How many times can you pour milk among 3 buckets?

This is a problem from usaco, which I solved, but I don't know how to estimate the number of times you can pour milk among the jugs, my solutions just uses a big enough number.

Here's the problem statement:


Farmer John has three milking buckets of capacity A, B, and C liters. Each of the numbers A, B, and C is an integer from 1 through 20, inclusive. Initially, buckets A and B are empty while bucket C is full of milk. Sometimes, FJ pours milk from one bucket to another until the second bucket is filled or the first bucket is empty. Once begun, a pour must be completed, of course. Being thrifty, no milk may be tossed out.


Write a program to help FJ determine what amounts of milk he can leave in bucket C when he begins with three buckets as above, pours milk among the buckets for a while, and then notes that bucket A is empty.


For example, if the input is 8, 9, 10 then the output is 1, 2, 8, 9, 10.


Another example: if the input is 2, 5, 10 then the output is 5, 6, 7, 8, 9, 10.

My approach is to recursively try all possible move sequences, keeping track of the contents of bucket C each time bucket A is empty. Currently I stop the recursion at the arbitrary depth 200. What is the correct depth to stop the recursion at?

Since the total amount of water is C liters (C being the capacity of bucket C), the number of possible states is at most the number of non-negative integer solutions of $x+y+z = C$, which is $\binom{C+2}{2}$. Any sequence of moves longer than that must repeat a state. Therefore it is enough to recurse up to level $\binom{C+2}{2}$.

StackExchange Computer Science Q#54445, answer score: 2