As the dimension of the board increases, it becomes increasingly difficult to checkmate the opposing king. Consider that a full-powered king (a piece which moves to any adjacent square) has $3^N-1$ possible moves when in the center of the board; that's 8 in 2D chess, 26 in 3D, 80 in 4D, 242 in 5D, etc. Even when backed into a corner, the full-powered king has $2^N-1$ moves; that's 3, 7, 15, 31, etc. Combine this with the rook's loss of blocking power, and the king seems nearly uncatchable. Several solutions have been proposed:
- Crippled King: We limit the directions of movement of the king, say to orthogonal moves only. Since then the king then cannot capture a piece immediately adjacent, perhaps we restrict only non-capturing moves, while allowing capture in all (or just more) directions.
- Bare Royals Loses: We call a baring of the opponent's king a win. Of course the queen is even more mobile than the king, so perhaps reducing an opponent's army to just K, Q should be a win.
- Fortress: As in Xiangqi, we could assign a certain block of cells as the king's fortress, which he may not leave. Depending on the size and location of the fortress, this sometimes allows draw by perpetual check when it might otherwise be escapable.
- Gladiator Kings: When a king comes to occupy some given region along with the opposing king (perhaps a common plane), the opposing king is "stuck" in that area until your king leaves the area. Perhaps both kings become stuck, and perhaps another piece may intervene and break the constraint.
- God Pieces: We introduce new pieces which are more adept at checkmating, e.g. the amazon (extended appropriately into multiple dimensions).
- Multiple Royals: Give each player more than one king, and force the players to keep all of their kings out of harm's way.
- Nothing: As slippery as a king can get in multiple dimensions, in 4x4x4x4 chess a king and queen against a lone king will be able to checkmate. I'm not sure how well this generalizes, but perhaps this is enough.