Ratings & Comments

I have two observations.  First is that I think wide boards actually
increase the relative value of diagonal movers, not orthogonal ones. 
Consider a very long narrow board.  It will take a diagonal mover many
moves before it can hit a square on the opponent's half of the board,
whereas an orthogonal mover can do so on turn one.  On a wide board, the
diagonal mover has more squares from which it can attack squares in the
opponent's camp.

Secondly, I agree about unit density.  I am currently working on a large,
complex ultima-like game with powerful unorthodox pieces.  I found that
the game only works on a 10x10 board with three rows: one row of pawns,
one row of guards (with value intermediate between pawns and pieces but
without the pawn's ability to promote) and one row of pieces.  Unit
density is 0.60.  Immediate development is slower than in orthochess, but
because units are more mobile than orthodox pieces, the game heats up
pretty quickly.  All of the games I have playtested, and there have been a
lot, have ended in less than a hundred moves.  The game isn't quite ready
for posting to these pages but if anyone is interested in seeing what I'm
talking about and wants to email me, I can send them a 'beta' version. 
It's quite playable and interesting, I think.  But you'll only like it if
you like games that are more complex and somewhat wilder than orthochess.

For comparison's sake, I quickly calculated some piece densities:

Shogi     49.4%
XiangQi   35.6%
Timur's   50.0%

The density of any 9x9 variant with an extra piece is 44.4%

ok, i'll have to come back and read more carefully later, but one thing i
noticed is something to the effect that wider boards help increase the
value of diagonal movers more than orthogonal movers.  i have had no
experience that would even remoately back up such a claim.  in david
short's doublechess, a game in which the board is 16x8, the bishop is
severely weakened by the width of the board.  it's well-known that
increasing the board size weakens the knight, but in doublechess the B is
hurt almost as much as the N by the board change (comparing to 8x8).  the
fact that it is more likely to attack the opponent's camp in 2 directions
rather than 1 is small compensation for the fact that it often takes 10
moves or move to get the bishop from side of the board to the other.  the
rook, on the other hand, is not affected at all.  in fact, when studying
the relative values of pieces on different sized boards, it is my claim
that all other things being equal, the rook is the most consistent piece
from board to board, and should be the baseline against which other pieces
are measured.

The measurement that was used by Gabriel Vincente Maura to justify the
design of his variant, Modern Chess (Ajedrez Moderno), 
http://www.chessvariants.com/large.dir/modern.html
is kind of interesting.  This is taken from the booklet that came with my
Modern Chess set, 'Mathematical Thesis of Modern Chess', 50 p., 2nd
English Edition Revised, 1974.

He defines the maximum mobility of each piece as the number of squares it
can move to from its best position on the board, that is:

K=8, Q=27, B=13, N=8, R=14, P=2

The maximum relatve mobility for the total of each player's pieces is the
sum of the maximum mobilities of all the pieces, divided by two, because
there are two players.  Thus:

(K+Q+2B+2N+2R+8P)/2 = (8+27+26+16+28+16)2 = 60.5

He defines the maximum mobility that the chessboard offers simply as the
number of squares.  He wants the maximum relative mobility of the pieces
(60.5) to be equal to the maximum mobility offered by the chess board
(64).  Since the numbers aren't equal, he declares FIDE Chess to be
defective.  Needless to say, for Modern Chess, with the addition of the
Marshall, both numbers work out to 81.

Some example calculations for other variants:

                  'mobility'   board
Grand Chess           98        100
Timur's Chess         86        112
Xiang Qi              59.5       90
Shogi(unpromoted)     45.5       81
Shogi(promoted)       75         81

I believe that this is little better than numerology, but it's still fun
to play with.

I think Gabriel is on the right track but needs an improved methodology.  I
would suggest using Betza's crowded board mobilty calculations.  To get
middle game figures, deflate the piece count by 40% and then calculate the
piece density.  For FIDE chess this gives a deflated piece density of 30%
and a square emptiness probability of 70%.  Then using these numbers
calculate the croweded board mobility of one army (for divergent pieces
such as pawns, just use the average of the mobility of the capturing and
non-capturing moves). As it happens this is quite close to 64 for FIDE
chess--so lets simplfy and say that that it is exactly 64 for a ratio of
mobility to number of squares of 1.0.

Having calculated the crowded board mobility of the army divide the square
of the number of squares by the mobitity. For FIDE chess, this is 64
squared divided by 64 = 64.  For a hypothetical 100 square game with a
whole army crowded board mobility of 125, this is 100 squared divided by
120 = 80, while an 81 square game with a whole army mobility of 72 = 91
1/8.  I would predict that the first hypothtical game would have a typical
number of moves close to FIDE chess than the second, even though it has
more squares. 

Final results significanlty greater than 64 indicate games that play
slower than FIDE Chess, results significantly less than 64 indicate games
that play faster than FIDE Chess.

Taking two real games as examples:

Betza's Tripunch Chess would play faster than FIDE Chess even if it were
played on a 10 by 10 board,

Feeble Los Alamos Chess will play slower than FIDE Chess even though it is
played on a 6 by 6 board.

There is no real need to do the actual calculations for purposes such as
time limits for tournaments--a good guess as to whether the game is faster
or slower than FIDE chess is adequate.  The relevant factors are number of
squares, piece density, and strength of pieces.

My s[w]eeping switchers army for chess with different
armies features three pieces with assymetric retreat.
All of them are bent riders: The panda (aka slip rook),
the erl queen (aka slip queen) and the unicorn.

Another one is the mao (xiangi horse) which is not a bent
rider.

--Jörg Knappen

The bent riders and lame leapers (like the Mao) are part of a larger class
that might be called 'multi-movers'.  These are pieces that can make two
(or more) geometrically different moves in the same turn.  The gryphon
moves as a Ferz and then (optionally) as a Rook; the Mao moves as a Wazir
then (mandatorily) as a Ferz. Any such piece will have the asymmetric
retreat property if the order of move types is not reversible. If the
gryphon could move Ferz then Rook or Rook then Ferz it would not have the
asymetric retreat property (and would be immensely powerful).

True leapers such as the Knight in a sense might be said to have the
asymmetric retreat property but it is irrelevant as they can jump over
occupied squares--I prefer to think of a leaper's move as a direct
point-to-point move that does not pass over interventing squares, in which
case the Knights retreat is not asymmetic.

I believe that multi-movers are the only type of pieces which have
symmetric movement patterns but asymmetric retreat. (OK everbody, please
prove me wrong if possible!)

well, besides multi-movers, leaping pieces such as grasshoppers have
symmetric move patterns but assymetric retreat.  and xiangqi cannon has
assymetric retreat when capturing but not when moving, which is one of the
things that makes it such a neat piece (and difficult to get used to).<P>

and incidentally any piece that move differently forwards than backwards
(these pieces don't have symmetric move patterns, at least not about the
x-axis) is going to have assymetric retreat.  this includes lots of betza
pieces such as fBbR, fRbB, etc etc (i could go on and on) and shogi pieces
(which can of course be easily described in betza notation) such as the
gold, silver, copper generals, the white horse and the whale, etc etc.

Thanks, Ben.  Cannon type pieces of course have asymmetic retreat (though
these could be arguably defined as a subtype of multi-movers). Indeed the
Grasshopper and the Cannon when capturing have a stronger form of
asymettric retreat.  Some definitions:

1. High-power Symmetric Retreat--the piece can alway return to its
starting square on the next move by reversing its path (unless prohibited
by the need to meet check, etc.) Example:  Knight. Nb1-c3 can always be
followed by Nc3-b1.

2. Low-power Symmetric Retreat--the piece can return to its starting
square on the next move by reversing its path unless the opponent has used
his turn to block it.
Example: Rook.  Rc3-h3 can be follewd by Rh3-c3 if opponent has not moved
a piece to d3,e3,f3 or g3.

3. Low-power Asymmetric Retreat--the piece cannot reverse its path but may
be able to return to its starting square on the next move if the alternate
retreat path is not blocked. Example: Gryphon.

4. High-power Asymmetric Retreat--the piece cannot return to its starting
square on the next move unless the opponent moves to facilitate it.
Example: Grasshopper.  Gc3-c7 cannot be followed by Gc7-c3 unless the
opponent moves a piece to c4.

Thank you all for your replies.

Might be interesting to have a large variant built on the general theme of
asymmetry: some pieces would have the asymmetric retreat property but have
symetric move patterns, some would also have asymmetric forward and
backward moves, some with asymmetric left and right moves, some with
divergent captures, etc. Perhaps a 11 by 11 game with strong pieces and a
strong, asymmetric King.

11 by 11?  Shouldn't a game with an asymmetry theme be played on an
asymmetrical board? :-)

Perhaps the 43-square contest would be perfect for an Asymmetry Chess...

I like the idea of the large board.

In addition to the general game with the ability to select armies, I feel
that it might be nice to define a game with preset armies and a
distinctive name as a particular case of the general format so that a
collection of games could hopefully be produced by various players.

Here's the rule from our FIDE laws page
(http://www.chessvariants.com/fidelaws.html):

10.10
The game is drawn, upon a claim by the player having the move, when the
same position, for the third time: 
(a) is about to appear, if he first writes the move on his scoresheet and
declares to the arbiter his intention of making this move; or 
(b) has just appeared, the same player having the move each time. 
The position is considered the same if pieces of the same kind and colour
occupy the same squares, and if all the possible moves of all the pieces
are the same, including the rights to castle [at some future time] or to
capture a pawn 'en passant'.