How Many Dice Rolls Would It Take to Hit Your Favorite Kind of Number?

Abstract

Noga Alon and Yaakov Malinovsky recently studied the following game: you start at 0, and keep rolling a fair standard die, and add the outcomes until the sum happens to be prime. We generalize this in several ways, illustrating the power of symbolic, rather than merely numeric, computation. We conclude with polemics why the beautiful rigorous error estimate of Alon and Malinovsky is only of theoretical interest, explaining why we were content, in our numerous extensions, with non-rigorous, but practically-certain, estimates.