Each Shuffle of a Deck of Cards is Probably Unique in History

Playing CardsWhat if I told you that each time you shuffle a deck of cards thoroughly, it is a near certainty that no 52 cards have ever been arranged in that order at any time in history?

Hard to believe? It’s true, and it is fairly easy to demonstrate. Let’s start with the number of ways that a deck of cards can be arranged. For a deck of 52, the number is 52! or 52 factorial. A factorial of a number is simply the number multiplied by all smaller integers. So 4! is 4 x 3 x 2 x 1 = 24. This is simple enough, but 52! is a very, very large number.

52 Factorial

An ordinary calculator can’t handle computing 52! but Wolfram Alpha can. The result of 52 x 51 x 50 x 49 x 48 x 47 x 46 x 45 x 44 x 43 x 42 x 41 x 40 x 39 x 38 x 37 x 36 x 35 x 34 x 33 x 32 x 31 x 30 x 29 x 28 x 27 x 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 is this number:

  • 80 658 175 170 943 878 571 660 636 856 403 766 975 289 505 440 883 277 824 000 000 000 000

Another way to express that number is 80,658 times a billion, times a billion, times a billion, times a billion, times a billion, times a billion, times a billion. It is also 80 unvigintillion, but let’s call it 8.0658 x 1067 or 8.0658e67. Clearly this is a very large number indeed, but how much larger is it than all the times that cards have been shuffled throughout history?

All the Shuffles in History

The earliest known playing cards appeared in the 9th century, but to cover all our bases, let’s assume that playing cards have been around as long as humans. Not everyone plays cards often, but some people play several times a day, so to be sure we’re not underestimating, let’s assume that everyone who has every lived shuffled a deck of cards once every day for their entire lives. I’ll also use the current world life expectancy of 71 years, even though people in the past didn’t live as long on average. So here are some round figures for our estimate:

  • Number of people who have ever lived: 108 billion
  • Number of days in an average life: 25,915

The result? Only 2,798,820,000,000,000, otherwise known as 2 quadrillion, 2.79882 x 1015, or 2.79882e15.

Your Odds of Duplicating a Shuffle

Dividing our estimate of the number of shuffles in history by the number of different ways a deck of cards can be ordered gives our odds that a thorough shuffle will duplicate an arrangement of cards that has already existed in history. Those odds are:

  • 1 in 28 818 573 541 706 860 271 102 652 118 169 924 509 892 074 660 704 013

That is, one in 28 sexdecillion.

I think the above calculations are an overestimate of how many shuffles have taken place in history, but perhaps you disagree. After all, avid card players may shuffle a deck dozens of times a day, maybe hundreds. Let’s go nuts and assume that every person who has ever lived shuffled a deck of cards once per second for their entire lives, even in their sleep. That’s 108 billion people times 25,915 days times 86,400 seconds in a day, giving us 241,818,048,000,000,000,000 shuffles. Our new odds:

  • 1 in 333 548 371 089 000 785 784 014 322 171 578 898 236 571 043 795

That is, one in 333 quattuordecillion.

What if all 108 billion people who have ever lived all started shuffling a deck of cards once per second at the moment of the Big Bang, 13.8 billion years ago, and continued until today? With only 435,196,800,000,000,000 or 4.351968e17 seconds having elapsed so far since the beginning of time, we would still only manage 4.7e28 shuffles, giving us these odds of duplicating a shuffle:

  • 1 in 1 716 127 659 574 467 779 169 859 921 401 280 862 936

That is, one in 1 duodecillion. Those are the best odds you’re going to get, and it means that every time you shuffle thoroughly, you are creating an arrangement of cards that almost certainly never existed before.

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