Game Routes - Core Idea

The state is ways[v], the number of ways to reach level $v$ from level $1$. The base case is ways[1] = 1 (one way to be at the start). The transition is: for each edge $u \to v$, add ways[u] to ways[v]. This says "every way to reach $u$ extends to a way to reach $v$". Why does this work?

Because the graph is a DAG, there are no cycles. Each path is counted exactly once. When you process $v$ in topological order, you've already counted all ways to reach its predecessors $u$, so you can sum them safely.