8 - How Deep Is Your Playbook?

Summary

Abstract

An American football season lasts a mere sixteen games played over seventeen weeks. This compact season leads to, fairly or unfairly, intense scrutiny of every player's performance and each coach's decision. Our goal in this paper is to determine a measure of complexity for the decision of choosing a defensive alignment on any given down. There are only four standard defensive formations, defined generally by the personnel on the field, but how a coach physically situates the players on the field can emphasize widely different defensive strengths and weaknesses. To describe the number of ways a coach achieves this goal, we utilize the notion of equivalence classes from abstract algebra to define classifications of defensive formations. Enumerative combinatorics is then necessary to count the number of fundamentally different defensive alignments through the application of binomial coefficients. Descriptions of the rules for the game and diagrams of different defensive alignments make this paper accessible to even the novice, or non-fan.

Introduction

Many people have fond memories of participating in team sports in elementary school, high school, and even college. We recall the thrill of a close game and the joy or sadness after a win or loss. As we age, less of our time is devoted to playing sports as other responsibilities take priority and our bodies simply are not capable of competing at the same level. Our role in sport transforms from being an active participant to being a spectator.