= Solution
Retain the ordered position state $(p,q)$, the same initial state and the <goal test> $p=q$, with unit joint-action cost. Replace the deterministic <search transition model> by a <set>-valued outcome relation. Let
$$
R(p,u)=
\begin{cases}
\{\delta(p,L),\delta(p,R)\},&u\in\{L,R\},\\
\{\delta(p,u)\},&u\in\{U,D,W\}.
\end{cases}
$$
Then
$$
\boxed{\operatorname{Results}((p,q),(u,v))=R(p,u)\times\{\delta(q,v)\}.}
$$
Here $\delta$ for an attempted move leaving the board returns the current room. This explicit boundary convention keeps every outcome inside the <state space>; alternatively disable a horizontal command unless both possible destinations are legal. Mrs Robot’s commands can still be restricted to legal moves as before.
The result is a fully observable <nondeterministic planning> problem, assuming that actual positions are observed after every step. Both possible horizontal outcomes must be represented, even though they may coincide at a boundary or when $n=1$. No outcome <probabilities> are supplied, so attaching equal <probabilities> would add an assumption. Its solution is a position-dependent plan or policy.
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