Would you like to play a game of chance? It’s very simple: we’re just going to roll a dice. We’ll even weigh it in your favor: we win if the dice rolls a one or a six, and you win if anything else comes up.
Here’s the catch: the dice we’re using looks like this.
Chances are, you saw that and said “no thanks, that’s obviously unfair,” and you’d be right: dice with one side that’s even just 5 percent longer than the others have been shown to land on the larger faces on more than half of throws.
But if not – if you said something like “certe, quod bene videtur,” for example – you might just be an Ancient Roman.
That’s because almost all the dice archaeologists have found from that era – a full four out of every five examples – are visibly asymmetric.
“[It] stand[s] out to us today because in western culture six-sided dice are highly symmetrical, and because we expect the die to roll ‘fairly,’ wherein each side has equal probability of being rolled,” explains a recent paper, published in the journal Archaeological and Anthropological Sciences, which sets out to finally answer the question that’s puzzled historians for decades: why?
The traditional answer is perhaps the most intuitive one: they were all dirty cheats. There’s a couple of things supporting this idea, not least of which is the fact that it just kind of makes sense. We’re all used to the concept of probability – if you want to skew a dice, making one dimension longer or shorter is an easy way to do it.
On top of that, there’s the fact that these rectangular dice seem to have one and six on the larger side noticeably more than would be expected by random chance alone – hinting that perhaps these dice were specially designed to score high or low more than any other value.
But through experimental research, the study authors figured out a much more innocent reason for that tendency: “We conducted an experiment with naïve die producers” – that is to say, a couple dozen college students – “to see how they configure pips on blank cubes with different degrees of asymmetry," the paper notes.
“If there is a strong tendency to configure pips in a certain way that is different from what we see in the archaeological record then the configuration on the archaeological dice is more likely to be intentional,” it explains. “However, if naïve die-producers show a strong production preference that is the same as the pattern in archaeological finds, then the predisposition may be explained by a production bias.”
And it turned out nearly all the dice set up in the classic “Sevens” figuration – the system favored by both the Ancient Romans and our modern dice, where opposite sides add up to seven – ended up with six and one on the two largest sides.
Maybe it was because the students started numbering on the larger faces; maybe it was because they felt that six, as the highest number, needed most space. Maybe it just “felt right” to them for some reason – whatever was behind this tendency, it doesn’t matter as much as what it tells us about Roman gamblers: that the higher frequencies of sixes and ones on larger faces isn’t a sign that they were cheating.
Besides, there’s a bigger problem with the cheating hypothesis – and you already figured it out yourself at the beginning of this article.
“If die asymmetry was the result of attempts to alter probability outcomes of certain rolls, without being noticeably and visibly asymmetrical, we would expect greater percentages of dice at or close to the 5 percent visibility threshold,” the authors point out.
“Instead, nearly half the asymmetrical dice are obviously lopsided, being 20 percent or more larger along one axis.”
So, if these dice were created specifically for cheating, they wouldn’t be very good at their job: nobody would play with them. There must have been another reason – were the dice created with specific games in mind, perhaps? Ones which required uneven probabilities?
Well, again, this theory runs into some issues. “We reasoned that if asymmetrical dice are a distinctive ‘type’, with a unique set of shapes set apart from other more cubic shapes, that their production and use was probably specialized,” the authors write.
But when the researchers analyzed a sample of 28 Roman-period dice, found across 13 sites in the Netherlands – an area chosen because, at the time, it was split between the Roman-ruled south and the Germanic Frisii-controlled north – they found no such pattern. Instead, dice asymmetry varied along a continuum, with no particular ratios or shapes favored over any other.
So: another explanation for the Romans’ wonky dice quashed. Maybe the answer was something simpler. Something you have to get into the head of a real Roman to understand.
See, the thing about Romans was, they didn’t know about probability – but they did know about Gods. “Romans probably did not think that die shape mattered, because even with a non-cubic die all sides can still be thrown,” Jelmer Eerkens, one of the two study authors, told Haaretz.
“Today we would say that, yes, each side can be thrown but with unequal probabilities – however, most people in Roman times probably would not understand that way of thinking.”
Instead, the paper explains, Roman rollers would have believed the outcome of the dice was up to fate, or the will of the gods – dice could even be used as a way to commune with the gods, or as a way of soothsaying, and if Fortuna wanted you to roll a six, or a three, or any other number, that’s what would come up.
“This does not mean that all individual Romans were oblivious to rolling frequencies,” the paper points out. “For example… Cicero’s writings indicate that he was frustrated by the more general Roman worldview that gods controlled all manner of human experience. He specifically uses die-rolling as an example to bring up the subject of chance and an albeit limited notion of probability, questioning whether dice rolls were always determined by the gods.”
But “the writings of Cicero are unlikely to have reached a wide audience,” they add. There may have been a few especially seasoned gamblers out there who realized that asymmetry affects the roll of a dice – but for most Romans, as long as you could throw it, it was “fair.” Or, at least, as fair as anything controlled by the notoriously capricious Roman pantheon.
“We can connect to these ancient cultures because we still use dice and recognize them,” Eerkens told Haaretz.
”But we can also see how thinking and understanding about the world has changed through little hints from these ancient objects.”