Triangular Numbers and the N-Sided Die Optimal Stopping Problem
Bennett Eisenberg · Mathematics Magazine · 2021
Summary–A fair die with sides numbered from one to N is rolled. You can either receive the value shown or else pay one and roll again. If you do not like the second roll, you can pay another one and roll again. This can be repeated as many times as you wish. The questions are, “When should you roll again?” and “What is the expected return from playing this game?” This is the N-sided die optimal stopping game. It has become a popular problem on the internet in the case N = 100 with several solutions given. There is a weakness in the solutions, however, in that they use numerical approximations. In this paper we find exact formulas for the solution for arbitrary N and show that the stopping rules and expected returns form an interesting pattern as N increases that is related to the sequence of triangular numbers.