Admittedly this is a problem I encountered from school but I cannot think of a proper proof solution. I thought about the logic that in order to cover all squares, there must be closed loops of movements. So in the easiest case, where there are only 2 squares, person in square A goes to square B and person in square B goes to square A. This means that for grids with even number rooms, it is possible. But how can I prove that for odd number rooms, there is no way to create closed loops of movements to cover all squares.
The question comes with the constraint that I have to use Pigeonhole Principle to solve but I am open to ideas.
Hints:
Colour the rooms alternately black and white like a chessboard starting with black in a corner
How many people are in black rooms and how many in white rooms?
They each move to an adjacent room, so after the move how many people are in black rooms and how many in white rooms?
Is this possibly consistent with each room having one person after the move?