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The dice used by gamers and gamblers in northern Europe became much more fair about 600 years ago. What鈥檚 more, the pattern on the faces changed, and these trends together might reflect people鈥檚 growing awareness that bets were decided by chance, not by the gods 鈥� even though probability theory was centuries away.

Dice often turn up in archaeological digs, but are rarely studied systematically. Now at the University of California Davis and at the American University of Natural History in New York have examined 110 dice from sites in the Netherlands.

Modern gamers wouldn鈥檛 like the earliest, Roman-era dice, some of which are 2000 years old. About 90 per cent are asymmetrical and wouldn鈥檛 roll randomly. Only from about AD 1450 were most dice more or less symmetrical.

Medieval Europeans may have been increasingly aware of the importance of chance. 鈥淧eople in Roman times could make symmetrical dice, and they often did,鈥� says Eerkens. 鈥淏ut there are a much higher percentage of non-symmetrical dice in the Roman period. That may mean that players in Roman times thought that other forces were mostly at work in determining the outcome of a roll.鈥� If the gods controlled the roll, why worry about unfair dice?

However, by the 15th century, gamers evidently demanded symmetrical dice.

Tumbling dice

But what really interested Eerkens is how the numbers were arranged. There are a variety of ways to arrange the numbers one to six on a die, and several are seen in Roman-era dice.

However, between the years 1250 and 1450 one arrangement was dominant. Of 41 dice from this time, 37 have 1 opposite 2, 3 opposite 4, and 5 opposite 6. This configuration is called 鈥減rimes鈥�, because opposite faces add up to prime numbers.

Then dice suddenly changed. In just 150 years, primes died out and was replaced by the modern configuration: 1 opposite 6, 2 opposite 5 and 3 opposite 4. This is called 鈥渟evens鈥�, as numbers on opposite faces always add up to seven.

When Eerkens and de Voogt looked at dice from the UK, they found a similar shift from primes to sevens, again beginning in the 1450s.

Shifting numbers

A standard arrangement of spots might have been a way to check a die was authentic. 鈥淚 think it鈥檚 easier for players to quickly inspect the die, to make sure it鈥檚 up to the standard, if there鈥檚 an easy pattern to remember,鈥� says Eerkens. 鈥淧rimes and sevens are easy ones.鈥�

The rise in popularity of the sevens configuration might be another sign of the growing demand for dice to be symmetrical. The primes configuration might have become unpopular because it was thought to be 鈥渦nbalanced鈥�: for instance, it has 1 opposite 2 鈥� adding up to 3 鈥� and 5 opposite 6 鈥� adding up to 11. The sevens configuration is balanced in the sense that opposite faces always add up to seven, which may have been thought to make it especially likely to roll in a fair way.

This implies Medieval Europeans 鈥� particularly gamblers 鈥� were starting to think about the factors that govern the outcome of rolls, long before .

Probably probability

鈥淚t has always surprised me that, at times when mathematicians were already able to produce deep arguments about many topics, probability seems little understood,鈥� says at the University of St Andrews, UK.

Robertson doubts trends in dice design led to a grasp of probability. But others are open to the idea.

鈥淚t does seem reasonable that, as dice design became more 鈥榬egular鈥�, successful players were better able to see patterns in play that might have led to early probabilistic thinking,鈥� says at Lake Forest College, Illinois. It may have gone both ways: perhaps 鈥渕ore regular dice stimulated probabilistic thought, which in turn stimulated the importance of even better dice construction.鈥�

Certainly, the early stirrings of probability theory emerged from gambling dens. , a 16th-Century polymath and gambler, wrote about the mathematics of games of chance a century before Pascal, albeit in a basic way.