Solution:

One way is to walk directly from room 10 to 20. Else, divide the rooms into 10 pairs $A_{0} = (10, 20), A_{1} = (1, 11), A_{2} = (2, 12), \ldots, A_{9} = (9, 19)$. Notice that
- each move is either between rooms in $A_{i}$ and $A_{(i+1) \bmod 10}$ for some $i \in \{0, 1, \ldots, 9\}$, or between rooms in the same pair, meaning that our path must pass through $A_{0}, A_{1}, \ldots, A_{9}$ in that order before coming back to room 20 in $A_{0}$,
- in each of the pairs $A_{1}, A_{2}, \ldots, A_{8}$, we can choose to walk between rooms in that pair 0 or 1 times, and
- we have to walk between rooms 9 and 19 if and only if we first reach $A_{9}$ at room 9 (so the choice of walking between $A_{9}$ is completely determined by previous choices).

Thus, the number of ways to walk from room 10 to 20 is $1 + 2^{8} = 257$.