Numerically Balanced Dice

We discuss what it might mean for a die to be numerically balanced, explore connections with magic squares, and present integer programming models that can be used to design numerically balanced dice. Introduction On most commercially available twenty-sided dice (d20s), the numbers on opposite sides sum to 21. This is true for the d20 shown in Figure 1(a), manufactured by Chessex, which has the 8 opposite the 13, the 20 opposite the 1, the 10 opposite the 11, and so on. Why do die manufacturers do this? To answer this question, let’s perform a thought experiment. Suppose we get our hands on a perfectly fair d20, a regular icosahedron. We heat it up to make it soft, and then we carefully position it in a vice or a clamp, as in Figure 1(b), making sure that one of the faces lies flat against one jaw and that the opposite face lies flat against the other jaw. Then, with every turn of the crank, the jaws move closer and closer together, squeezing the two “jaw faces” closer and closer to each other, and making the once-fair d20 flatter and flatter, less and less of a regular icosahedron, and more and more like a coin. And with every turn of the crank, the d20 will become less fair. If we are careful with the squeezing, each jaw face will continue to be as likely to turn up as the other jaw face, but each jaw face will also become more likely to turn up than any of the 18 non-jaw faces. Still, if the numbers on opposite faces sum to 21, then the expected value of a single roll of the squashed d20 will remain 10.5, the value for a fair d20 (the average of the numbers 1 through 20). So the squashed d20 won’t systematically favor higher-than-average or lower-than-average numbers. Figure 1: (a) a d20 and its reflection in a mirror, (b) squeezing a d20. Our thought experiment reveals that the opposite numbers convention is a form of “numerical balancing” that serves to promote fairness, giving die manufacturers a way of hedging against the possibility that their d20s are not perfectly formed regular isocahedra. Are there additional measures that they could take? Is it possible to make d20s that are even more numerically balanced? Vertex Sums For each vertex of a d20, we can compute the vertex sum—the sum of the numbers on the five faces that meet at that vertex. If a d20’s numbers were uniformly spread amongst its twenty faces, then each vertex sum would be 52.5 = 5 × 10.5, five times the average of the numbers on the die. If all twelve of a d20’s vertex sums are as close to 52.5 as possible (i.e., all of its vertex sums are 52 or 53), we say that it has numerically balanced vertices. Having numerically balanced vertices could provide additional protection against certain manufacturing imperfections and irregularities. Let’s perform a second thought experiment: Suppose that the d20 shown in Figure 1 contains a large air bubble that lies just beneath the vertex formed by the faces numbered 1, 13, 5, 15, and 7. With the bubble-vertex side of the d20 being less dense than the rest of the die, its faces may be more likely to turn up than the others. If so, the die would tend to produce lower numbers than a fair d20 would. The reason why is that the bubble vertex’s vertex sum, 1 + 13 + 5 + 15 + 7 = 41, is considerably lower than the ideal vertex-sum value, 52.5. Face Sums Similarly, for each face of a d20, we can compute the face sum—the sum of the numbers on the three faces that are adjacent to that face. If a d20’s numbers were uniformly spread amongst its twenty faces, then each face sum would be 31.5, three times the average of the numbers on the die. If all twenty of a d20’s face sums are as close to 31.5 as possible (i.e., all of its face sums are 31 or 32), we say that it has numerically balanced faces. Having numerically balanced faces could provide even more protection against manufacturing defects. For a third thought experiment, suppose that an air bubble lies just below the face numbered 13 and suppose in addition that there is a small bump on the surface of the opposite face. These two defects may cause the numbers adjacent to the 13—the 1, the 11, and the 5—to turn up with greater frequency than the other numbers. If so, the die will once again favor lower numbers. Here, the reason is that the face sum of 1 + 11 + 5 = 17 is substantially lower than the ideal face-sum value, 31.5. Full Face Sums Alternatively, for each face of a d20, we could compute the full face sum—the sum of the numbers on the three faces adjacent to that face plus the number on that face itself. Although it may seem preferable— for aesthetic or geometric reasons—to use full face sums when defining numerically balanced faces, doing so turns out to be problematic. Consider a cube, and consider numbering it to form a d6. If we follow the opposite numbers convention, keeping the 6 opposite the 1, the 5 opposite the 2, and the 4 opposite the 3, then there are two possible numberings, and they are mirror images of each other. In each of these numberings, each face has the same face-sum value, 14, as the four adjacent faces will always contain two pairs of opposite faces whose numbers sum to 7. But in each of these numbering, each face does not have the same full-face-sum value. Instead, there are six different full-face-sum values: 15, 16, 17, 18, 19, and 20. (The 15 is the full-face sum for the 1, the 16 is the full-face sum for the 2, and so on.) Thus, for a d6, if we are committed to following the opposite numbers convention, there is no way of making all of its full face sums the same. This is also true for a d20. The methods we describe later in this paper can be used to show that if we number a d20 in accordance with the opposite numbers convention, there is no way of making all of its full face sums the same. Equatorial Bands If we select a vertex to serve as a d20’s north pole, then its opposite vertex must be its south pole. The five faces adjacent to the north pole form the northern polar cap, and the five adjacent to the south pole form the southern polar cap. The remaining ten faces—five pairs of opposite faces—form an equatorial band that encircles the d20. Figure 2 displays two views of one of the ten possible equatorial bands. Figure 2: Two views of an equatorial band. A net enables us to see all twenty faces of a d20 at once. In Figure 3, the top and bottom rows of the net are the two polar caps shown in Figure 2, and the middle row is the corresponding equatorial band. For each possible equatorial band, we can compute the band sum—the sum of the numbers on the faces that make up the band. But in this case we don’t need to do so. If a d20 obeys the opposite numbers convention, then each of its band sums must be 105, as each band is made of five pairs of opposite faces whose two numbers sum to 21. In other words, if a d20 obeys the opposite numbers convention, then it must have numerically balanced equatorial bands. And numerically balanced equatorial bands are desirable. Let’s perform one final thought experiment. Suppose that during the manufaturing process, one particular north-pole-south-pole pair is farther apart that it should be, making the d20 somewhat cigar shaped. If so, the faces on the equatorial band would turn up more often the others. But if each band sum is 105, then the expected value of a single roll will remain 10.5.