Factoring in the Chicken McNugget monoid

TL;DR

This paper explores the Chicken McNugget problem, analyzing which quantities can be purchased using specific pack sizes, and connects it to broader questions in the theory of non-unique factorization.

Contribution

It provides an accessible introduction to the McNugget problem and extends the discussion to related questions in non-unique factorization theory.

Findings

Identifies which numbers of Chicken McNuggets can be formed with packs of 6, 9, and 20.

Connects the problem to broader mathematical questions in non-unique factorization.

Offers insights into the structure of the McNugget monoid.

Abstract

Every day, 34 million Chicken McNuggets are sold worldwide. At most McDonalds locations in the United States today, Chicken McNuggets are sold in packs of 4, 6, 10, 20, 40, and 50 pieces. However, shortly after their introduction in 1979 they were sold in packs of 6, 9, and 20. The use of these latter three numbers spawned the so-called Chicken McNugget problem, which asks: "what numbers of Chicken McNuggets can be ordered using only packs with 6, 9, or 20 pieces?" In this paper, we present an accessible introduction to this problem, as well as several related questions whose motivation comes from the theory of non-unique factorization.