billion years. Great! Now, another step. You take one step every billion years of shuffling until you’ve circumvented the globe. That’s 40,000 kilometers by the way. 25,000 miles. A thousand marathons, if you will.
Eventually, you’ll be back in San Diego. Are you done? Haha — no. After you arrive from your slow trip around the world, you take a pipette and remove one drop from the Pacific Ocean. Then, you repeat. Not just the drop removal. The whole thing. Hey! Keep shuffling! One step every billion years, one trip around the world, one drop from the Ocean. At 700 million cubic kilometers of water, that too should take a while.
Once you’re done and the Pacific Ocean is empty, you’ll be rewarded with a fantastic prize: A piece of paper. You take the paper, and you put it right next to you, flat on the ground. Shuffle. Shuffle. Shuffle. You see where this is going, don’t you? You’re gonna have to fill this Ocean back up.
One step, one billion years, one trip around the world, one drop restored. When the Ocean is full again, you get another sheet of paper. You put it on top of the other one. Shuffle. Shuffle. Every time you’ve emptied or refilled the ocean, you get another piece of paper. And you’ll continue stacking them.
How high will your stack have to reach? The Empire State Building? Plane height? The moon? No. Your stack of paper will have to reach the sun. That’s an easygoing 150 million kilometers. Fortunately, you’ll only have to build 3,000 of those stacks until, finally, your card configurations will have to start repeating.
One shuffle every second. One billion years, one step. One trip around the globe, one drop from the ocean. One complete ocean, one piece of paper. One paper stack to the sun, repeated 3,000 times. That’s how long it’ll take until you find an arrangement of a 52-card deck that has existed before.
We humans are a tiny part in an infinite universe. Compared to its vastness, we are small, and our problems seem even smaller. Despite our smallness, however, we can contemplate the universe. We can comprehend a great deal of it, and we can definitely solve all our problems. Isn’t that amazing?
The universe is bigger than we can ever imagine — and yet, we can hold the entire universe in a single hand. All we have to do is shuffle a deck of cards.