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George Duke wrote on Mon, Oct 11, 2010 11:46 PM UTC:
Piece density was a topic 8 years ago Aronson started,
http://www.chessvariants.org/index/displaycomment.php?commentid=23951.
F.i.d.e. is 50% there. Piece-type density often stays very close to 10%, much as
that F.i.d.e. is 6/64 there being RNBKQP; but the type-density topic is
scattered around.  At one extreme Battle Chieftain works with 1/84
piece-type density. Of the 24 leading NextChesses, only Schoolbook and Centennial are exactly 10%. However, Black Ghost would be within a hair's breadth.  (Note piece density of Centennial is 52%, so only Schoolbook has 50%-10%, but so do most of the other Carreras of 80 squares, of which Schoolbook is representative.) The twenty-four there are: Bifurcators 7/80 and 7/64; Great Shatranj 7/80; Mastodon 7/80; Three-Player 6/96; Unicorn Great 9/100; Big Board 6/100;
Sissa 7/81; Eurasian 8/100; Schoolbook 8/80; Fischer Random 6/64; Bilateral 10/84; Centennial 10/100; Kings Court 8/96; Wildebeest 8/110; Transactional(#15) 6/64; Fantasy Grand, as low as 7/100 and as high as 20/100, where it requires at least one druid army to go over 9%; Black Ghost 6.5/64; Eight-Stone 7/72 with stones as one piece-type; Modern 7/81; Melee 9/81; Templar 7/72; Courier de la Dama 9/96; Switching 6/64; Seirawan 8/64 counting each deposited piece full unit, the 6+1+1, as virtually certain entry.

Joe Joyce wrote on Tue, Oct 12, 2010 08:28 PM UTC:
Interesting observation, George. Putting the numbers together like that
certainly shows a strong trend. With the caveat that this result is for 2D
games with board sizes very near 8x8, the result may tell us something
interesting about chess in general. A number of very different people have
contributed to this project in one way or another. That the piece-type
density falls out so close to 10% in all these games, even when the pieces
themselves may be very different from FIDE standard must mean something,
but what? 

It does seem as if most variants follow this 10% rule, but it could be
equally true that 5-10 piece types in a chess variant is the default human
standard, that we find a handful or two of piece kinds just about perfect,
regardless of size. Then we could quantify the rule as: 10 +/- 5 is the
number of pieces humans like in a chess game. In that case, I suspect it
would be a playability issue. Humans can keep only so many things in mind
at one time. As boards go to significantly larger sizes, they become less
playable if a 10% ratio is maintained. Some of the shogis should illustrate
this. 

But turn it around and look at smaller boards. Many smaller games reduce
the total number of pieces considerably, but the piece-types by much less.
Often one piece only is dropped. And some keep everything on a much smaller
board. An excellent example of this is LL Smith's One Ring Chess. With 32
squares, it has all 8 standard pieces and 4 pawns/side, jumping the ratio
to 20%. 

If the second idea is right, an examination of very large and very small
chesses should show them not holding at 10%, but 'cheating' toward a
particular [small] absolute number of pieces.

George Duke wrote on Wed, Oct 13, 2010 10:51 PM UTC:
CVs can go from 1% to 100% piece-type density. It is hard to imagine
over 100% without double occupancy. There should be better example of high 50%
than Dave's Sample Game, which is 2/4:
http://www.chessvariants.org/index/msdisplay.php?itemid=MSdavesexamplega.
Battle Chieftain, a playable cv, is not the only one exhibiting 1%,
http://www.chessvariants.org/large.dir/contest84/battlechieftan.html,
which 1/84 is there.  The Druids of Hatch's Fantasy Grand are 20% if both sides have the same mix, each piece-type one only, so keeping rather typical 40% piece density 10x10: 
http://www.chessvariants.org/large.dir/druid.html. Just remove Pawns and have no doubled pieces to get high piece-type density, but that aesthetic seems to be avoided.  Some other famous ''tens,'' that is cvs of approximately ten percent piece-type density will follow.
Now one example is year 1999 contest winner Vyremorn, 
http://www.chessvariants.org/large.dir/contest/vyr-chess.html, of 14/132.

H. G. Muller wrote on Thu, Oct 14, 2010 09:06 AM UTC:
Mini-Shogi must score pretty high in terms of piece-type density. It starts
out with 24%, but 4 of the 6 pieces can promote, two of them to essentially
new types. I am not sure how promotions to Gold should be counted; in Shogi
a promoted Pawn (say) is considered a different piece type than a Gold,
(with a different representation), despite the fact that it moves the same.

George Duke wrote on Thu, Oct 14, 2010 10:30 PM UTC:
Rococo is another famous CV that we arrived at a piece-type density of 10%:
http://www.chessvariants.org/index/displaycomment.php?commentid=5613. Rim squares are available to both sides and not really occupied half the time at all, but would seem to have half the impact in strategic planning; so naturally count them as 1/2.  
http://www.chessvariants.org/other.dir/rococo.html.
If happening to count them as 0.4 or 0.6, the calculations they appear in are not changed that much. Piece-values estimates, like mine here for Rococo or M. Nelson's in another comment of ''all messages,'' have in a way been overtaken by potential of computer analyses, if they were not already in 2004.  The now six-year-old formula for game length ought to be revisited.  In the expanded form, for clarity to compare any cv with any other cv, M, Moves expected, depends on board size, piece-type density, power density, and exchange gradient. The formula is a reason we want to get piece-type densities.

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