In From knights to kings and From Knights to Kings on a rectangle we tried to help knights be next to their friends after they were promoted to kings. What happens if we have kings who are deposed and become knights?
We start off with an $N\times M$ board filled with kings. A king is friends with all the kings on squares that he could move to, i.e all adjacent kings. After they are deposed, they want to have all their friends still be in squares that they could get to in a single move. So two kings A and B:
A B . . . .
would want to be like this after becoming knights:
A . . . . B
Unfortunately A's two other friends (originally in the two spaces below him) cannot be put anywhere that allows A to still reach them. So a $3\times 2$ board doesn't work.
A $1\times 1$ board is trivially solved - no friends either before or after the change.
Any $n\times 1$ board (where $n>1$) is obviously impossible - each king has one friend if he is at the end, or two friends if he isn't. After they become knights, none of them can move so none of them can be a knight's move away from their friends.
Are there any board sizes that allow the kings to be a knight's move away from all of their friends?