Codeforces 478D Red Green Towers - Finding the Height

To find $h$, solve $\frac{h(h+1)}{2} \le r + g$. You can iterate: start from $h = 1$ and keep incrementing while $\frac{h(h+1)}{2} \le r + g$. The largest such $h$ is your tower height. Let $S = \frac{h(h+1)}{2}$.

Now count how many ways to assign colors to each level such that you use at most $r$ red blocks and at most $g$ green blocks, with exactly $S$ total blocks used. This is where the DP comes in.