This chart compares the fraction of squares attacked vs. distance for selected pieces. All the FIDE pieces are here, along with the ancient "building block" pieces, and several newer pieces, most of them shortrange leapers, some of my own design.
The idea for this came from 2 sources. 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 for this chart.
Joe Joyce
| dist | K-G | F-W | A-D | W-E | N | FAD | H-M | J | Sl | Sb | Kz | 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 |
| TotSq | 8 | 4 | 4 | 8 | 8 | 12 | 16 | 16 | 16 | 24 | 24 |
| 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 |
| Tot | 12 | 20 | 20 | 16 | 32 | 32 | 32 | 64 | 32 | 64 |
Legend
In the first column, dist is distance and Tot is total squares attacked by the piece[s] in that column over the entire piece range.
K-G = king and guard
F-W = ferz and wazir
A-D = alfil and dabbabah
W-E = warmachine and elephant = DW & AF
N = knight
FAD = "war elephant"
H-M = High priestess and Minister = NAF & NDW
J = DWAF; known as the squire, jumping general, and mammoth/mastodon in various games
Sl = linear sliding general or Q2 [2-square queen]
Sb = bent sliding general
Kz = Kozune = NDWAF
H-S l = [D+W]&[A+F] linear hero and shaman
H-S b = [D+/-W] bent hero
S b = bent shaman = [A+/-F]
L-O = lightningwarmachine and oliphant = [DW+DW]&[AF+AF]
P = parallel general = [DWAF+DWAF]
T = twisted knight = [AF +/- AF]
X = flexible knight = [DW +/- DW]
Z = zigzag general = [DWAF +/- DWAF]
B-R = bishop and rook
Q = queen
What does all this mean? Darned if I know yet, but there are some interesting patterns we can look at.
| 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 | bent Sliding general, Kozune - 2 |
| 32 = 4x8 | PG, TN, XN - 4; B, R - 8 |
| 64 = 4x16 | ZZG - 4; Q - 8 |
Notes
1] If 2 pieces attack the same number of squares, and one has a shorter range than the other, that piece is more powerful. This is known in most cases, but indicates the FAD is stronger than the linear Hero and Shaman.
EDITS by author: The above statement was initially based on the values of the man [nonroyal king] and knight, which I took to be 4 and 3. Based on a value of about 3 for both of them, I no longer believe this statement of relative power is always true, merely sometimes true. The chart is a better indicator of piece power, and it's always true that the mean free path of game pieces favors infinite sliders the longer the game goes on and the longer that path gets.
2] If 2 pieces attack the same number of squares and have the same range, and one leaps, that piece is more powerful - trivial & obvious.
3] 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.
4] 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.
5] The 4xN column implies there is a piece with N = 7 that would be a moderate-range uneven stepper - at a guess, a "super-bison" with step lengths of 2 and 3.
6] Pieces with N = 9 to 15 would fill in the zeros [or zeroes, if you prefer] in the lower left of the table - again, trivial and obvious.
7] Most pieces' fractions go down as the distance goes up. The exceptions are the "interesting" pieces.
8] Sure looks like you could "plug in" fractions to build a wide range of theoretical pieces, doesn't it?
9] That super-bison [N = 7] piece, if bent, should exhibit the same "fraction behavior" as the bent hero and shaman; ie: the fraction increases and then goes back down as the distance increases.
10] 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.
11] Note number 10, right above, 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.
Interesting stuff.
I'm not a huge fan of divergent pieces (like the Orthodox pawn, or every piece from Ultima), but I wonder how two pieces with opposite movement and capture would compare.
Example: a Keen moves like a King but captures like a Queen; A Quing moves like a Queen but captures like a King. The Keen is slow, but can threaten at long range; the Quing is fast, but only threatens neighbors. Which of these units is more useful?
What about a Wazir/Knight vs a Knight/Wazir?
Think I could say with fair certainty that the Keen is a superior piece. Its attack fraction is noticeably higher than the quing's after the first square. ;-) The superior mobility of the quing does not offset the severe problem with capture. And the wazir/knight is a better piece than the N/W, for the same reason. The mWcN attacks 8 squares at range 2, vs the mNcW, which attacks only 4 squares at range 1. The mKcQ attacks 8 squares at each and every range, even though it positions itself slowly. The mQcK, even though it can move to anywhere it can "see", it only captures on 8 squares, no matter how big or small the board is. And the ability to capture without getting captured first is valuable. The quing telegraphs her punches; the keen potshoots his victims. Imagine a superlarge board with lots of pieces scattered around. Which piece is more valuable there?
Sound reasoning…but then there's also the fact that the Keen cannot move to support troops on the other side of the board without capturing. If it is unable to support from a distance (because of pieces in the way) and unable to capture-move (because all pieces are protected), then it is very slow to come to the aid of others - and the larger the board, the larger this liability. The Quing, meanwhile, can travel anywhere to offer short-range support.
Perhaps this could be similar to the Bishop/Knight dichotomy, with the Keen strong on an empty board and the Quing strong on a crowded one? Might be interesting to play a variant with both pieces to test this out.
Just out of curiosity I ran the pieces through my latest version of CAST:
Groovy. That seems to support Joe's contention that the Keen is far stronger than the Quing.
On the other hand, CAST doesn't factor in a board's current piece density (yet?). What do you think of my question above? Might a Quing be better than a Keen on a crowded board?
The current version of CAST does include my initial attempts at factoring in board density by introducing a blocking value. I've run a few scenarios with varying densities and what seems to happen is that as the density increases stepping and sliding pieces grow weaker and leapers grow stronger - which is about what I'd expect anyway.
So the Quing and Keen both diminish in value but at about the same approximate percentage rate - so while the absolute gap between them is reduced their relative strengths remain similar at just under 2:1.