Patient 13 is cured. He wants to pass by all other patients before leaving. But he can't see the same patient twice or he will get sick again. Can he leave from the door in Room 4?

Followup: Instead of a door in room 4, if there were a helicopter in some other room, is it possible to exit the hospital by visiting all other patients but no patient twice?

Heard from Himanshu Nautiyal, my classmate from IIT Delhi (1991-1995), in 2020

If the first few steps in the sequence are 13 → 9 → 13 → 14, the rest of the sequence is easy to construct :) And it's easy to see that the helicopter problem is also solvable for any square.