Calculating Risk at the Dicing Table

This paper begins from a surprising coincidence. There were two types of dice that Greeks regularly gambled with: six-sided cubic dice, and four-sided knucklebone dice. Usually in Greece three dice were thrown in dice games with cubic die, while five knucklebones were thrown in games with knucklebones. What is a remarkable coincidence is the number of combinations that these two types of dice—five knucklebones and three dice—produce. When one throws three cubic dice the number of possible permutations is 216 (6x6x6). The number of permutations, however, are less important in dice games because it does not matter which die bears which number, but rather simply which numbers one has rolled. That is, one needs to know the total number of combinations, not permutations. Calculating this is a little more difficult—one can either write out all the permutations and cross out redundancies, or one can use something like the factorial and calculate the combinations with an equation. The number of possible combinations that three six-sided dice yield is thus not 216 but 56.

Here is where things get strange: the number of possible permutations from five four-sided knucklebone dice is 1024 (4x4x4x4x4)—almost five times the number that three six-sided dice would produce. So, for example, it is almost five times less likely to throw all high numbers in a game of knucklebones (known as the “Euripides” throw in the fourth century BCE) as it is to roll a triple-six in a game of cubic dice. Yet when one calculates the number of combinations, that is, when one crosses off all redundancies, one reaches precisely the same number of combinations, that is, 56. This is a remarkable coincidence, yet considering the math involved one would hardly expect it to be noticed by inebriated gamblers at the dicing table.

And yet it does seem to have been noticed: although much later, there are lists of 56 fortunes that have survived in manuscripts, inscriptions, and recently, papyri, which can be accessed either by using three cubic dice or five knucklebones. On the western side of the tradition is the well-known Sortes Sanctorum of Roman Gaul which used three cubic dice. On the eastern side of the tradition are the lesser-known Astraglomanteia inscriptions of Asia Minor which used five knucklebones. That both originate from a single Greek source has been proven by a recent papyrus publication.[1] Thus, at some point it was understood that the three cubic dice produced the same number of combinations that five knucklebones do: the question is how early?

In this paper, I will explore these two types of dice games (and if time, other gambling games) first showing the way these games were played (e.g., three cubic dice and five knucklebones) and then posing the question of how aware Greeks were in the mathematical risks of gambling. Although mathematical calculations of probability and risk are usually said to begin with Cardano’s Liber de ludo aleae in the 16^th century, remarkable coincidences like the present one suggest a level of mathematical sophistication at the ancient dicing table as well. But how much? Were ancient gamblers aware, for example, just how low a chance there was in getting the famous “Euripides” throw as opposed to a “triple-six”? Were they aware of the uneven probabilities that arise from the unevenly-shaped knucklebone, and were they actively trying to “fix” these problems? Ultimately, how much risk were ancient gamblers able to calculate ahead of time before putting money on the line? To know the answer would add an interesting chapter to the history of probability.