# How to prove that an armchair rotated 90 degrees can/cannot reach a position adjacent to starting position

I am trying to solve the following problem from a book:

A very heavy armchair needs to be moved, but the only possible movement is to rotate it through 90 degrees about any of its corners. Can it be moved so that it is exactly beside its starting position and facing the same way?

I tried with a lego piece and markers and it looks like this is not doable. But I am looking for a more rigorous way of proving this.

I tried with labeling the corners of the chair as {FL, FR, BL, BR} representing front-left, front-right, back-left and back-right respectively, and figuring out how the corners change in a coordinate system.

Assuming initial position for {FL, FR, BL, BR} as {(0,0), (1,0), (0,1), (1,1)}, a 90 degree clockwise rotation on the FR corner will transform the corners to {(1,1), (1,0), (2,1), (2,0)}.

Now I am stuck with not having a strategy to prove that the target state {(1,0), (2,0), (1,1), (2,1)} can/cannot be reached.

May be there is a much simpler way to reason as well?

Show that if FL is at $$(x,y)$$, then $$x+y$$ is even