The Problem With Baseball Hats

Abstract

A young baseball player stacks n baseball hats by each door to his home. Every time he leaves the house to go to practice, he grabs a hat from the stack by the door he exits; then when he returns to his home after practice, he leaves his hat on the stack by the door he enters. In our problem we consider how many times, on average, the young baseball player will go out to practice and back into his house before his stack of baseball hats by the door he exits runs out. We will begin with an examination of two doors that start out with n hats by each to determine a formula that calculates how many cycles the boy will run through before he goes to grab a hat as he leaves the house, but instead finds an empty stack. We will then broaden our focus as we start to consider the implications other such nuances and alterations might bring to the problem, looking at what happens when we evaluate the variance surrounding our average (that is, the greatest and least number of times before he runs out of hats), add additional doors, introduce the very likely probability that the boy could lose a hat (or come back with extra hats!), and so forth. Thus we will see the problem with baseball hats as we seek to find the solution—though perhaps not quite the solution an actual baseball player (or his mom) might be looking for.