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Shuffle like a pro!

By David, 8 September 2014 Activity

You will need

Someone is holding a deck of cards between their thumb and fingers.

Hold the deck in your right hand.

  • Two decks of cards. Younger mathematicians with small hands might have difficulty with a full sized deck of cards. Using smaller cards might be easier. If you’re still having difficulty, try asking an adult for help with shuffling.

Overhand shuffle (for right-handed people)

If you are left-handed, click here to see the left-handed instructions.

  1. Put a deck of cards face down on a table.
  2. Use your right hand for this step. Put your thumb on one short edge, and your fingers on the other short edge. Then pick up the deck and turn it face up.
  3. Use your left hand to turn the deck of cards over so you can’t see the faces while keeping the deck between your thumb and fingers.
    Someone has cupped their left hand around the deck of cards and is shuffling them.

    Take a few cards off the top of the deck with your left thumb.

  4. Keep a hold of the deck with your right hand, but cup your left hand under the deck of cards.
  5. Use your left thumb to pull a few cards off the top of the deck and into your left hand. Repeat this process over and over until all the cards are in your left hand.
  6. It takes a few goes to shuffle a deck properly, so put the cards back into your right hand and repeat this process a couple of times.

Riffle shuffle

  1. Put a deck of cards on the table.
  2. Divide the cards into two roughly equal piles.
  3. Use your right hand for this step. Choose a pile. Put your thumb on one short edge, and your fingers on the other short edge. Pick up the pile.
    Someone is holding a pile of cards in each hand. Their thumbs are close together.

    Bring the two piles close together.

  4. Still with your right hand, take your pointer finger off the end of the pile, bend it and rest it on the middle of the cards.
  5. Pick up the other pile with your left hand in the same manner.
  6. Put both piles down on the table, putting your thumbs close together.
  7. Squeeze the piles tightly, and push down with your pointer fingers to bend the cards.
  8. Slide the piles together as close as you can.
  9. Slowly lift your thumbs up. The bottom few cards of each pile will slap down onto the table. If your piles are close enough, they will overlap. This is exactly what we want to happen. Keep going until all the cards have slapped down.
    Someone is bending both piles of cards and flicking them down so they interleaver.

    Gradually release the cards from both piles simultaneously.

  10. You can now push the cards together to make a shuffled pile. For a cool way of joining the piles, squeeze the place where the cards overlap, and use your palms to bend the ends of the piles toward each other. Slowly release with your thumbs and hopefully the cards will come together. (But they might fly everywhere instead!)

Compare the pair!

  1. Take two decks of cards. Sort them, putting all the cards in order.
  2. Shuffle one of the decks using the overhand shuffle.
  3. Shuffle the other deck using the riffle shuffle.
  4. Look for patterns in the two decks. Is one deck more shuffled than the other?
Two sequences of cards.

Compare the two shuffles – can you find patterns in them?

What’s happening?

Shuffling will take an ordered deck of cards, and make it more random. This process removes patterns from the order of the cards. If you play cards with a badly shuffled deck, you might spot patterns in the order of the cards. You could use those patterns to guess the next card in the deck, which would give you an unfair advantage.

Some methods of shuffling are better than others. It only takes a few minutes of riffle shuffles to randomise a deck of cards. It takes much longer if you use an overhand shuffle. Most casinos use machines to shuffle their cards, because they are quick and people think they do a good job. But machine shufflers are not perfect. In 2000, a mathematician named Persi Diaconis wrote a report on shuffling. As a result, many casinos redesigned their shuffling machines!

More information

A slightly different way of doing a riffle shuffle
A proof showing why five riffle shuffles doesn’t completely randomise a deck of cards

Did you like this activity? Find out more about shuffling.

Overhand shuffle (for left-handed people)

Someone is holding a deck of cards between their fingers and thumb.

Hold the deck in your left hand.

  1. Put a deck of cards face down on a table.
  2. Use your left hand for this step. Put your thumb on one short edge, and your fingers on the other short edge. Then pick up the deck and turn it face up.
  3. Use your right hand to turn the deck of cards over so you can’t see the faces while keeping the deck between your thumb and fingers.
  4. Keep a hold of the deck with your left hand, but cup your right hand under the deck of cards.
  5. Use your right thumb to pull a few cards off the top of the deck and into your right hand. Repeat this process over and over until all the cards are in your right hand.
  6. It takes a few goes to shuffle a deck properly, so put the cards back into your left hand and repeat this process a couple of times.

    Someone has cupped their right hand under the deck of cards and is shuffling.

    Use your right thumb to take a few cards off the top of the pile.

Head back to the rest of the activity by clicking here!

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