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Why is it called Chicken McNugget Theorem?

The Chicken McNugget Theorem is a playful name for the Frobenius coin problem when applied to the specific case of two coin denominations. The name derives from a real-world example involving the purchase of Chicken McNuggets at a fast-food restaurant.

Origin of the Name:

  1. Real-World Application: The name "Chicken McNugget Theorem" comes from the problem of determining the largest number of McNuggets that cannot be purchased with given box sizes of Chicken McNuggets. For example, if McNuggets are sold in boxes of 6 and 9, the theorem helps find the largest number of McNuggets that cannot be bought exactly with these sizes.

  2. Mathematical Context: The problem is a special case of the more general Frobenius coin problem in number theory. When applied to two relatively prime numbers (coin denominations), the Frobenius number (the largest value that cannot be expressed as a non-negative integer combination of these denominations) has a simple formula: \( g(a, b) = ab - a - b \).

  3. Popularization: The term "Chicken McNugget Theorem" was popularized by mathematicians and educators as a way to make the mathematical concept more relatable and engaging, particularly for students. The connection to a familiar and everyday item (McNuggets) makes the abstract problem more concrete and easier to understand.

Example:

If McNuggets are sold in boxes of 6 and 9, the largest number of McNuggets that cannot be purchased exactly using these box sizes is:

\[ g(6, 9) = 6 \times 9 - 6 - 9 = 54 - 6 - 9 = 39 \]

So, 39 is the largest number of McNuggets you cannot buy with boxes of 6 or 9.

The playful name helps to illustrate a serious mathematical concept in a fun and accessible way.