After one or two riffle shuffles, some cards will remain in their original sequence. Even after four or five shuffles – far more than most casinos typically use – the deck will retain some trace of order. But once you shuffle the deck seven times, the cards become truly mixed, at least as far as most statistical tests can prove. Beyond that point, further mixing will not do much. "It's just as close to random as can be," Diaconis says.
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Markov chains have no "memory" of what came before. This is a pretty good model for shuffling cards, says Assaf. The result of the seventh shuffle depends only on the order of the cards after the sixth shuffle, not on how the deck was shuffled the five times prior to that.
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the Markov chain describing riffle shuffles has a sharp transition from ordered to random after seven shuffles. This behaviour, known to mathematicians as a cut-off phenomenon, is a common feature of problems involving mixing. Think of stirring cream into coffee: as you stir, the cream forms thin white streaks in the black coffee before they suddenly, and irreversibly, become mixed.
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