On McNugget math (and being less helpful)

Another great Pi Club (still a working title) meeting.

Today, we worked on the McNugget problem (a case of the Frobenius/coin problem), which goes something like this:

Chicken McNuggets come in packages of 6, 9, and 20. Assuming money is not a factor and that you can only buy full packages, what is the largest number of Chicken McNuggets that you cannot buy?


(Of course, I brought McNuggets for the occasion. You know, as manipulatives…)

This was the hardest problem I have posed to these students so far, and when they weren’t getting very far after about 25 minutes, I was starting to feel nervous. This problem is too hard, I was thinking. Should I step in? They’ll never come back if I just torture them like this!


And yet – after 20, 25, 30 minutes – they never asked for help, never asked for a hint. Not even once. Now that‘s resilience.


I have to admit, it’s often hard for me to not be helpful. Patience is a virtue that I’m still working on… and therefore I know how hard it can be to not know, to not have the answer. Seeing my students struggle with this can be just as heart wrenching.

I did throw them a small lifeline after some time, when it really seemed like they didn’t know which road to take – or if there even was a road. And then… I watched and waited. And waited.


I thought back to SUM 2015, when Dr. Ruth Parker gave us all a neat little problem to solve – and then, even when she moved on with her talk, didn’t give us the answer. She said: “I know how satisfying it is to solve a problem – and the harder the problem, the more satisfied you feel. So I won’t take that away from you.”

And that’s what eventually eased my mind: realizing that, by being too “helpful”, I wouldn’t be helpful at all – in fact, I would be stealing something essential and wonderful away from the students. Worse, I would send the message that I didn’t think they were capable of solving it.

Which, of course, they were. And they did.

And man, it felt so good.


In the spirit of being less helpful, I won’t explain their solution. However, you could work out the answer from the previous picture.

Once the students solved this problem, I asked how it would change if you could also buy groups of 4 McNuggets (which is where the blue numbers come from in their list), as well as why the problem wouldn’t be as interesting if McDonald’s only sold packages of 4, 6, 10, and 20. Which, as I found out today, is the case in Canada – how mathematically unfortunate!

P.S. Watch a couple of the Numberphile guys try to order the highest non-McNugget number of chicken McNuggets here. Spoiler alert!

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