Comments/Ratings for a Single Item

May I bring to the table what I call 'The Chiral Rook'. The Chiral Rook
is the same as the normal Rook, but it can only access the left or right
side of the board, which determined by its initial placement. This Rook
can access only half of the board, and thus is similar to a Bishop, which
is often called a colourbound Rook. What is the value of such a piece? How
would it change when there is rotational symmetry?

Knights have a little secret. And that secret is: they're almost doubly
colourbound. A narrow Knight is, a wide Knight is, and a certain
configuration of Knight moves is even bound to 1/5th of the board. Knights
alternate 1/4 bindings whenever they move. Thus, I propose the
Quadrant-Changing Rook. The Quadrant-Changing Rook is the same as a normal
Rook, but it must change what quadrant of the board it resides. This piece
is absolutely horrid in development, and awkward in the endgame. What is
this piece's strength?

In other words, if we have an empty 8x8 board and a white Chiral Marshall on the D1 square, this piece can move to B2, C3, D5, D6, D7, D8 (the four rook moves which must end on the opponent's side of the board), E3, and F2.

The same Chiral Marshall on D8 can move to A8, B8, C8, D8, E8, F8, G8, H8 (rook move), B7, C6 (knight moves), D7, D6, D5, D4 (rook move again), E6, and F7 (knight moves).

The black Chiral Marshall can only make a rook move ending on White's half of the board (A1-H4)

I like this because it encourages more aggressive play; by making the pieces more powerful on the opponent's side of the board, it makes passive play less fruitful and should make games more exciting.

- Sam

Just as a Bishop is colourbound, a Rook is squarebound. A piece even less
bound than the Rook would be a Bishop that can stop on the corners of the
squares, preferably only a certain type of corner, e.g. the lower left
corners of the squares, though for symmetry in a FIDE-like set-up, a
toroidal 64 point grid is recommended. And, yes, even this piece is bound
in some way, for it cannot access the edges of the squares.

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.