An argument from continuity

Two sophomores approached my colleague Josh with a question: “How can we construct a fair 5-sided die?”

Josh posed a prior question: Is it even possible to construct such a die? He fashioned an interesting argument from continuity: Consider two square pyramids with the same height — one with a tiny base (much smaller than the lateral faces) and one with a very large base (much larger than the lateral faces). Clearly the first has a very small chance of landing on its base and the second has a very large chance of doing so. By gradually moving from the one to the other, there must be an intermediate point that makes the pyramid a fair die.

Josh used this teachable moment to explain the difference between an existence proof and a constructive proof. We now know that a fair 5-sided die is possible, but we have no idea how to construct one!

Categories: Math, Teaching & Learning