The Intermath problem asks a few questions about the number of McNuggets we can buy if Macdonald's sells its McNuggets in boxes of sizes 6, 9 and 20.

I'm going to start by looking at an easier problem
- suppose McDonalds sells McNuggets in only two sizes.

Here are some questions similar to the Intermath
questions, and also some different ones:

Suppose I know the sizes of the two boxes.

Are there only a finite number of sizes I cannot get, or are there infinitely many sizes I can't get? How do I tell the difference between these two cases?

Is there a formula for describing exactly which
number of sizes you cannot order?

Is there a formula for describing exactly how
many different sizes you cannot order?

Is there a formula for describing exactly the
biggest number that I cannot order?

If I give you a number, is there a way
of describing how many different ways (0,1,2,3, etc.) I can order that
number of McNuggets?

Are there any clever ways to approach any of the above questions?

If I know the answers to all of the above questions for two sizes of McNuggets boxes, can I use this information to get the answers for 3 sizes?

Here is a spreadsheet that gives some idea of what is going on!!

Here is a link to a working spreadsheet (I hopeP.

working spreadsheet