Consider an odd chessboard with an odd number of squares. A king is placed on each square of the board, then the kings are picked up and placed on each square of the board again. Can this be done in a way that every king is in a square next to its original position?
Note: we do not follow the rules of chess in this question, it is practical
No, because every pawn will change the color of the square it stays on, and there is different number of light and dark squares on the odd-sized board.
Edit: Adding a new point into the question is a bad practice, see https://meta.stackexchange.com/questions/43478/exit-strategies-for-chameleon-questions