Solution of an Odds Inversion Problem

Abstract

–“From a bag containing red and blue balls, two are removed at random. The chances are 50-50 that they will differ in color. What were the possible numbers of balls initially in the bag?” This problem appeared in the National Museum of Mathematics’ Varsity Math puzzle, week 117. It is quite easy to solve, but what if we generalize to arbitrary odds? In this article, we characterize the solutions of the general case. We show that for most odds values that are at most 50-50, there is an infinite number of solutions, while for a certain well-defined class of odds below 50-50 and for any odds greater than 50-50, the number of solutions is zero or finite. We also explore some other interesting and surprising properties of this problem.