HexaCube Math

HexaCube d4s are, like all HexaCube dice, chamfered cubes, a shape which also goes by the impressive name of tetratruncated rhombic dodecahedrons. A chamfered cube is a cube with the edges sliced off to make 12 new faces. This means that a chamfered cube has 18 sides. Six of the sides are square faces. (They are smaller versions of the original cube’s sides). The faces that have replaced the cube’s edges are 12 hexagons. The hexagons can have all the same lengths, or be irregular. HexaCubes use irregular hexagons, with two sides longer than the rest. However, each of the 12 hexagons on a die are exactly the same. To avoid confusion, I refer to these irregular hexagons as “hex faces.”

A HexaCube d4

The key to understanding the theory behind HexaCubes lies in two concepts. The first is that the relative size of the two kinds of faces can be adjusted so that it is more, or less, likely that the die lands on a particular kind of face. For example, if the square faces are made larger in relation to the hex faces, it will become more likely that a rolled die will land with a square face up. The second concept is that manipulating these probabilities can be used to make fair dice that have a different amount of sides than outcomes.

A square result is more likely for the shape on the left

To illustrate this second point, consider the d4. Using conventional methods, it is impossible to make a 4-outcome 18-sided die. Each value of 1 through 4 can be placed on 4 sides, but there will be two sides left. These sides would either have to be blank sides that would have to be re-rolled or have some other method of picking 1 through 4. Either way, these two sides would not be one of the 4 outcomes. One way of looking at the problem is to note that 18 cannot be divided by 4 evenly.

Now, let’s say that size of the squares is adjusted, relative to the hexes, so that the squares land twice as often as the hexes. This means that if I had a square marked “1” and a hex marked “2,” the “1” outcome would occur twice as much as the “2” outcome. In order to balance the two outcomes, I could place a second “2” on another hex. Therefore, you can say that a square outcome is equal to two hex outcomes. If you count the 6 square faces twice and the 12 hex faces once, the number you come up with is “24.” Four goes into 24 evenly. if you examine the d4 illustration above and count the quantity of each outcome, counting the square faces twice, you will note that each outcome adds up to 6. This is because 24 divided by 4 is 6.

And the Rest is Just Details…

Once the d4 is understood, the other dice are just variations on the idea. The HexaCube d6, d8 and d12 are exactly the same proportions as the d4. This is because these numbers multiply into 24 evenly. Ten is not a factor of 24, so you will note that the d10 and the ‘tens’ percentile die have slightly larger square faces in relation to their hex faces. These square faces come up to 3 times as often as the hexes. This means that numbers that are a factor of 30 work on this type of die (3 x 6 square sides = 18 + 12 hex sides = 30). The binary die is most often paired with the d10 for d20 rolls, so its proportions match its mate. The binary die’s two outcomes would work on either type of die.

There is a little more information on how the numbers were arranged on the dice to maximize their performance. That information is at the end of Are HexaCubes Fair? section.