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In chess, castling can't be done if the king or relevant rook have moved. How would the basic strategy of the game change if the king and rook could castle after moving? Also, how would the game change if the queen castle as well?

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In most cases, I'd imagine the queen wouldn't castle, as it would mean it would have a harder time getting out. The only useful case I can think of would be if you developed your bishop to e3 and "castled" queenside with your queen, putting it on c1, preparing for Bh6 to potentially trade off a fianchettoed bishop while also bringing your a1 rook to an active square. Being able to castle with your king or rook after you moved would change chess quite substantially, and it'd be difficult to quantify all the changes, but if I had to guess, kings would stay in the center longer and rooks would move around more in the middlegame.

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