Baltic Way 2023 Shortlist

Let's define the coordinate system with the origin point at the home base and vertical-horizontal axes. W.l.o.g. assume that the first move was east and the path had length of $n$. Then each odd move changed the $x$ coordinate of the robot by $1$ and each even move changed the $y$ coordinate by $1$.

At the end of the day both coordinates were equal to zero again, so there had to be an even number of odd and an even number of even moves. That implies that only $n$ divisible by $4$ can fulfill the conditions.

For $n=4$ we have a square path. For $n=8$ we had $4$ changes of $x$ coordinate and $4$ changes of $y$, so the whole path was inside some

For $n=12$ there is a path in the shape of "+" with first $4$ moves like $(\to ,\uparrow ,\to ,\uparrow )$. Now we can change the middle $(\uparrow ,\to )$ sequence by $(\downarrow ,\to ,\uparrow ,\to ,\uparrow ,\leftarrow )$. Thanks to this change the robot explored new territory south-east from the one before explored. We got $+4$ of length of the path. There we can do it again and again, reaching any length of $4k+8$ for all $k\in {\mathbb{Z}}^{+}$.