Math Games

A Fair 3-sided Die

Ed Pegg Jr., May 16, 2005

"THE CHANCE ELEMENT in thousands of indoor games is introduced by a variety of simple random-number generators. The most popular of such devices, ever since the time of ancient Egypt, have been cubical dice. Why cubical? Because of their symmetry, any of the five regular solids can be and have been used as gaming dice, but the cube has certain obvious advantages over the other four solids. It is the easiest to make, its six sides accommodate a set of numbers neither too large nor too small, and it rolls easily enough but not too easily." (Martin Gardner, Mathematical Magic Show)

The quote above is taken from Martin Gardner's Mathematical Games: The Entire Collection of his Scientific American Columns. This CD collection of all 15 books is fully searchable, and is sold by the very site you're visiting, The Mathematical Association of America. In his column on dice, he mentions that an icosahedron is inside the Magic 8-ball, and that dice date back to 2000 BC. You can see these via Arjan Verweij's gallery of ancient dice. In 2003, the auction house Christie's of New York offered a roman glass gaming die from the 2nd century AD.

Figure 1. An ancient icosahedron.

For more history, I will defer to the excellent Wikipedia article on dice. One question -- are the dice fair? Does each face occur an equal number of times?

The cube (d6) and icosahedron (d20) are two possibilities for fair dice. Most people are familiar with the platonic solids.

All of the platonic solids are isohedral, which means all the faces are the same, and that all faces have the same relationship with other faces. There are 25 isohedra, and 5 infinite classes of isohedra. There are also such things as spheres (d1), lenses (d2), and rolling logs (dn). Here's a chart of all of them.

Figure 2. All the isohedra.

Of these, only a handful have been manufactured. 5 and 7 sided dice can be seen in this picture, I'll get to those later. In addition to the below, Kevin Cook's collection contains manufactured examples of the rhombic dodecagon (d12), the isosceles tetrahedron (d4), and a variety of the 2n forms. He also has some prototypes and paper models. As an exercise, make 8 copies of any triangle, the more irregular the better (preferably drawn with a ruler). Now, assemble them to make a d8. For actual dice you can buy easily, I believe the following picture is a complete set.

Figure 3. All the manufactured isohedra.

But are they really fair dice? I talked with Lou Zocchi about how dice are manufactured. He photographed icosahedra made by various companies, stacked 10 high. If the dice were perfectly symmetrical, the heights would be identical -- but they were not. The slight irregularities allowed for a predictable bias, with the favored number depending on the manufacturer. He suggested I use a set of calipers on my own dice, which I did. The white d20 above ranges in height from .7 inches to .72 inches, depending on which face is up. It looks like a fair die, but it isn't. Some of the cubic dice I measured are similarly lopsided.

How can one measure fairness? Perhaps the center of gravity (the centroid) can be used. In the figure below, the edges of a cube have been projected onto a sphere, centered on the center of gravity of the cube. In the middle projection, the cube is fair, and all the resulting spherical polygons have equal area. In the last projection, the die is weighted to one side. The resulting spherical polygons have different areas, and suggest that the weighted side would most often be the bottom side.

Figure 4. The Geometric Model applied to a cube.

With this model, it is easy to calculate the heights of "fair" prisms and anti-prisms.

Figure 5. Heights of prisms fair under Geometric Model
unit sided n-gon
3 4 5 6 7 8 9 10 11 12
Prism Height .5336 1.000 1.5060 2.0598 2.6602 3.3049 3.9916 4.7181 5.4824 6.2828
Antiprism Height .8165 1.4953 2.2270 3.0244 3.8864 4.8100 5.7921 6.8297 7.9202 9.0613

What height should a coin be to have a 1/3 chance of landing on Sides? John von Neumann used the Geometric Model to solve this problem in his head, and provided an answer to three decimal places in half a minute. Can you match his feat?

I once put a lot of faith in this geometric model, until the I enthusiastically explained it to my professor. At the last minute, I'd built a model of an irregular heptahedra I'd done calculations for. My explanation went pretty well, for awhile. "The chance it will come to rest on this side is 10 percent," I concluded. To my horror, my model toppled over onto another side. I tried to make it stay on the one face, but it kept toppling. It wasn't a stable face. Later, I learned of unistable polyhedra that are stable on only one face.

Figure 6. A unistable polyhedron with the fewest known faces, and a cross-section.

I realized that any realistic model for dice would need to take into account the the amount of energy required to topple from one face to another. A nickel is 2mm thick, with a diameter of 21mm. Using trigonometry, the topple heights are .094mm and 21.095mm -- it doesn't take much energy to knock an nickel off edge.

The Energy State Model works with the concept that dice bounce. Depending on the type of die, and the surface, the die will bounce a different number of times. Before each bounce, the Geometric Model can be used to determine which face of the die has current influence. During each bounce, a matrix can be set up to determine the relative probabilities that other faces will inherit influence for the next bounce. Thus, a series of matrices can be used to model what is happening to a die. All of these matrices are different, because the amount of energy possessed by the die decreases geometrically with each bounce . In order for the model to work, some nth bounce must relate to an identity matrix, relating to a state where there isn’t enough energy left to shift the die from any face to any other face. Bouncing dice are somewhat similar to a Markov process.

Energy State Model: Let qjk = (radians swept from point under centroid on face j by edge k)/2π, b = bounciness, e = kinetic energy, n = bounce number, hjk = Topple height from side j to side k, f(x) = 0 if x≤0, 1 if x>1, else x.

Ajk = qjk (1 - f( hjk /ebn ))
Ajj = 1 - (sum of other entries in j column)
= Matrix of the A entries.

The initial state vector x can be determined by the Geometric model. The final probability distribution y is given by A6 A5 A4 A3 A2 A1 x = y. As n approaches infinity, An approaches the identity matrix. Usually, this happens before n reaches 10. With the red d6 cube above, material has been removed from the cube to make the pips, and the center of gravity was slightly skewed. My now fellow columnist Ivars Peterson wrote about my calculation in his column Unfair Dice. The poster Experimentally Obtained Statistics of Dice Rolls actually rolled the dice, for some interesting results.

Figure 7. Computed results for 10,000 tosses of a die with drilled dimples.

A nickel is 2mm thick, with a diameter of 21mm. Other values can be assigned arbitrarily: e (kinetic energy, or force of flip) = 200, b (bounciness or elasticity) = .2. The elasticity matters a lot. Flipping a nickel onto mud might have elasticity 0.01, whereas glass might be .4 With the setup here, a nickel lands on edge roughly 15 times out of every hundred million tosses. It's happened several times in recorded history. On December 1, 1707, composer Jeremiah Clarke mentioned an occurence in his suicide note (his coin flip to pick the suicide method landed edge first in the mud). In mud, a coin hardly bounces, so it will land on edge much more often.

Dan Murray did calculations for a coin landing on edge for Physical Review E. He later built a dice-rolling machine, and used it to test Vegas dice (purportedly manufactured to a tolerance of 1/5000th of an inch) and other shape. One object he tested was cylinders of various heights. As might be expected from the Energy State model, the coefficient of restitution (bounciness) has a large effect on behavior.

Figure 8. From Experiments with Cylindrical Dice.

A thick coin cannot be made perfectly fair as a 3-sided die. It depends too much on how you're tossing it, and the surface the coin lands on. If you want something fair, it has to be isohedral.


In 1975, Lou Zocchi started making nice dice.



I'd like to say again how excited I am to hear back from you on the topic of dice.

I have spent some time updating my webpage on the topic:

In particular, I have included the poster that I presented at
a conference in Germany back in 2001:


D. Christie, R. Glasheen, C. Hamilton, M. Imoto, P. Matthews, J. Moffat, T. Monajemi, D. B. Murray, J. Nelson and A. Sturm, "Experimentally Obtained Statistics of Dice Rolls," 6th Experimental Chaos Conference, July 22-26, 2001, Potsdam, Germany,

Kevin Cook, "What Shapes Do Dice have?"

Dice-play, "Crooked Dice,", "Mystery of the Magic Eight Ball Revealed,"

Martin Gardner,

H. Croft, K. Falconer, and R. K. Guy, Problem B12 in Unsolved Problems in Geometry New York: Springer-Verlag, p. 61, 1991. {0387975063}

Mitch Klink, "What Other Shapes of Dice are There?"

Dan Murray, "The Physics of Dice,"

Daniel B. Murray and Scott W. Teare, "Probability of a tossed coin landing on edge", Physical Review E, volume 48, p. 2547-2552, 1993.

Justin Smith, "Constructing a fair 3 sided coin,"

Arjan Verweij, "Gallery of Ancient Dice,"

Eric W. Weisstein. "Geometric Centroid, Isohedron, Spherical Polygon, Unistable Polyhedron." From MathWorld--A Wolfram Web Resource.

David Wells, The Penguin Dictionary of Curious and Interesting Geometry, London: Penguin, p.~265, 1991. {0140118136}

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Ed Pegg Jr. is the webmaster for He works at Wolfram Research, Inc. as an associate editor of MathWorld, and as administrator of the Mathematica Information Center.