Detecting Devious Dice

by MXSavant 05/13/05

Readers may recall my earlier article that dealt with loaded or gaffed dice (see Unfair Dice: Loads, Gaffes, Sliders and Other Polyhedral Perils), in which I went into some detail about all the many ways there are to make dice non-random number generators. This week, I’m going to talk about the ways one can spot “cheater dice” to make sure your gaming group or tournament is above reproach.

Now let me concede something up front; I don’t think that the use of unfair dice is exactly endemic to the gaming community and that will continue to be the case, in my opinion, until or unless wagering becomes involved. However, there is enough of a market for unfair dice that companies manufacture and sell them for players who aren’t able or disposed to make their own. (Irony alert: some of the same companies who make unfair dice also sell dice in quantity for use in tournaments) So gaffed dice and the people who use them are out there. Before we begin, I encourage anyone who has not yet read my above-mentioned article to do so before proceeding further. What follows will make a lot more sense.

If you suspect a die is crooked, take a close look at it.1 Most wargaming and RPG cheater dice available commercially are mispots. So start by looking at each face and make sure all the numbers are there.

Most dice that are unbalanced will reveal their nature if you drop them in a glass of water. Loaded dice will turn over as they sink and come to rest with the loaded side down. Floaters, or dice that lighten one side will usually float, hence their name. A balanced die will sink without turning. The taller the water column through which the die sinks, the more reliable the test.

To detect bevels, hold the suspect die against a square one and see if they wobble. Be sure to try all sides. If you hold them up to the light, look for a hairline of light. You can also spot suction dice in this way.

Capped dice can be identified by scratching all sides with a fingernail to see if it leaves a mark. A capped side or edge will more likely show a mark. A capped die will also feel slightly tacky when it’s rubbed against a regular die.

Edge work dice can usually be found by inspecting the edges and looking for rough spots, irregularities shaving or sanding.

The best way to detect flats is with a precision caliper or micrometer to see if any material has been removed. Flats that are more crudely made can be detected by placing them against an honest die of the same (alleged) dimensions. Slick dice can also be found this way, or just by carefully inspecting each side.

An Experiment

I decided to test the claim that “drug store dice” suffer from irregularities based on their design and manufacture. A D6 with recessed pips will not be perfectly balanced because the mass that is removed to make the pips on the six face makes it lighter on that side. At the same time, the opposite one face has much less mass removed, so it is heavier. The question is whether these irregularities are enough to bias a die.

I pulled a D6 at random from my dice bag. I had purchased this one as a pack of five from a grocery store many years ago. It measures 5/8” on a side.

The results of rolling this D6 180 times gave me the following results:
Number of outcomes:

Number of outcomes:
32
22
24
27
28
47
Outcome:
1
2
3
4
5
6

Here is a graph of the results:

MX009-01

Results of 180 rolls of one “drug store” D6.

The main thing that caught my eye was the large number of sixes rolled. The question is, does this reflect a real bias or not? To answer that question, one must turn to statistics.

There is a kind of statistical distribution called the Poisson Distribution that is a good tool for the kinds of problems represented by this question.2 What we want to find out is whether or not these data show the number of sixes rolled to be radically divergent from what we expect.

Warning: The following discussion carries mild risk of exposure to mathematical and statistical jargon.

In order to find out how far afield these results are, we need to find the Standard Deviation (or s) for this data set. Put simply, the Standard Deviation is a way of quantifying how far something is from the mean. In this case, the mean number of sixes would be 30, since in 180 throws, there is (theoretically) a one in six chance of scoring a six. To calculate s for a Poisson Distribution, we can use the following formula:

MX009-02

Where n is the number of trials (180), and p is the probability for any one outcome, in this case 1/6. The result is 5.

Now, let’s subtract the number of sixes we got (47) from the number of sixes we expected to get (30). The result is 17. This is called the displacement. If you divide the displacement by the Standard Deviation, you get 3.4, which is the number of standard deviations (or “sigmas”) our results were away from the mean.

Is that significant? Probably. A physicist friend of mine with whom I consulted on this question said that 3.4 sigmas is enough to notice but will not convince everyone that something is going on. In order to really settle the question, I would need to run my 180 trials at least another thirty times, preferably many more. But even this one trial is suggestive that there is in fact a bias on this die.

Now, if it bothers you that your recessed pip D6s might be slightly biased, you can use dice with imprinted numerals instead of pips, which will probably reduce the mass difference to a negligible amount. Another solution is to buy used casino dice, which have printed spots and are so carefully machined and balanced that they are quite good random number generators. There are a number of web sites that sell them. They will cost a little more, but they are extremely good dice.

A cheaper alternative might be to use a dice cup. I repeated my experiment, but this time I put the die inside a glass jar with the lid on. I “rolled” the die by shaking the jar and making sure the die bounced off the lid at least once. The distribution that resulted looked a lot more normal normal and the number of sixes, while still higher than expected, were only 1.6 sigmas off the mean. That will probably be good enough for most players.

Notes:

  1. See Chapter 13 of A. D. Livingston, Dealing with Cheats. Illustrated Methods of Cardsharps, Dice Hustlers and Other Gambling Swindlers. J. B. Lippencott Co., 1973.
  2. I wish to thank my good friend Dr. Shawn Carlson, Executive Director of the Society for Amateur Scientists, for his assistance in this subject. Any errors made in this section are mine alone.