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In the past few games I've played, when someone tried to pass constructively, a piece fell off when the player removed the piece off the bottom. We played that the person who tried to pass progressively kept the piece they removed and the piece that fell off, but we weren't sure where the next player could place a their piece.
Any help would be appreciated, and thanks in advance!
Yes, the rules say, "Play a piece in the same section as the last player." I think you are asking, Where does one place their piece if the last player didn't play a piece? (Due to the situation you described).
Interpret the line, "... as the last player" to mean "... as the last player to place a piece." As such, the next player places a piece in the same section as the last player to play a piece (which was the second-last player) or in one of the two next free sections.