Author Topic: Piece Comparisons: Attack Fraction vs. Distance  (Read 26 times)

joejoyce

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Piece Comparisons: Attack Fraction vs. Distance
« on: April 07, 2018, 12:20:23 am »
Some time ago, I got interested in shatranj-style pieces and how they compared with other shatranj pieces and more modern ones. Another member here, Greg Strong, an apparently no longer active chess variant designer, David Paulowich, and I kicked around ideas for a while. From my write-up of that time: "I was looking at the total number of squares attacked by several shortrange pieces and finding them grouped in multiples of 4 when I received an email from David Paulowich containing his figures on the average number of the immediately adjacent 8 squares attacked by the pieces in several games, ranging from 2.5 in shatranj and his Shatranj Kamil X to 5.0 for my Lemurian Shatranj [FIDE is 4.0], and his conclusions. I extended his idea and combined it with my numbers" to make a chart of the percentage of squares attacked at each range for a number of short range pieces and the standard long range sliders. (The board is assumed to be big enough to show the full footprint of each piece out to the range specified.

dist    K-M   F-W    A-D    DW-AF    N    FAD    HP-Mn DWAF    Q2 M+/-M NDWAF   dist    #sq    totsq
  1       1      0.5      0        0.5         0     0.5    0.5      1          1         1        1        1       8         8
  2       0      0        0.25   0.25      0.5    0.5    0.75    0.5      0.5       1        1        2     16       24
  3       0      0         0        0           0      0       0         0         0          0        0        3     24       48
  4       0      0         0        0           0      0       0         0         0          0        0        4     32       80
FtPt     8      4        4        8           8      12      16      16       16       24      24          

K- standard King
M- Man, non-royal king
F-Ferz, moves 1 square diagonally
W- Wazir, moves 1 square orthogonally
A- Alfil, jumps 2 squares diagonally
D- Dabbaba, jumps 2 squares orthogonally
N- standard kNight
HP- High Priestess - NAF
Mn- Minister - NDW
Q2- Queen that slides only 1 or 2 squares/turn

dist - distance piece has moved from original square
#sq - number of squares at that distance only
totsq - total number of squares a piece could move to at each distance (not counting the original square)
FtPt - footprint, the total number of squares a piece could hit in a single move

Notes:

M+/-M is a 2 step "bent" Man, meaning it can change directions (indicated by the "+/-" symbol) between its first and second step. It may not return to its original square, however. In this symbology, a Q2 (2 step queen) would be M+M, as the piece can only move in a straight line and not change direction during a move.

All the pieces here are very short range, and are used in short range variants played on the chessvariants.com website. My next comment will add some 3 and 4 square ranged pieces along with the linear sliders for comparison. And a little discussion of what some of this might mean.

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joejoyce

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Re: Piece Comparisons: Attack Fraction vs. Distance
« Reply #1 on: April 07, 2018, 03:00:36 am »
dist    H-S l    Hb      Sb      L-O     P         T         X         Z     B-R      Q      dist #sq    totsq
1       0.5      0.5      0.5     0.5     1         0.5      1         1     0.5      1         1     8      8
2       0.25    0.75    0.25  0.25    0.5      0.5      1         1     0.25    0.5      2    16    24
3       0.17    0.17    0.5    0.17    0.33    0.5      0.17    1     0.17    0.33    3    24    48
4       0         0         0       0.13    0.25    0.25    0.13    0.5  0.13    0.25    4    32    80
5       0         0         0       0         0          0         0        0     0.1      0.2      5    40    120
6       0         0         0       0         0          0         0        0     0.09    0.17    6    48    168
7       0         0         0       0         0          0         0        0     0.07    0.14    7    56    264
8       0         0         0       0         0          0         0        0     0.07    0.13    8    64    328
FtPt 12      20       20      16       32       32         32      64     32      64          

H-S l - linear Hero (D+W) and Shaman (A+F)
Hb - "bent" Hero (D+/-W)
Sb - "bent" Shaman (A+/-F)
L-O - Lightningwarmachine (DW + DW) and Oliphant (AF + AF)
P - Parallel general (L/O)
T - Twisted knight (AF+/-AF)
X - fleXible knight (DW+/-DW)
Z - Zigzag general (DWAF+/-DWAF)

One more chart:

tot sq att'kd    piece&range
4 = 4x1        W and F - 1; A and D - 2
8 = 4x2        K or G - 1; [DW] and [AF] - 2
12 = 4x3      FAD - 2; linear Hero and Shaman - 3
16 = 4x4      HiP, Min, JG, Sliding general - 2; L, O - 4
20 = 4x5      bent Hero and Shaman - 3
24 = 4x6      M+/-M, NDWAF - 2
32 = 4x8      P, T, X - 4; B, R - 8
64 = 4x16    Z - 4; Q - 8

The orthogonal and diagonal pieces may be considered as pairs, with wazir and ferz being the 1st pair, then dabbabah, alfil, all the way up to rook and bishop. These pieces are all linear movers, and have exactly the same movement patterns, rotated 45 degrees. When you make a bent 2-step piece, this splits the pairs. The ortho partner becomes much stronger up close, and the diag partner becomes stronger at a distance - see the Hero-Shaman pairs, and the Lightningwarmachine-Oliphant vs fleXible-Twisted kNight pairs.

The "4xN" column just above these comments points up the difference between 2-stepped pieces with [the ability to choose] even steps compared to [forced] uneven steps. If N and range are odd, the steps are uneven. If N is odd and range is even (FAD), the piece is a single step piece with a choice of components. The king, N = 1, is a "collapsed" case, and drops out here.

If you stop the calcs for the bishop-rook pair and the queen at a distance of 4, which is the maximum I've used for "short" range pieces, their total squares attacked numbers become 16 and 32, and they don't leap. This implies that the "Mean Free Path" is the critical value factor for infinite sliders. For pieces, especially shortrange ones, that may leap at different points in their moves, the mean free path value has [greatly] reduced significance. Density approaching saturation and clustering of pieces would seem to be more determining factors here. In fact, the actual values of pieces changing as the game goes on, as the pieces in the game change, and as the board changes, are actually the only things I can be reasonably sure are true.

This is the 'cleaned up' version of what I did a decade ago. I found it interesting and potentially useful. Comments are welcome.