Choosing coins with least or no change given

Question

I am making a game which consists of coin denominations of $10, $5, $3, and $1. The player may have 0 or more of each type of currency in his inventory with a maximum of 15 coin

Answer 1

The problem can be defined as:

Return a subset of items where the sum is closest to x, but >= x.

This problem is called the subset sum problem. It is NP-complete. You won't find a perfect algorithm that runs in pseudo-polynomial time, only imperfect heuristics.

However, if the number of coins is very small, then an exhaustive search of the solution space will certainly work.

If the number of coins is larger, then you should look at Wikipedia for an overview: https://en.wikipedia.org/wiki/Subset_sum_problem#Polynomial_time_approximate_algorithm