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by Dean Malmgren, Bo Peng, and Vlad Seghete

We recently taught a data literacy training class where we asked participants to roll 6-sided dice that were peculiarly shaped: some were elongated and rod-like, some were stumpy, some basically looked like square shaped coins and a few were perfect cubes. The course was fun and people learned a lot, but we had an ulterior motive: we were curious to understand how the shape of these elongated dice affected the odds of particular outcomes.

We made the elongated dice from wooden rods in which the cross section was square, but we cut the rod at differing lengths. We stuck with the same labelling convention of regular dice: that opposite faces sum to 7, but we made sure to label the two square faces with a 1 or a 6 like this:

Note that all of these dice have different heights but are otherwise shaped very similarly. Since they are all cut from the same rod, one face was always a 1” square and the main difference was in aspect ratio—the ratio of the height to the length of one the edges on the square faces:

Since we were already collecting the data for educational reasons, several of the participants in the dice rolling game were kind enough to share their spreadsheets with us after the fact. We then aggregated all of the data in a separate spreadsheet to investigate how the aspect ratio of the dice is related to the likelihood of different outcomes. Specifically, what is the probability of a 1 or a 6—a square side—being rolled, given the aspect ratio of that die?

In this plot, we are showing the probability of rolling a 1 or a 6 as a function of the aspect ratio of the die. The squares denote our best estimate of the probability of rolling a 1 or a 6 and the error bars denote our uncertainty arising from sampling error.

If each outcome (1, 2, 3, 4, 5, and 6) were equally likely, the probability of rolling a 1 or a 6 would be exactly 1/3 (dashed grey line). This is exactly what we observe when the die has an aspect ratio of 1—meaning that all of the sides have the same dimension—which is exactly what you would theoretically expect.

For dice with a small aspect ratio (dice that are basically square coins), we would theoretically expect them to always land with a 1 or a 6. After all, how many times have you ever flipped a penny and seen anything other than heads or tails happen?

For dice with a large aspect ratio (dice that are basically square rods), we would expect them to basically never land with a 1 or a 6. Afterall, how many times have you ever dropped a pin and seen it not land on its side?

In these extreme circumstances, it’s relatively easy to reason out exactly what should happen. But what if you’re given a die with an aspect ratio of 1.2 or 0.53? Or, more generally, how does the probability of landing with a 1 or a 6 change as a function of the aspect ratio? Empirically this curve looks quite smooth and it definitely feels like something that can be theoretically solved (we got nerd sniped and tried a few quick things but we unfortunately didn’t get very far).

Without knowing the functional form though, it’s pretty clear that there are two outliers (blue squares and error bars) and, because we were in the room with the people that were rolling the dice, we know the origin of the differences. The outliers have nothing to do with faulty dice or anything like that; these dice were outliers because the individuals rolling them were deliberately rolling them in non-traditional ways. Although it’s surprising how ineffective the strategy was, Die O was “rolled” by dropping it such that it would likely land with a 1 or a 6 every time. Die P, on the other hand, was “rolled” by rotating it rapidly end-over-end so that it had a lot of angular momentum to encourage it to land with a 2, 3, 4 or 5 every time.

The fact that the rolling mechanics can play such a big role (yuk yuk) in the outcomes leads us to believe that being able to theoretically predict the odds on an elongated die may be quite difficult, indeed. Of course, if you have any ideas for how to do this, please fork our gist!