Codeforces 478D Red Green Towers - Transition

Level $i$ needs exactly $i$ blocks, all one color. From state $dp[i-1][j]$ (built $i-1$ levels, used $j$ red), you have two choices:

$1.$ Make level $i$ red: Use $i$ more red blocks. Go to $dp[i][j + i]$.

$2.$ Make level $i$ green: Use $i$ green blocks. Go to $dp[i][j]$. The transition (how we move between states) adds the number of ways: $dp[i][j] += dp[i-1][j]$ (green) and $dp[i][j+i] += dp[i-1][j]$ (red). This transition captures the relationship between subproblems.