## February 01, 2006

### The Chicken Nugget Problem

Here's a question: McDonald's sells Chicken McNuggets in quantities of 6, 9, and 20. If you want a specific number of nuggets you can usually order it...for example if you want 35 you can order a 6 pack and a 9 pack and a 20 pack, if you want 63 you can order 7 9-packs, etc. But you can't order some numbers: there's no way to construct 25 as a combination of 6, 9, and 20. So the question is, what is the largest number of nuggets that you can't order?

OK, so that's just a basic math brainteaser. It's been discussed here and here and here and here and here (which points out you can now get 4 McNuggets in a Happy Meal, which throws everything into a cocked hat). It's even on Wikipedia claiming that "McNugget numbers" is a general term for sets of numbers made up of combinations of a fixed set of other numbers (the search for references is complicated by the existence of the Denver Nuggets basketball team, who have an annoying habit of employing 6 foot 9 basketball players who score 20 points in 43 minutes of playing time).

Now, where did this problem originate? Well, check out out this issue of Focus, a publication of the Mathematical Association of America. Scroll down to page 23 (the 22nd page of the PDF) and look at the letter to the editor from Michael Barr, a mathematician at McGill University who also happens to be my father.

That's me with the summer job at McGill 20 years ago, walking into work with my father. The story he relates did happen; I remember walking in to McGill and discussing this problem as we crossed the Rockland overpass between Mount Royal and Outremont (or maybe we cut through the fence and walked across the train tracks). We certainly thought it up on our own, but what's not clear is whether someone else had independently come up with the same problem earlier. McNuggets were introduced in 1983 and I graduated from high school in 1984, and in-between I managed to develop quite the McNuggets habit. My friend Val and I would talk about "splitting 20" -- meaning ordering a box of 20 McNuggets and then each eating half (one time we split 20 and then about an hour later went back and split 20 more -- the mind boggles). The summer job he refers to was indeed in 1986 so that would have been 3 years for someone else to spend enough time in a McNuggets-induced haze to come up with the problem. It's possible, I suppose, but like my father I choose to believe we were the first.

Posted by AdamBa at February 1, 2006 08:40 PM

http://proudlyserving.com/cgi-bin/mt-tb.cgi/395

yeah, thing is, now the 9 is gone and is replaced by 10 piece and if you want the 6 you have to order the happy meal :)

Posted by: Steve at February 2, 2006 05:36 AM

Equally taxing is figuring out what percentage of a McNugget is actually real chicken :)

Posted by: anonymouse at February 2, 2006 10:52 PM

"you want the 6 you have to order the happy meal"

That may explain America's obesity problem -- here in the UK the Happy Meal serving is 4 nuggets... :)

Posted by: Mat Hall at February 7, 2006 07:27 AM