McNugget numbers

McNugget numbers – The Question

This post is about McNugget numbers and a small mental math exercise I gave my self this week while driving on the high way.

McNuggetsTo start with the beginning of the story. I was driving for quite a while this Friday – actually for the better part of 4 hours. Which I in many ways consider a waste of time and energy, but I had to for several reasons. At one point I stopped at McDonald’s to get a cup of coffee and stretch my legs.

While waiting in the line I was watching the prices for the many delicious offerings…. And I saw they had McDonald’s Chicken McNuggets in the four quantities 4,6,9 and 20. That’s when I realised that I had bumped into the curious little phenomenon called McNugget numbers at some point and decided I wanted to think this through my self.

A McNugget number is a number which cannot be made up from ordering McNuggets in the available quantities, 1,2 and 3 are obviously McNugget numbers since the least you can order is four.

I asked my self what are the McNugget numbers. All of them…

What is the largest number, and how can I be sure that there are no larger number than what I have already found. Or if you take a “brute force” approach to checking McNugget numbers, what is the stopping criteria.

The question is relatively easy to answer and the highest number is rather low after they added theĀ  4-piece McNugget box. What happens if you remove that, so the question is what McNugget numbers are there based on 6,9 and 20 McNugget boxes.

It is in reality a stupid little question, but it kept me entertained for some minutes of boring highway. I think what I really wanted to say, is that there is math every where and you can pose these small questions for your self to improve your mathematical thinking and to keep your self entertained. I know google answers the question for you in a matter of seconds, but try to make it a mental exercise, or using pen and paper if you like.

And let me hear you comments, solutions questions and so on. I will post the answer on Sunday June 26.

Posted by Kristian


I created a table 1..35 on a piece of paper and marked 6, 9 and 20 as McNugget numbers. Then I went through all the McNugget numbers n and marked n + 6, n + 9 and n + 20 as McNugget numbers (basic sieving). Once I filled the table I tried finding patterns and did a lot of thinking. After some time and very little progress I made a program that did the sieving for me and found the largest non-McNugget number. Only if I had made the original table a bit larger…

But I’m looking forward to your solution. There must be some easy way to do this.

And by the way, this was a lot of fun. If you ever have any problems like this, please post them! I’m also going to try to come up with my own problems and then try to solve them. That’s probably a good way to train my math skills.

I don’t think there is a particular easy way of doing this. But thinking about it leads to some quite interesting thoughts. Reading about it later on will also open up for some rather interesting results within number theory and combinatorics.

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