A Relatively Modern Idea: Fairness in Dice Rolling

[Image courtesy of Larsdatter.com.]

This may come as a surprise to you, fellow puzzlers, but fairness was not always a priority when it came to rolling dice.

Nowadays, whether you’re going after that elusive Yahtzee, hoping for doubles to earn another roll in Monopoly, or trying to roll sevens in a game of craps, the basic concept behind throwing dice is that every outcome of a six-sided die has an equal chance to appear. Unless you’re dealing with loaded or gimmicked dice, your odds should be 1 in 6.

But a recent study by researchers from the American Museum of Natural History and the University of California, Davis, has revealed that fairness in dice rolling didn’t really become a concern for dice users until the Renaissance. Researchers gathered dice spanning 2000 years of human history to explore why this was the case.

[Image courtesy of Wikipedia.]

From an article on Science Alert:

Roman-era dice, the researchers found, were a mess when it came to shape. They were made from a variety of materials, such as metal, bone and clay, and no two were shaped entirely alike. Many were visibly lumpy and lopsided, with the 1 and 6 on opposite sides that were more likely to roll up.

In fact, it seems like variety was the name of the game in Roman times, since the number configurations, shape, and size were inconsistent across the board, although dice were fairly common in the time period.

[Image courtesy of Pinterest.]

The Dark Ages led to a downturn in dice frequency, as they become very rare between the years 400CE and 1100CE.

The use of dice rebounds after 1100, and are most commonly found in ancient Mesopotamia and Egypt in what is known as the primes configuration, meaning that opposite numbers add up to prime numbers. 1 pairs with 2, 3 pairs with 4, and 5 pairs with 6.

There was a reinvigorated focus on the mechanics of chance and calculating probability, thanks to names like Galileo and Pascal, as well as a spirit of greater scientific understanding overall. Those Renaissance influences led to both a standardized shape for dice and a change in the numbering system. At this point, most dice convert to the sevens configuration, where opposite sides add up to seven (1 pairs with 6, 2 pairs with 5, and 3 pairs with 4).

[Image courtesy of Smithsonian.com.]

And according to lead researcher Jelmer Eerkens, cheating may have been on the mind of manufacturers going forward. “Standardizing the attributes of a die, like symmetry and the arrangement of numbers, may have been one method to decrease the likelihood that an unscrupulous player had manipulated the dice to change the odds of a particular roll.”

That change from variable shapes, sizes, and designs reflects a sea change in thinking towards dice and chance. Before, the shape didn’t matter because the results were attributed to Fate or some greater outside force, but later on, an understanding of chance and probability pushed standardization of dice forward.

In the end, it’s amazing how much of our culture and worldview, both past and present, can be revealed by exploring how we solve puzzles and play games.

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Keep your Get Out of Jail Free card handy.

Recently, friend of the blog Chris Begley sent me an article about some interesting math facts. Most of them were about probability and how many people misinterpret the likelihood of various events happening based on bad assumptions about probability.

For instance, since you have a 50/50 chance of heads or tails when you flip a coin, it seems logical that if you flipped a coin ten times, you’d get heads five times and tails five times, whereas in reality, it’s common to have runs of one result or the other that fly in the face of that simple 50/50 assumption.

But that wasn’t the fact that caught my eye. I thought it was much more intriguing that not all spaces on a Monopoly board have an equal likelihood of being landed on.

And that can affect how you play. For instance, if you believe each spot has an equal chance of being landed on — 1 in 40, given the 40 squares on the board — you might opt to buy all three colors in a given area to give yourself a 3/40 chance (7.5%), or you might go for all 4 railroads to give yourself a 4/40 chance (10%).

[A breakdown of spaces and likelihood of landing, based on the UK version.
(Chance and Community Chest cards differ between UK and US versions,
though probabilities for spaces in the US version are quite similar.)]

But that’s not how Monopoly actually works. Some spaces are far more likely than others. This is partly due to rolling two dice every time you move (which makes 6, 7, or 8 spaces the most likely results). There are also rule cards that make some squares more likely than others.

The most common space to land on is Jail (due in no small part to the Go to Jail square and where Chance and Community Chest cards send you). The most common PROPERTY to land on is Illinois Avenue, followed by B&O Railroad, Tennessee Avenue, New York Avenue, and Reading Railroad.

[A breakdown of space probabilities for the US version of the game.]

On the flip side, Mediterranean Avenue is the least likely to be landed on, followed by Baltic Avenue, Luxury Tax, Park Place, and Oriental Avenue. (Again, the Go to Jail square comes into play, as Park Place is seven squares away and the most common dice roll is 7.)

I like that a little properly applied math might make you a better Monopoly player. (Though if I’m going to walk the Boardwalk, I’d rather be playing The Doom That Came to Atlantic City.)

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