In games with playing cards, you are usually asked to shuffle the cards beforehand. What is the most efficient technique to get this mixed up as well as possible? Some shuffle in different ways, in 2 halves, throw the cards on the table,….

## Answer

What a cool question! Mathematicians deal with countless – sometimes very crazy – topics, and shuffling a deck of cards is also one of those topics.

First of all, it is important to consider what it means for a deck of playing cards to be shuffled enough. A standard deck of playing cards consists of 52 cards (we’ll leave out the jokers for a moment), which means there are a whopping 52! (that’s 52 factorial, a number equal to 52*51*50*49*…*3*2*1, an incredibly large number) Configurations exist for the order of those cards. I won’t go into unnecessary detail, but mathematically we call a deck of playing cards sufficiently shuffled if, after shuffling, the chance for that deck of cards to end up in any of those many configurations is the same.

There are many shaking techniques, each with their advantages and disadvantages, I will describe the three best known.

- Overhand shuffling: hold the deck of cards in your left hand, with your right hand you take a deck of cards from the deck of which you repeatedly place a small number of the top cards on top of the original deck. This action is often performed in quick succession. To obtain a sufficiently shuffled pack of cards using this technique, you have to perform this set of actions no less than 2500 times in succession.
- The ‘riffle’: You divide the deck of cards into two piles – one in each hand – and weave the cards together by letting them release from both thumbs. In 1992, two mathematicians – Bayer and Diaconis – proved that a deck of cards is sufficiently shuffled in this way after only… 7 times.
- Spreading the cards on the table and mixing: it may look less professional, but this technique isn’t bad either. If you keep mixing for at least 1 minute, you will also achieve a sufficiently shuffled deck!

Once a deck of cards has been sufficiently shuffled, you can rest assured that no one in the world has ever achieved your exact configuration.

## Answered by

#### Lins Denaux

Combinatorics, Finite Geometry and Discrete Mathematics Algorithms and Computational Mathematics Cryptography and Coding Theory

http://www.ugent.be

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