Half Bughouse is better known as Anywhere Chess. You find here two different descriptions of the same game.
A legal move in Half Bughouse consists of EITHER:
1. Any legal chess move (including captures, Pawn promotions, and castling); OR:
2. The transfer of any piece or Pawn (already on the board) to any VACANT square, except that:
2a. King moves must be normal, legal chess King moves.
2b. Captures must be normal, legal chess captures.
2c. Pawn promotions must be normal, legal chess Pawn promotions. A Pawn may not be placed on the first rank or the eighth rank, except during a normal promotion.
Of course, checks must be respected. Rules regarding check and mate are the same as in chess.
Threatening 2 gxf7+ Kxf7 3 Nb1-g5+ Kf6 [...Ke8?? 4 Qf7 mate] 4 Qf7+ with mate soon to follow. If 1...fxg6, 2 a2-f7+ with similar disastrous consequences.
1. .... hxg6
Black sacs a Rook for the tempo.
2. Rxh8 a7-g3
Black adopts a strategy similar to White's, but White does not have the option of hxg3 so he must defend:
3. Bc1-e3 b7-f4
This is only a two-move threat, so White has time to launch something of his own:
4. Rxg8 fxe3
5. Rxf8+ Kxf8
6. Ra1-h8+ Bc8-g8
7. Nb1-h7+ Ke8
8. Rh8-f8 mate