February 2, 2021

The Die Is Cast

Attributed by Suetonius to Julius Caesar (100-44 B.C.)

A die tumbles out of a cupped hand, bounces on the carpet a few times, rolls a short distance, then teeters to a stop. The uppermost face of the white cube shows four black dots arranged in a square.

Grinning, a child briskly moves a red token four squares to the right along the bottom row of a large checkerboard grid. The token lands on a square marked with the foot of a ladder. The player immediately takes the shortcut, advancing the token up the rungs to a higher row. Just ahead lies a square ominously marked with the upper end of a chute—the start of a costly detour.

With moves governed entirely by the roll of a die, Chutes and Ladders (earlier versions featured snakes and ladders) is a racecourse on which children of different ages and their elders can meet on an equal footing. Physical prowess and breadth of knowledge are immaterial on such a field. Only luck comes into play.

The playing of games has a long history. We can imagine the earliest humans engaged in contests of physical strength and endurance, with children racing about playing tag and great heroes (such as Heracles) struggling against daunting obstacles, as recorded in ancient myths.

Written references to games go back thousands of years, and archaeologists have recovered a wide variety of relics that they interpret as gaming boards and pieces.

In the year 1283, when the king of Castile, Alfonso X (1224-1284), compiled the first book of games in European literature, he testified to the importance of games-playing in medieval society.

"God has intended men to enjoy themselves with many games," he declared in the book's introduction. Such entertainments "bring them comfort and dispel their boredom."

Even in Alfonso's time, many of the board games he described were already hundreds of years old. Chess, the king's favorite, had been developed in India centuries earlier. Backgammon, one of the great entertainments of thirteenth-century nobility, had evolved from the Roman game tabula.

Succeeding centuries brought new amusements, along with variations on old ones. Each age and place had its particular favorites: the dice-and-counter game of pachisi in India, the coin-sliding game of shove ha'penny in William Shakespeare's England, the ancient game of go in China and Japan, and the card game cribbage in seventeenth-century Europe and America. In the Victorian era in Great Britain, nearly every parlor featured a wooden board of holes and pegs for the game of peg solitaire.

Amusement remains the motivation underlying the explosion of ingenuity that has created a bewildering array of addictive computer, video, and arcade games, various forms of online and casino gambling, and new sports ranging from beach volleyball to snowboarding, along with novel board games and puzzles to tickle the mind.

"With their simple and unequivocal rules, [games] are like so many islands of order in the vague untidy chaos of experience," author Aldous Huxley (1894-1963) wrote several decades ago. "When we play games, or even when we watch them being played by others, we pass from the incomprehensible universe of given reality into a neat little man-made world, where everything is clear, purposive and easy to understand."

In these miniature worlds, competition brings excitement. Randomness serves as an equalizer. Chance introduces an element of suspense. Risk amplifies the thrill of play to an intoxicating level.

These tiny microcosms also attract mathematicians, who can't resist the distinctively human pleasure of learning the secrets of games. Who stands to win? What's the best move? Is there an optimal strategy? How long is a game likely to take? How do rules combine with chance to produce various outcomes? How are fairness and randomness linked?

In games of chance, each roll of a die, toss of a coin, turn of a card, or spin of a wheel springs a delicious surprise. Anyone can play. Anyone can win—or lose. Mathematics helps dispel some of the mystery surrounding unpredictable outcomes. It embodies an ever-present urge to tame the unruliness of Lady Fortune.

Using mathematical reasoning, we can't predict the outcome of a single roll of a die, but we can alter our expectations in the light of such an analysis. We may take comfort in the notion that, if the die is fair, each face of it will come up equally often in the long run.

More generally, we can begin to make sense of and exploit the patterns that inevitably appear and disappear among the infinite possibilities offered by random choices.

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