When a gang of gambling cheats sussed out how to beat the house, they inadvertently highlighted a loophole from a shuffled deck. It took a magician-turned-mathematician to reveal how.
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.
[...]
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.
[...]
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.
Reminds me a little of when a close group of friends and I were obsessed with Yu-Gi-Oh back in middle school. At sleepovers, we'd constantly duel each other for hours and refine our decks to no end.
To make it "more fun" (at least, from our perspectives), we'd do a limited kind of sorting on our decks before each game and stack the decks in the pattern "monster, monster, non-monster" so that even after our opponents shuffled our decks for us, they'd still be in a relatively optimal configuration.
> To make it "more fun" (at least, from our perspectives),
It definitely is more fun. As a kid, I won a Pokemon TCG match because my opponent never got a card they could play and forfeited (although it turns out we both just didn't know the actual rules for this). It was a little funny, but not fun to win, and I can't imagine he enjoyed it either. There's a lot of sub-optimal shuffles where the winner is so obviously pure luck it makes you wonder why you ever even considered strategy in the first place and IMO those are really unfun.
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[ 0.24 ms ] story [ 31.2 ms ] threadAfter 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.
[...]
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.
[...]
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.
"
To make it "more fun" (at least, from our perspectives), we'd do a limited kind of sorting on our decks before each game and stack the decks in the pattern "monster, monster, non-monster" so that even after our opponents shuffled our decks for us, they'd still be in a relatively optimal configuration.
It definitely is more fun. As a kid, I won a Pokemon TCG match because my opponent never got a card they could play and forfeited (although it turns out we both just didn't know the actual rules for this). It was a little funny, but not fun to win, and I can't imagine he enjoyed it either. There's a lot of sub-optimal shuffles where the winner is so obviously pure luck it makes you wonder why you ever even considered strategy in the first place and IMO those are really unfun.