Question 197
A museum has a square plan and is divided into $n^{2}$ square rooms that are all the same (with $n\gt 1$). Each pair of adjacent rooms (i.e. with a shared wall) communicates via a door (as shown in the example alongside for $n=4$). The night watchman wants to organize his inspection tour in such a way as to respect the following rules: the watchman starts from a certain room, where he stays for a minute, after which he moves to an adjacent room, where he stays for another minute; the route
it continues by connecting adjacent rooms, in each of which the guardian always stays for exactly one minute before moving.
It is allowed to pass through the same room several times, but at the end of the journey (which is not necessarily in the starting room) the guardian must have been in each of the $n^{2}$ rooms for exactly $k$ minutes.
Determine for which positive integers $n$ and $k$ it is possible to organize the path respecting these rules.
it continues by connecting adjacent rooms, in each of which the guardian always stays for exactly one minute before moving.
It is allowed to pass through the same room several times, but at the end of the journey (which is not necessarily in the starting room) the guardian must have been in each of the $n^{2}$ rooms for exactly $k$ minutes.
Determine for which positive integers $n$ and $k$ it is possible to organize the path respecting these rules.