Last month’s puzzle was about the probabilities of various outcomes of the throw of two dice. The dice we used were the standard cubes. But dice can be made in many differenct shapes. For example, we could make dice out of octahedrons.

An octahedron has eight faces, each an equilateral triangle. As in a cube, the faces of the octahedron are all the same regular polygon. The angles at which these faces meet at the corners and edges are everywhere the same. Because of this regularity, it is usable as a die since all faces have an equal probability of coming up on top.

The octahedron is one of the five polyhedrons, known as Platonic Solids, that have this regularity. See Truncating An Icosahedron for more information about them.

With a pair of cubical dice we found that the number most likely to come up was 7 because there are more combinations that equal 7 than equal any other number. In the table below, the roll of one die is shown in red across the top. The roll of the second die is in blue down the left side. The table shows all the possible combinations that can come up when you roll two dice. Six of those combinations equal 7. So the probability of rolling a 7 is 6/36 or 1/6.

What is the most common outcome when you roll two octahedral dice?

After you’ve thought about it check our solution.

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