Last Bridge Home Local Next

Five shuffles enough for random pack of cards,
say scientists

by Robert Uhlig, Technology Correspondent

Electronic Telegraph, London, Tuesday 10 October 2000


The number of times that bridge or poker players need to shuffle a pack of cards to ensure its order is fully random has been calculated by two mathematicians.

The study by a father and son found that few players shuffle the deck enough times to be sure their opponents might not remember the sequence of some of the cards. But it also discovered that, to ensure total randomness, the pack does not need to be shuffled as many times as previously thought.

Nick Trefethen, professor of numerical analysis at Oxford University, found that five riffle shuffles - where the pack is split and the cards flapped together at their edges - will make a pack fully random. His research, undertaken with his father, Lloyd Trefethen, professor emeritus of mechanical engineering at Tufts University, challenged previous investigations which claimed that total randomness required at least seven riffle shuffles.

The Trefethens used information theory, a branch of mathematics concerned with storing, transferring and decoding data. According to their report in the Proceedings of the Royal Society, an unshuffled deck contains 225 bits of information.

Prof Nick Trefethen said: "Each time you shuffle, the information becomes more vague, until it is obscure enough to be considered random. If you know the exact order of the cards, you have all 225 bits. If the pack is shuffled until none of the original order remains, the information you have is zero."

Using a computer program to shuffle the cards, the Trefethens found that after one shuffle, 173.58 bits of information remained in the deck. After two shuffles, 121.58 bits remained. Prof Nick Trefethen said: "By the fifth or sixth shuffle, the deck has lost 99 per cent of its information, which is enough to consider it random."

The research suggested that few card schools or casinos shuffled their cards enough to guarantee that none of the order of previous hands remained, but it did offer one consolation to players. The popular "overhand shuffle", where small groups of cards are dropped back into the rest of the pack in reverse order, needs at least 2,700 shuffles and takes about 20 minutes to ensure total randomness.


Editor's note:

Last Bridge Home Top Local Next
Top of