### A Mind-Boggling Case of Shuffled Cards

Just for the Hell of It Dept.

Just recently I saw a little article describing how people on Twitter were in a mild uproar over the “paradox” or “theory” that one 18” pizza actually contains more pizza than two 12” ones. (It’s not a “theory” but simple geometry and algebra: A=πr², so an 18” pizza would have a total area of around 254 square inches, and two 12” ones would have a combined area of about 226 square inches.) The article describing this furor emphasized the dumbing down of the American populace, with some aspersions cast towards an educational system that lately seems more intent on indoctrinating young people into the cult of politically correct progressivism, than in actually educating them.

Even so, I have occasionally been mind-screwed by odd bits of factual information, and I would like to share one that leaves the pizza paradox far behind, where it belongs. It involves playing cards.

How many possible configurations of shuffled playing cards do you suppose there could be? Guess. No, come on, guess. A million? Ten million? As much as a billion? Even if there were as many as a billion possible orders of cards in a shuffled deck, it would be very unlikely that in your lifetime you would even come across two well-shuffled decks of cards with all the cards in the same order. And if you consult tarot cards the odds would be even less, since there are 78 of those in a standard deck, if I remember correctly. Anyway, have you guessed yet? No? Guess.

Let’s calculate it out here. If there are any of you who passionately hate math, and were astonished by the pizza paradox, I apologize. Go away.

First let’s ignore the jokers and assume that a deck of standard non-pinochle playing cards contains 52 of them. So, there are 52 possible first cards in a well-shuffled deck. For each of these possible first cards there are 51 possible cards remaining to be the second card. Thus there are 52×51 possible combinations for the first two cards, equalling 2652. For each of these combinations there are 50 possible third cards, and for each of those combinations there are 49 cards remaining to be the fourth, and so on. When we reach the 51st card there are only two options for the next one as there are only two cards remaining; and of course the last card adds no new possibilities since there is only one left for each combination of 51. So the total number of possible shuffled card decks would be 52×51×50×49…3×2×1, or, in the notation of math geeks, 52!, or fifty-two factorial. So, the total number of possible shuffled decks of 52 cards would be 8.066×10⁶⁷, or, in ordinary non-abbreviated notation, approximately 80,660,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. The number is so big that I have no idea that it would be called. I just can’t count that high. It’s somewhere up in the zillions, maybe even the gajillions.

To give you some idea of the unimaginably huge number of possible card sequences in a shuffled deck, the number is larger than the estimated total number of atoms in our entire freaking galaxy. Not just the total number of atoms in our planet earth, mind you, or in our entire solar system either—the total number of atoms in the Milky Way Galaxy, containing, as Carl Sagan used to say, billions and billions of stars. There may be more possible combinations of playing cards than all the atoms in the Milky Way, the galaxy of Andromeda, plus the Magellanic Clouds, assuming that astrophysicists know what they’re talking about. (If you don’t believe me on this, feel free to Google it. My source on the total number of atoms in our galaxy is taken from HERE . Don’t confuse our galaxy with the entire astronomical universe, though; take the total number of atoms in the universe and divide by a hundred billion. Fortunately for the health of my mind, there are several orders of magnitude more atoms in the whole universe than there are possible card decks.)

As if that weren’t weird enough, one implication of this astonishingly huge number of possible shuffled decks is that it is probable that in all the history of the world and of playing cards, there have never been two well-shuffled decks of cards that have been identical. It may never happen, even if people continue to play cards for thousands of years. Even if it has happened through random chance or some mysterious synchronistic phenomenon of human luck, even so the same sequence of cards has not come up more than a very few times, assuming that the cards really were randomized and nobody was cheating. It boggles the mind.

Here’s another strange one that I find more weird than one medium pizza being bigger than two small ones—it’s one that I learned long ago, as a student studying marine biology. Let’s say that you scoop a cupful of water out of the ocean; it doesn’t matter which ocean (pick an ocean, any ocean…); and then you write your initials on every water molecule in that cup of seawater. (Let’s say you have an ultrafine-point pen.) Now let’s say that you pour your initialed water molecules back into the ocean, and stir the oceans well. Assuming that your cupful of water has been thoroughly mixed in with all the other water molecules in the oceans of the world, you could then scoop a cupful of water anywhere where there is seawater, and the number of molecules with your initials on them would be somewhere in the neighborhood of two hundred.

Finally, here’s one that I worked out on my own, while living in a cave in Burma and arguably having too much free time on my hands. You’ve heard of the song “A Hundred Bottles of Beer on the Wall,” right? Sure you have. Maybe you’ve even sung it with other people, say, on a long and boring bus ride when you were a kid. OK. Now, how long do you suppose it would take to sing “A Trillion Bottles of Beer on the Wall”—you know, “A trillion bottles of beer on the wall, a trillion bottles of beer, take one down, pass it around, nine hundred and ninety-nine billion nine hundred and ninety-nine million nine hundred and ninety-nine thousand nine hundred and and ninety-nine bottles of beer on the wall…”? How long would it take to sing the whole thing all the way down to zero? Guess. No, come on, guess. A century, you say? Bullshit. According to my calculations it would take approximately 250,000 years. And that’s without taking any breaks. So now you know. Use this knowledge wisely.