What’s the perfect number of cards for playing poker? According to a new mathematical discovery, the answer is the traditional 52. Almost.
In Texas Hold’em poker, players wager on the best five-card hand they can make among the two cards in their hand and the communal ones on the table. Hands are ranked based on their probability of occurring. A full house, for example, with three cards of the same value (fives or kings, for instance) and two cards of another, is less likely than a flush with any five cards of the same suit. A full house therefore beats a flush. In short-deck poker, a variant that removes cards numbered 2 through 5 (called “ranks 2–5”), there are fewer cards of each suit and a flush becomes less likely than a full house. In a recent paper posted to the preprint server arXiv.org, computer scientist Christopher Williamson examines how the game of poker changes as the number of cards per suit increases or decreases. Depending on that number, an interesting paradox can occur, Williamson found—one that’s nearly always avoided in games with a 52-card deck with 13 in each suit.
After a bidding process, poker players will declare their best hands in what’s called the “showdown.” Consider two hands: a one pair, which has one pair of cards with the same value, and a two pair, with two pairs of such cards. In short-deck poker, although a two pair is less likely to appear than a one pair, its showdown probability—the likelihood that it will be the best hand someone has—is actually higher than that of a one pair. If the hand rankings were adjusted to account for this, no one would ever play a two pair over a one pair, which would change its showdown probability once again.
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“Thirteen is kind of the sweet spot,” Williamson says regarding how many cards a deck should have in each suit. It’s the smallest number for which the showdown ranking actually matches the standard ranking for all hands other than a single high card (the lowest-ranking hand). Although the usual number of cards in a deck is purely historical—it’s said to symbolize 52 weeks in the year and the four seasons—Williamson was delighted to find a more mathematical justification. There is only one superior setup, which would keep all rankings aligned to their showdown probabilities, including the high card: you’d need 23 cards in each suit, nearly twice as many as in the traditional deck.
Nikita Luther, a top professional poker player, finds the result fascinating. “It’s so complex, the way the variables interact with each other,” she says. Luther says she hasn’t deeply explored variants beyond Texas Hold’em; even there, “I could spend the rest of my life just trying to understand this game.”
Try a related math puzzle here.
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