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I am not quite sure if this is correct, but when moving from 3 players to 2 players, we play the BB is the person who was not the BB last hand (or the hand before if neither were the BB). This way the same player does not get the BB twice in a row. The button goes to the other player.

Is this correct? It seems fair but not sure if correct.

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This is exactly how we do it... the BB moves to the next player, and the other player is the dealer.