Chess Give It Time To Think It Over

"Okay, now correct me if I'm wrong, but would it be legal to move a king next to another king in this variant? For example, suppose white has Ra1 and Kc6 and Black has Ka6, so Black (to move) is checkmated by the rules of normal chess. Could black play Kb6+ (moving out of the old check, into a new check, and putting white in check at the same time)? If so, then KR vs K would be a draw in this variant! Actually so would KQ vs K because even if one side mated in the corner, the other side could simply take the queen. But if this is not allowed, then all the KX v K endings are the same as normal chess as the lone king will not be able to give check, right? Come on, Mom, I don't have all damn day."