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A piece with a constant (psition-independent) gait that has only two
targets, located in an inversion-symmetric way, will be able to move back
and forth along a line. Moves in one direction will exactly cancel moves
in the other direction, so that only the difference determines where the
piece is. This means all possible long-term destinations can be reached by
moving only in one direction. If in this process the piece skips over a
square, this square is unreachable.

If a piece has 4 moves in an inversion-symmetric pattern, such as narrow
or chiral Knights, but of course also Alfil and Dababba, the moves can be
grouped in pairs of opposing moves. For each pair the same situation as
above exists. All long-term targets can be reached through N moves in one
direction of the first pair, and then M moves in one direction of the
other pair. The targets can thus be mapped onto a two-dimensional grid,
which in general will be a subset of the board. The Wazir is the only
inversion-symmetric piece with 4 destinations that can access the entire
board.

With 6 or 8 moves and inversion symmetry, the destinations logically map
onto 3- or 4-dimensional grids, but as the board is two-dimensional, you
will see a projection of such grids on the board. Such a projection can
quite easily acces every square, as the number of grid-points in a
three-or more- dimensional grid is so much larger than the number of
squares on a to-dimensional board. So color-boundedness is the exception,
rather than the norm, in inversion-symmetric pieces with more than 4
destinations.

For pieces that do not have inversion symmetry the situation is different.
On a two-dimensional board you need at least 3 moves to lift
color-boundedness. With 2 moves, the piece is either restricted to (a
subset of) a line, or is irreversible an cannot return to its original
position after it moves. The 'Y-piece' (fFbW) is an example of a piece
with 3 moves that can acess every square of the board reversibly.