The Fascinating Mathematics of Chicken McNuggets

Humphrey Huang

According to Google, a chicken nugget is a “fast food product consisting of a small piece of deboned chicken meat that is breaded or battered, then deep-fried or baked”. Its golden brown crispy coating provides an enjoyable texture, while the soft chicken interior creates a satisfying bite. This delicious combination has allowed chicken nuggets to be almost universally loved, and around 2.3 billion servings are consumed yearly in the US alone. Chicken nuggets are extremely popular in fast food restaurants, as seen with the rise of the McDonald’s Chicken McNugget. Surprisingly, the Chicken McNugget has an unexpected connection to an interesting mathematical concept.

Many stories surround the origin of the Chicken McNugget theorem, the most popular coming from McDonald’s famous nugget. Originally, McDonald’s only had nuggets in packs of 9 and 20. At the time, math enthusiasts were interested in calculating the highest amount of nuggets a person could not buy with these packs. For example, a person could not order 8 nuggets, as no combination of packs of 9 and 20 could yield that number. But exactly how did the mathematicians calculate the maximum number of chicken nuggets someone could not order?

Suppose there are two relatively prime positive integers, m and n (relatively prime means the m and n don’t have any common factors except for 1). Given that a and b are nonnegative integers, combinations of m and n can be written in the following expression:

In the case of the McDonald’s chicken nugget problem, a and b are represented with the number of packs of 9 nuggets and 20 nuggets someone buys, and m and n are represented by the packs of 9 nuggets and 20 nuggets themselves. For instance, someone could order 29 chicken nuggets by getting 1 pack of 9 nuggets and 1 pack of 20 nuggets. The Chicken McNugget theorem evaluates the greatest integer that cannot be written in the form of the expression above. Through number theory, the following expression is yielded:

Using the Chicken McNugget theorem, the McDonald’s chicken nugget problem can be solved. In this case, the nuggets in packs of 9 and 20 represent the integers m and n, which can be plugged into the expression above. Since chicken nuggets are in packs of 9 and 20, the greatest number of nuggets one cannot order is 151 nuggets!

Indeed, there is a mathematical theorem for almost everything, including a crispy golden chicken nugget.