A king and a rook are on opposite corners of an 8x8 chessboard. The king is trying to avoid being checked by the rook. Normally the rook would check the king in one move but this rook is restricted - his moves are limited to up to two spaces, that also applies to his ability to make the check.
Can the king avoid check indefinitely? If not, how long can he last?
(In reply to Open question
by Steve Herman)
The rook can check by moving onto an immediately adjacent square to the king.
As you acknowledge, the problem is harder with the restriction that the rook cannot put itself in danger.