Comments/Ratings for a Single Item

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.

Following in Peter Aronson's footsteps, I am going to ramble on some more. How crowded the chessboard feels depends on several factors, including the number of pieces on the board and their Attack Density: calculated by adding up the number of adjacent squares each 'nonpawn piece' attacks and then dividing by the number of nonpawn pieces. Ruggero Micheletto's Ultra Chess achieves the maximum 8.00 by adding the King's move to every piece in the game (except Pawns).

Looking at some of my recent 10x10 variants: Opulent Lemurian Shatranj has Attack Density 4.00 (same as standard chess) and Shatranj Kamil X has Attack Density 2.50 (same as Shatranj). The latter game includes Cannons, which attack opposing Pawns in the initial setup. I don't know if that makes this variant more 'crowded', but it certainly adds to the number of threats on the board.

Personally, I prefer the term 'defence density' for this heuristic, since King safety is more difficult in Capablanca Chess than in FIDE Chess.

Hey, Sam. The FIDE attack density is 4.0, check your numbers. ;-)

This balances the attack density of FIDE with Capa. So what accounts for
the extra strength of Capa, compared to FIDE? There are a number of factors
that might enter into this. Compression, for example, the ratio of width to
depth. For FIDE, 8/8 = 1.0, for Capa, 10/8 = 1.25. The opposing pieces are
relatively squeezed together. And that's why Grand Chess, 10/10 = 1, feels
so open compared to Capa. At least it does for me. Even though the extra
squares are 'behind' the starting arrays. Elongation is a good name for
this, and it has its own effects. Gary Gifford used them in his ShortRange
Project games. The Capa - Grand Chess pair is a very good example of
elongation.

But that's just one part of the [very complex, non-linear] equation. With
FIDE, Capa [and Grand, all] having the same attack density, why is Capa
'stronger' than FIDE? Consider the long range pieces. FIDE has 5/8th of
its pieces long range, 62.5%; Capa has 7/10th, 70%. With the individual
rook or bishop types, FIDE is 3/8, 37.5%, to Capa's 4/10, 40%. This
doesn't quite seem to be enough to explain all the difference between the
2 games. ;-)

Okay, here I'll admit that FIDE has one grade A power piece, of 8, 12.5%.
Capa has 3 of 10, 30%. This does explain why the games feel different, but
it doesn't give a lot of insight into the exact how. Just how are they
different? Can we put numbers to this, that have predictive value for other
games, with less-familiar pieces? So we should keep looking around.

The mean free path is another concept that keeps popping up as relevant.
How far can a piece move before it bumps into another piece? This area has
a lot of wrinkles to it. Enough so they might wait for the next post.

To continue expanding David's concept of attack density out further [and
incidentally explain why I too call it attack density rather than defensive
density], I'll bring up attack fraction once again, this time in
conjunction with mean free path ideas. 

The attack fraction is the percentage of squares, at each incremental
range, that the piece may attack. A rook or bishop attacks 4 squares at any
range, but at range 1, next to them, that's 4 of 8 squares. At range 2,
that's 4 of 16 squares, and at range 10, that would be 4 of 88 squares.
And the slider is blockable by 1 piece. So its attack fraction would be:
.5, .25, .17, .13, .1, .09, .07... at range 1, 2, 3...

Let's look at the knight. Its attack fraction at range 1 is 0.0, at range
2 it's 0.5, at range 3 and beyond, it's 0. At range 2, the knight is
twice as good as the rook or bishop. And it's unblockable except by a
friendly piece in the target square. And, since self-capture games are
rare, the knight is as unblockable as they get. A slider is not guaranteed
to go 2 squares. 

Okay, now consider the mean free path of a piece in the midgame, then look
at the [bent] hero and shaman. The hero's attack fraction is .5; .75; .17,
and the shaman's is .5; .25; .5, for ranges 1, 2, and 3. Each attacks [up
to] 20 squares, 8 unblockably, and the remaining 12 have 2 paths to each
destination. The hero attacks 4 squares at range 1, 12 at range 2, and 4 at
range 3. The shaman attacks 4 squares at range 1, 4 at range 2, and 12 at
range 3. This explains why, even with a range limited to 3 squares, the
bent hero and shaman are more valuable than a rook on an 8x8. 

**********************************************************
Somewhere earlier in this thread, I believe it was commented that larger
boards may require a lower piece density. I would like to second that
statement, and suggest that after considerable experimentation with large
variants of different sorts, I'm getting a sneaking suspicion that at
least half the very best games for very large boards will have very low
densities.

Piece density should be no absolute and instead there should be clues from the rules and the piece mix which way to go from conventional 50%. This truism is backdrop for Joyce's more technical attack density etc. Do small boards then generally work better with higher than 50% piece density? Let's
see, Los Alamos 6x6 has 50%. Centennial is great at 52% piece density on 100. I think Grand and the Canadian Omega are fast dying out partly because of low piece density -- as well as Carrera compounds in the one case -- making boring games. Reading this very thread 7 years ago, I coined ''piece-type density,'' which conventionally stays around 10% (meaning 6/64 for example), but adhering to the rule is the exception. Only a plurality of cases are 9%, 10%, 11% piece-types.

Hey, George, it's been a while since we argued opposite sides of a
question. This seems like a fairly interesting one to kick around a little.
But I'd like to start by looking at some supersized games, or at least
game boards.

On a 12x16, there are 2 ways to orient the board. With 50% starting
density, the simplest setups for the 2 orientations are 3 rows of 16 per
side, or 4 rows of 12. Using the 10% rule, 192 squares gives 19 different
kinds of pieces. One to three pawns leaves 16 - 18 piece-types. We'd want
3 or 4 colorbound pieces, anyway, just because. Even with weak pieces, the
total power on board adds up when you start with 96 squares occupied. So
you want to mitigate some of it. Besides, there are some neat colorbound
pieces besides the bishop and the Omega wizard. I'd recommend John Ayer's
Duke, a colorbound queen-type slider as a third, and something like the
modern elephant [ferz + alfil] or another short-range leaper [but not the
camel by itself - that piece is just awkward, and should be restricted to
those games where it is designed in as an integral part of the game, like R
Wayne Schmittberger's Wildebeest Chess] of range 3 or so. 

That still leaves 13, give or take, pieces to go. The king leaves a dozen
or so pieces, and we've used up 4 colorbound pieces already. Well, what
sliders do we use? Bishop's already used, so we have the rook and queen -
down to 10. 

Let's hold onto the knight for a while, since it is so shortrange. We may
have to get creative here, but for now, I'll note the knight covers that
missing piece if there is only 1 type of pawn in the game, leaving 18 more
pieces to find.

Okay, I've kept you in suspense long enough, George; we can use the
falcon as a shortrange slider. I'd also recommend the nahbi, Uri Bruck's
sort-of knight analog, which slides 2 diagonally, then steps 1
orthogonally. It visits the 8 knight squares plus the 8 squares diagonally
out from those 8 knight squares. Now we're down to 8 pieces. 

Now I'm getting desperate. The BN and RN. The bent hero and shaman [and
adding 1 more colorbound, but strong, piece]. Four pieces to go... help. 

Four more piece-types to go, and that's if you like my choices so far. I
could see anybody knocking out maybe a half dozen of my choices. This is my
first move, arguing by counter-example. Or, if you prefer, arguing by
example. 

Enjoy.

ps: The original value I saw for the Mann [non-royal king, aka: guard,
prince...] was 4, so a 3.88 piece would be the mann without the straight
back move. However, HG Muller comes up with a much lower figure for this
piece, so add a second step forward only, or a forward-only dabbabah leap.
Or you could try dumping the mann's backwards move and adding the 2
forwardmost knight moves, either lame or not. It seems to me that for
pieces of less than rook value, you could play around with the mann [a
superset of gold and silver], knight, dabbabah, and alfil and pretty much
dial up a piece of a certain value. Of course, these pieces would very
likely be infuriating to use en masse.

The CV Alphanumeric has 100% piece density and it stays that way throughout the game. 
http://www.chessvariants.org/index/displaycomment.php?commentid=23477

Okay, in this conversation which has rambled over a few different threads
now, you mentioned my 12x16s. I've done, basically, 3 different games that
are different kinds of games. One came out quite good, and spawned larger
[and smaller] variants. One is fascinating, but is more potential than
finished game at this point, and that potential hints at different kinds of
chess games. The third is a bust, one of my biggest flops, ranking right up
there with goChess as a game that looks so good in theory and so bad in
practice. That would be SpaceWar, a game in the Games Garage [as is
goChess], and deservedly so.*

All three games have 3 major things in common: first, they all have 32
pieces/side, and second, can be played face to face using 2 standard chess
sets. For some games, the pieces may need to be marked with colored twist
ties, unless the players have several colored sets. Third, they are all
multi-movers. The one major difference between SpaceWar and the other games
is that SpaceWar is the only one that does not use leaders of some sort.

So for starters, the density is 33%. This is enough to speed up the game
all by itself; you don't have so many pieces to wade through, compared to
the 48 pieces per side with the 50% rule. But the major difference is the
piece-type density: it's 5 with the most successful of the 3 games,
Chieftain. Not even close to 19. [I do go up to 7 piece-types on the 15x30
version, but that's a board with 450 squares.] 

The Chieftain games all tend to be decided, if not done, within 40 turns,
partly because they are multi-move. However, the pieces only move 1 to 3 or
4 squares maximum per turn, so this is obviously not the full answer. The
other half of the equation is moves per pieces per turn. To illustrate: in
FIDE chess, there are 16 pieces to start, and the players move 1 piece per
turn, 1/16th of their army, or just over 6% of that army each turn. This
percentage increases steadily as pieces are lost, but the first loss takes
you down to moving 1/15th of the army, the next, to 1/14th, or about 7%. By
the time you're through the opening and into midgame, you might have lost
6 pieces, so you are moving 1/10th, or 10% of your army. When you've got
your king and only 4 other pieces left, you're moving 1/5th, or 20%, of
your army. This is not a very high number. 

Let's look at Chieftain [effectively all versions from the largest to the
smallest]. At start, 1 in 8 pieces is a leader. So, 1/8th of your army is
moving each turn, or 12.5% That's a high number to start with. You move 4
pieces/turn, and your entire army of 32 pieces can be moved in 8 turns. And
the mechanics of the game are such that players tend to trade 2 or more
pieces/turn, when actively fighting each other. After 5 turns of combat, at
2 pieces lost/turn, your 32 piece army has shrunk to 22 pieces, but you are
[most likely] still moving 4 pieces/turn [or you are losing.] This is
roughly 18% of your army per turn, in early midgame, and it only goes up as
the game progresses. The pace of the game is twice as fast as a standard
chess game. Basically, in midgame, you can move your entire army every 4-5
turns. This gives a bit of a different character to the game, because the
individual units are very slow, but the army is very fast. I think this is
the effect that allows Chieftain games to finish in less turns [but not
fewer individual piece moves, which are 2-4 times as numerous in
Chieftain.] These games emphasize the wargame aspects of chess.

The concept with fascinating potential is the chesimals series of games. A
chesimal is a second-order chesspiece, with a variable footprint. It is
made out of a number of individual chesspieces of various types, that are
required by rule, even though each piece moves individually, to effectively
move as a unit. The 32 pieces per side are assembled into 4 chesimals.
There are, so far, 3 different types of chesimals, each with its own unique
characteristics. I'd originally thought of these higher-level pieces as
like army corps, but in practice, the game feels like a combination of
chess with a wargame touch and Conway's Game of Life, which leaves me
wondering what will evolve out of it. 

There you go, George, my 3 [more or less] 12x16s, complete with critique.
I figure one good, one bad, and one, who knows, maybe 'interesting'. [And
maybe just 'ugly'.]

*SpaceWar has attracted a guest mechanic, John Smith, who was intent on
souping it up quite a bit. The results are in the garage.