Question

Problem set 2, problem 2 - I have no problem proving that "If it is possible to buy X, X+1..." but I am completely at a loss to explain why (as P2 asks). I'm even just looking for a nudge in the right direction here...

Answer 1

Do you mean that you can prove it (mathematically) but you can't explain it in english ?
Basically, it is just a proof by recursion. I would explain it this way in english:

Say I want to buy an arbitrary number x of nuggets and i know that it is possible to buy x, x+1,..., x+5 nuggets.
I can always buy c boxes of 20 (c >=0) plus n others (0 < n < 20). Now I just need to demonstrate that can I buy n nuggets
I can buy x + 6, x + 11 just by adding a box of 6.
I can buy x + 12, ... x + 14 by adding a box of 9.
I can buy x + 15, ..., x + 19 by adding a box of 6 and a box of 9.
I can buy x + 20 by adding a box of 20.
For any number greater than x + 20, I can buy a number of boxes containing 20 nuggets, and go back to the previous demonstration.
So I can buy any number of nuggets greater than x.

Maybe not the more elegant proof, but I think it's very visual.

Answer 2

You can also expand it to be more general. For any combination of sizes of packets, where n is the smallest size, and I know that I can buy x, x+1.... to x+n-1 nuggets, I know that I can buy any number greater than x because I can just add a multiple of n to one of those initial numbers.

i.e. any number larger than x can be written in the form A + B*n, where A is a whole number between x and x+n-1 and B is any non-negative integer.

Answer 3

Thank you, this was what I was looking for - I basically wasn't sure how to express it in words... Got the rest of the assignment done without having this part figured, so I guess I understood it haha