McNuggets were originally served in packs of 6, 9 or 20 pieces. Having lunch with his son at McDonald’s in the 1980s, mathematician Henri Picciotto wondered what numbers of Chicken McNuggets could not be purchased with a combination of these three packs. His list contained the numbers 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, and 43.
All other numbers of nuggets could be obtained and would be known, from that day on, as “McNugget numbers.” In other words, a “McNumber” is defined as the number of nuggets that can be purchased by combining 6, 9, and 20 pieces. For example, 44 is a McNumber because 44 nuggets can be obtained by purchasing four packs of 6 and one pack of 20. So is 45: just order five packs of 9. 46 nuggets can be obtained by purchasing one pack of 6 and two packs of 20, 47 with three packs of 9 and one pack of 20, 48 with eight packs of 6, 49 with one pack of 9 and two packs of 20, and so on.
The largest non-McNumber number is 43, since it is impossible to solve the Diophantine equation 43=6x+9y+20z (with x, y, and z being natural numbers), as can be verified (by combining the three packs of Chicken McNuggets in every possible way, you will never get 43 nuggets!). The largest number that cannot be obtained with multiples of a given set of positive integers is called the “Frobenius number”; so 43 was the Frobenius number for Chicken McNuggets.
I said “was” because, unfortunately, since McDonald’s started selling 4-packs of chicken nuggets, the Frobenius number has dropped sharply to 11.
https://extremelyinterestingfacts.quora.com/Have-you-ever-tried-to-order-43-Chicken-McNuggets-from-McDonald-s
