If you thought that your prayers had more to do with a roll’s outcome than anything else, perhaps a perfect shape wasn’t required. Eerkens says, “Some of the non-symmetry that we see in the earlier dice might be a by-product [of the idea] that it wasn’t thought to be very important in the function of the dice—that it didn’t matter too much, because other things were controlling whether you would win or lose the game.”

After the fall of the Roman Empire, dice largely disappear from the record in the Netherlands. Then, when they come back on the scene in about 1100 AD, the way the numbers are arranged around the die has changed. Rather than using the arrangement called sevens, where one and six are on opposite sides, as are three and four, and five and two, so each set of sides adds up to seven, the dice now use an arrangement called primes. One and two are opposite, as are three and four, and five and six: The sets each add up to a different prime number. “Suddenly, they’re all in this primes configuration,” says Eerkens. “We don’t know where that idea came from.”

At the same time, dice tend to get smaller, perhaps to make them easier to hide, as gambling was not favored by the increasingly powerful religious authorities. Some of the study’s dice from this era, in fact, were found all together in a small hole in a garbage heap, including one cheater’s die with an extra three. Could it be someone saw the light and forswore dice?

Still, despite official disdain, by the 13th century, at least in some parts of Europe, people begin to write in a systematic way about why dice games work the way they do, as the dice themselves grow more and more uniform. In the 17th century, even Galileo writes about why, in a game with three dice, the number 10 should come up more than the number 9. It’s an observation that one would only be able to detect after thousands of rolls, but the reason behind it involves how many combinations of numbers can add up to each different option.

The dice even switch back to the sevens configuration, a move that Eerkens suggests may have something to do with a growing sense that dice must be balanced, both physically and conceptually. (Lest you think one of these orientations is somehow the default, in an earlier experiment, the researchers asked schoolchildren to number the sides of paper cubes, and showed that neither primes nor sevens was intuitive. Instead, the kids wrote one on one face, turned the cube 90 degrees to write two on the next face, and so on, resulting in a configuration the researchers call turned.)

All these changes in dice come about, says Eerkens, “as different astronomers are coming up with new ideas about the world, and mathematicians are starting to understand numbers and probability.” Which came first: Did people begin to intuitively understand what true chance felt like, and adjusted dice accordingly, or did it trickle out from what would eventually become known as the scientific community? It isn’t clear, but to Eerkens, the story told by the dice is of a rising awareness of randomness. “It’s so embedded in the way we think [now],” he says. In our current conception, even the weather, once interpreted as a foregone conclusion thanks to divine intervention, is random. The very idea that there is a 50 percent chance of rain is a radical change from the past.

And it does make you wonder: Would you know that fair dice should fall equally on each side, or that in a three-dice game, it’s better to bet on 10 than nine, if someone hadn’t told you?

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