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You're right; there aren't strict subsets and we don't know all the
different possible piece types. When I heard multi-positional, I was
thinking of something like a Wall that takes up two squares. Perhaps you
mean a piece that can move from multiple positions as origin, viz
Fourriere's Wizard. This brings us to a new division, variable-destination
pieces and invariable-destination pieces. An independent-destination piece
could always move to the same squares. Dependent-destination pieces move to
different squares based on certain variable statistics. The most obvious
dependent pieces are those that have movement trace a path from their
current square. An alternative piece would be the Loner, which moves to any
squares that have a piece density of <25% as calculated from an average of
all possible 3x3 rectangles on the board. It becomes trivially more
powerful toward the endgame.