ANSWER MAN
After last week's best-of-baseball-answers extravaganza — which some of you loved, some of you didn't — we're back to all new questions this week with a little something for everyone: hard science, linguistics, television, gambling and... well, of course, baseball. But just a little!
When I was a kid I played a lot of cards with my card sharp grandmother. She taught me to always shuffle a deck of cards seven times to ensure that the cards were thoroughly mixed. She spent a lot of winters playing cards in Las Vegas, so I've always assumed that she knew what she was talking about. I've always followed her advice and it always seems to do a good job. My daughter and I now play a lot of cards, and as I was passing this esteemed knowledge on to her, she asked me two questions I couldn't answer.
1. Why seven times? I.e., why not five or nine times?
2. If you were to shuffle the cards 14 times, would you end up with cards that were poorly mixed? I.e., they were well mixed at seven and at 14 are now more or less back where you started.
- Ian
Your card sharp grandma is indeed sharp about cards! Seven seems to be the magic number.
In 1990, the New York Times reported on a mathematical proof that showed that it takes a minimum of seven shuffles of a 52-card deck to make it sufficiently close to random. "Random" in this case means that every single possible arrangement of the 52 cards — and there are about 8 x 10^67 possible arrangements — has an equal possibility of occurring. Shuffling the deck fewer than seven times leaves recognizable artifacts of its original state. Shuffling more than seven times adds to the randomness, but only negligibly. (For games like blackjack, where the suit doesn't matter, four shuffles is sufficient.)
This discovery was made by Dr. Persi Diaconis and Dr. Dave Bayer, who at the time of the study were math professors at Harvard and Columbia, respectively. (Diaconis is also a magician.) I am no mathematician, so I won't attempt to explain it, but you can check out some of Dr. Diaconis' papers if you want to see what's behind the proof.
Note that this derivation assumes (and requires) ordinary, imperfect shuffles. If you can reliably perform "perfect shuffles" — dividing the deck into two halves of exactly 26 cards, then doing a "riffle" shuffle so that the cards fall in a perfectly alternating left-right-left-right pattern — a 52-card deck will actually be perfectly recycled after eight shuffles. That is, it will be in the exact same order it was before the shuffling began! Oddly enough, with a 54-card deck (add jokers), it will take 52 perfect shuffles before the deck is recycled; with a 64-card deck, it will take only six!
Sources: New York Times, Clay Mathematics Institute, How Many Times Should You Shuffle a Deck of Cards?
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