Enter Your Reply

:-) Getting more into multiform [and referencing one of my designs again!],
I wonder how many games use the lamed knights, and how effective they are.
I used the 'L'-shaped, doubly-blockable move for the knight in my first
chess variant, a 4D game I called Hyperchess, played on a 4x4x4x4 board.
[To make the game playable, I eliminated all of the 3D and 4D diagonal
moves.] Amazingly, the knight in this game is on par with the queen [if
I'm remembering the numbers right, the Q attacks 22 squares max, the N,
23] and is truly not inhibited by being blockable [considering it has 2
paths to any destination square]. In fact, I think it's necessary for
balance that the N be blockable or it has too much power in hype. [This
way of using the knight practically makes it a Jetan piece.]

The more general point is that the geometry of the playing field, and
certainly its dimensionality, has the 'final say' in how powerful a
piece is. The simplest example is 2 pieces and 2 boards. Pieces: a rook
and a bishop; boards: both 1D, one an orthogonal row of squares and the
other, a diagonal row. Question [for the mathematical among us, I
suppose]: is it the case that for any [class of] chesspiece, there exists
a board that increases the piece power, and one that decreases it? Beyond
the trivial example of boards 'cut out' to mirror that particular
piece's move. The 1D boards provide 2 effectively trivial examples,
though you can't get a lot more from 1D.[.. unless you're Larry, or Dan,
or...] Another trivial example would be a disconnected board, with squares
radiating a knight's move apart from the origin. [What dimensionality is
that?] 

The N/4x4x4x4 combo above provides a non-trivial example, I believe. The
infinite 2D board, or a very large one, gives the sliders B, R, Q a value
approaching infinity/continuously increasing, while the N on that board
approaches a value of 4. Other short range pieces also plateau in value as
a 2D board increases in size. That's if it's continuous. Poke holes in
that 2D board, make it a semi-regular lattice, and the situation changes.
Short range leapers, or double [and triple...] leapers lose less mobility,
while 'infinite' sliders have their paths shrunk to shorter and shorter
distances. Is this fit of board and piece the case for all [types of]
pieces and non-trivial boards?

Edit: Hmm, this fits in well with George's Tessellations comment and Rich's on 91.5... just below. Lucky, I guess. Or maybe chess *is* geometry.