In recreational mathematics, the McNugget numbers are an example of a set of numbers obtained by linear combination of a fixed set of other numbers, similar to the numbers involved in the coin problem. A McNugget number is the total number of McDonald's Chicken McNuggets in any number of boxes. The original boxes were of 6, 9, and 20 nuggets. Since the Happy Meal-sized nugget boxes of 4 can now be purchased separately, the modern McNugget numbers are linear combinations of 4, 6, 9, and 20. According to Schur's theorem, since 6, 9, and 20 are relatively prime (6=2*3, 9=3*3, and 20=2*2*5; therefore, their greatest common divisor is 1) any sufficiently large number can be expressed as a linear combination of these numbers. Therefore, there exists a largest non-McNugget number, and all numbers larger than it are McNugget numbers. For the modern version of the McNugget numbers, 4=2*2 and 9=3*3 suffice for the theorem to apply. The McNugget numbers are as follows:

Original McNugget numbers: all integers except 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, and 43.

Modern McNugget numbers: all integers except 1, 2, 3, 5, 7, and 11.

A related question is how many different sets of boxes can be used to make a given number. This is a question of combinatorics that can be solved using generating functions.

See also

• Coin problem

External links

• McNugget Number at Mathworld (Wolfram Research)

Integer sequences