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I came across this question - Formal rules for two-player bughouse without a clock - which asks if there is a formal way of breaking deadlock situations where one player on each side doesn't want to move, but nothing about formal or official rules. Do they exist? Has anybody tried to control this chess variant?

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I found two sources. For fans of the USCF they have produced their US CHESS SCHOLASTIC BUGHOUSE RULES but not a grown up version that I can find.

The rather more serious Czech federation holds an annual bughouse championship in conjunction with their regular national championships as part of an annual chess festival. They publish their formal rules here.

These appear more serious and handle more difficult problems like formalizing storage of captured pieces and the way they are handled:

1.6 When a piece is captured, the captured piece is passed to the partner stuff-place immediately after the move is completed, not later than before the player's next turn. A player's "stuff-place" is an area reserved for pieces that the player can drop on the board. The player's stuff-place must be between his opponent's eyes and the player's hands. So the stuff-place is usually between board and player's hands.

Before the game the players and arbiter should agree on the position of the stuff-place

1.7 A player cannot attempt to hide pieces from the opponent. During the whole game the players shall manipulate the pieces in such a way that both opponents are able to see all pieces. Only when captured pieces are transferred to partner's stuff-place or in the moment when player is moving or dropping piece on chessboard, a piece may be partially hidden in a players hand. However it must be clear to his opponents which piece is partially hidden and this moment must be as short as possible.

1.8 a) If a player hides pieces in an improper way (rule 1.7), his opponent (or partner of his opponent) is entitled to warn him to "show pieces".
b) If a player drops a hidden piece, his opponent may restrict its use by the message "invisible", e.g. "invisible bishop". The opponent may then immediately start the player's clock and force him to make another move.
c) A piece restricted in this way may not be used until the opponent allows (by the message "bishop visible now", for example). The opponent should allow it as soon as possible, in good sportsmanship.
d) If a player repeatedly hides his pieces, his opponent (or partner of his opponent) may claim "repeatedly hiding" and the arbiter may impose some sanction. (rule 9.4.)
e) If a piece is hidden through no fault of the player, and this drastically alters the game, the arbiter may order a new game.

Contentious issues around simultaneous checkmate are also handled:

1.9 When player is checkmated after taking any piece then if his partner has the move, he may drop this piece as his last move.

1.10 If both games finish at the same time, the results on both chessboards are counted up; the team which wins on both chessboards wins the bughouse game. Otherwise the bughouse game is draw.

2.2a The bughouse game is draw when player is taking a piece, then (in the next opponent move) is checkmated, his partner's move is in this moment, his partner is by dropping this piece (before his time is out) checkmating opponent or making draw by achievement of 3x repeated position (See 2.2.b).

There is no mention of stalemate for the obvious reason that a player can always put a piece on from the "stuff-place" or wait for a piece from his/her partner.

Also no mention of 50 move draw rule because, let's face it, pieces are always going to get taken and pawns are going to be moved and recycled and very quickly.

There is, though, a 3-fold repetition draw law.

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