I'm trying to use fairy max to play a game with 4 new pieces at the corners (the new piece is a hawk and it jumps 2 or 3 squares in any direction). The game definition is below (in fmax.ini). When I run this game black's pieces are shown in wrong locations, and the game immedietly results in a stalemate. (I have legality checking turned off, but I get errors either way). Can anyone let me know what I'm doing wrong? thanks.

// variant - [new piece at corners]
Game: fun01 # PNBRQ.....AKpnbrq.....ak # fairy
8x8
8 4 5 7 3 5 4 8
8 4 5 7 3 5 4 8
p:74 -16,24 -16,6 -15,5 -17,5
p:74  16,24 16,6 15,5 17,5
k:-1  1,34 -1,34 1,7 16,7 15,7 17,7 -1,7 -16,7 -15,7 -17,7
n:259 14,7 31,7 33,7 18,7 -14,7 -31,7 -33,7 -18,7
b:296 15,3 17,3 -15,3 -17,3
R:444 1,3 16,3 -1,3 -16,3
Q:851 1,3 16,3 15,3 17,3 -1,3 -16,3 -15,3 -17,3
H:900 -16,7 -32,7 1,7 2,7 16,7 32,7 -1,7 -2,7 -15,7 -30,7 17,7 34,7 15,7 30,7 -17,7 -34,7
 

The Mann is also used in Waterloo Chess and Amsterdam Medieval Chess (called spy) and "Chess on an Infinite Plane" and "Bulldog Chess" (called guard). I'm in a few games using a guard but no one is sure what it's worth. Fair exchange for a bishop or knight, or is it worth more than that?
 

In the section "One Set, Four Boards" the author wrote "Seems like a silly idea, but the small number of pieces means that the game won't take so many moves, and the position of the Kings means that they will never be safe -- and so there will be many short games with exciting attacks.".

This is interesting. It's one of the most extreme examples I've seen where piece values are not a good indicator of one side's advantage in chess.

Thanks for sharing.:)

How do you post a diagram? I tried to post an image of the guard in one thread, but it doesn't show immediately on the page. An icon needs to be clicked again to see the graphic.

Wow, this rule change makes a big difference to the game of chess!

The sample games were finished in 4, 7, 18, and 13 moves - each move certainly has much more influence on the game play.

I've been trying to think of ways to "add power" to the game "Chess on an Infinite Plane" without adding more or stronger pieces. This might be a good way to do it (but maybe with some limitations).

Is there anyone who would like to try such a game? I'm open to any new and innovative ideas. If you have any favorite pieces, we can try those also (but I'm looking for more than just a mix of new pieces).

The games in play for Chess on an Infinite Plane are going well so far. I'm just interested in a version which might somewhat amplify the game power a little.

Ok, Thanks Kevin. I didn't know there was a diagram maker on this site. I'm impressed that it can make diagrams with hexagonal shapes. Your diagram also appears to be an interesting start to a new game. Thanks for the info.

Thanks Kevin, that helps a lot!

Here's my first chess diagram made by HTML code (which I'm pretty sure is an illegal position):

<IMG SRC="/play/pbm/drawdiagram.php?code=RnbqkbnRpPpPpPpP32pPpPpPpPrNBQKBNr">

Your information is a big help.:)

"Chess on an Infinite Plane"

Introduction: "Chess on an Infinite Plane" is a chess variant played on a chessboard of infinite size. This version uses Fergus Duniho's innovative and abstract piece images. The game starts with additional material not just for variety, but to help make enough material available in the endgame so that one player can force a win. Since there are no borders, pieces (and in particular the king) cannot be trapped at edges or in corners. The chancellors (R+N) provide immediate ability to create long distance attacks. There are also additional pawns, including some in rearward formations (called jäger units), which typically enter play late in the midgame, and can create the threat of additional promoted pieces. There are hawks centered in the jäger units, which protect the rearmost pawns, but can leave their formations and enter into spectacular play. Each player also has two guards, which can be used as king defenders or for other strategic play. Setup: Pieces: Black and White each have the following pieces: 1 king, 1 queen, 2 chancellors, 2 rooks, 2 bishops, 2 knights, 2 guards, 2 hawks, and 24 pawns. All pieces move as in classical chess, with the "extra" three piece types moving as follows: Chancellor (C) - Moves and captures as rook + knight.
Hawk (H) - Leaps exactly 2 or 3 squares in any orthogonal or diagonal direction. The leaping move means it can jump over other pieces.
Guard (G) - Moves and captures the same as a king but is not affected by check.
Pawns play the same and promote at the same rank as in classical chess. (Using the board coordinates shown in the diagram, white pawns promote at rank 15, and black pawns promote at rank 8). Pawns can promote to chancellor, hawk, or guard in addition to queen, rook, bishop, or knight. Pawns may capture en passant with the same rules as in classical chess. There is no castling. There is no fifty-move rule. Draws can only occur from stalemate, threefold repetition, agreement, or a proven case of insufficient material to force checkmate. All other rules are the same as in classical (FIDE) chess. The Chessboard: 1. Board for OTB Play: A playing area should be setup with at least 22 ranks and 20 files. Ensure provisions are available to expand the board if play requires. If this becomes inconvenient due to far-away pieces, a display board is used to indicate the location of remote pieces. If there is interesting play in small but remote areas, other playing areas can be labeled and used separately from the main board. 2. Diagram for Online Play: A chess diagram is used to indicate the position of pieces either after each move by white, or each move by black. The diagram should include 22 ranks and 20 files. If any pieces are moved outside of this area, the diagram is expanded or notes are shared to indicate the location of far-away pieces.  If there is interesting play in small but remote areas, other diagrams can be used to show piece positions separately from the main diagram. Ranks are numbered just as in classical chess; ranks 16 and higher are deeper into black's side; ranks 7 and less are deeper into white's side. Files visible in the initial diagram are labeled a - t. From white's view, they continue to the right u - z, and then aa,ab,ac, and so on. The file immediately to the left of "a" is "phi" ("φ"). Files further to the left are -a,-b,-c, and so on to -z, and then -aa, -ab, -ac, and so on. For convenience, the "φ" file can be written "0". (The zero file = "φ" = "0") Parenthesis are used around each square identification. Some examples of move notations are as follows (view from white): 1) A rook moving to the left from e8 to a8: R(e8)-(a8) or R(a8) (With a classical board, this would be "Ra8") 2) A rook moving to the left one square from the a8 square: R(a8)-(φ8) or R(φ8) or R(08) 3) A rook moving to the left three squares from the a8 square: R(a8)-(-b8) or R(-b8) 4) A rook moving vertically three squares from e8 farther back into white: R(e8)-(e5) or R(e5) (The game can also be played using numerical coordinates for both files and ranks, as is usually played at chess.com) Notes: Other sub-variants exist including: 1) Chess on an Infinite Plane with Huygens Option - The huygens is a piece which leaps 5, 7, 11, 13, and all larger prime numbers of squares in orthogonal directions. The huygens are initially located to protect one of the pawns in the jäger units, thus, making it safer for the hawks to leave their positions and join other battles. (as a side effect, the huygens may ensure that this variant of chess will never be "solved" by computers, as the complete set of prime numbers is itself unknown). 2) Formation Chess/Infinite Plane - The Battle of Kadesh - Each player starts with a large number of knights (usually twelve or more). The knights can join into 2x2 formations, gaining the ability to move together as a queen. The group must stop moving when at least one of the members encounters an occupied square. A group can capture an opponent’s piece, but only one and not more than one per move. If the group falls apart, or one member is captured, the remainder play again as knights. (This hybrid version was invented by EvertVB). Thanks to Fergus Duniho for making Diagram Designer available, and also the abstract piece collection. Also thanks to Kevin Pacey for helping with Diagram Designer. by Vickalan. I will also be happy to consider other variations of the rules, or to play with other pieces. Please contact me if interested in playing a game.

Thanks Kevin, you've been a big help! I just added the game to a new thread. I was even able to switch the square colors so the queens are now on her own color. Fergus Duniho's abstract (and cubist) piece images were also very helpful to make the chess diagram have a nice appearance.

I have many questions about this game, which I hope people on these forums might be able to answer. (For example what are some example of pieces required to checkmate on an infinite plane?)

And I also hope someone will play me in a game.

Thanks again!

I'm now posting the rules for a sub-variant of "Chess on an Infinite Plane:

"Chess on an Infinite Plane with Huygens Option"

Introduction: "Chess on an Infinite Plane with Huygens Option" is a chess variant played on a chessboard of infinite size. This version uses Fergus Duniho's innovative and abstract piece images. The game includes two huygens for each color. Huygens can jump 5, 7, 11 and other prime numbers of squares in orthogonal directions (i.e. 5, 7, 11, 13, 17, 19, 23, 29, 31...). Since it jumps in prime numbers, other jumpers such as the knight and hawk, need to make an inefficient maneuver when pursuing a huygens, causing wasted moves. The huygens do not add a lot of strength to the army, but may have some effect on play. They are initially located to protect one of the pawns in each of the rearward formations (jäger units). Thus, it is safer for the hawks to leave their positions and join other battles. (The huygens is named after Christiaan Huygens, a Dutch mathematician who studied the rings of Saturn and invented the pendulum clock). As with the base version of this game, other new pieces are included not just for variety, but to help make enough material available in the endgame so that one player can force a win. Since there are no borders, the king cannot be trapped in a corner. The chancellors (R+N) provide immediate ability to create long distance attacks. There are also additional pawns, including those in the jäger units, which can create a threat of additional promoted pieces. A hawk is centered in each jäger unit, initially protecting the rearmost pawns, but can leave its formation and enter into spectacular play. Each player also has two guards, which can be used as king defenders or for other strategic play. Setup: Pieces: Black and White each have the following pieces: 1 king, 1 queen, 2 chancellors, 2 rooks, 2 bishops, 2 knights, 2 guards, 2 hawks, 2 huygens, and 22 pawns. All pieces move as in classical chess, with the "extra" four piece types moving as follows: Chancellor - Moves and captures as rook + knight.
Hawk - Leaps exactly 2 or 3 squares in any orthogonal or diagonal direction. The leaping move means it can jump over other pieces.
Guard - Moves and captures the same as a king but is not affected by check.
Huygens - Jumps 5, 7, 11 and other prime numbers of squares in orthogonal directions (5, 7, 11, 13, 17, 19, 23, 29...).
Pawns play the same and promote at the same rank as in classical chess. (Using the board coordinates shown in the diagram, white pawns promote at rank 14, and black pawns promote at rank 7). Pawns can promote to chancellor, hawk, or guard in addition to queen, rook, bishop, or knight. Pawns may capture en passant with the same rules as in classical chess. There is no castling. There is no fifty-move rule. Draws can only occur from stalemate, threefold repetition, agreement, or a proven case of insufficient material to force checkmate. All other rules are the same as in classical (FIDE) chess. The Chessboard: 1. Board for OTB Play: A playing area should be setup with at least 22 ranks and 20 files. Ensure provisions are available to expand the board if play requires. If this becomes inconvenient due to far-away pieces, a display board is used to indicate the location of remote pieces. If there is interesting play in small but remote areas, other playing areas can be labeled and used separately from the main board. 2. Diagram for Online Play: A chess diagram is used to indicate the position of pieces either after each move by white, or each move by black. The diagram should include 22 ranks and 20 files. If any pieces are moved outside of this area, the diagram is expanded or notes are shared to indicate the location of far-away pieces.  If there is interesting play in small but remote areas, other diagrams can be used to show piece positions separately from the main diagram. Ranks are numbered just as in classical chess; ranks 15 and higher are deeper into black's side; ranks 6 and less are deeper into white's side. Files visible in the initial diagram are labeled a - t. From white's view, they continue to the right u - z, and then aa,ab,ac, and so on. The file immediately to the left of "a" is "phi" ("φ"). Files further to the left are -a,-b,-c, and so on to -z, and then -aa, -ab, -ac, and so on. For convenience, the "φ" file can be written "0". (The zero file = "φ" = "0") Parenthesis are used around each square identification. Some examples of move notations are as follows (view from white): 1) A rook moving to the left from e8 to a8: R(e8)-(a8) or R(a8) (With a classical board, this would be "Ra8") 2) A rook moving to the left one square from the a8 square: R(a8)-(φ8) or R(φ8) or R(08) 3) A rook moving to the left three squares from the a8 square: R(a8)-(-b8) or R(-b8) 4) A rook moving vertically three squares from e8 farther back into white: R(e8)-(e5) or R(e5) (The game can also be played using numerical coordinates for both files and ranks, as is usually played at chess.com) Notes: Thanks to Fergus Duniho for making Diagram Designer available, and also the abstract piece collection. Also thanks to Kevin Pacey for helping with Diagram Designer. by Vickalan. I will also be happy to consider other variations of the rules, or to play with other pieces. Please contact me if interested in playing a game.

Thanks Aurelian! I read about your variant Enep. I like the short and succint name. I know it means Enhanced Knight Exrtra Pawn, but the name also sounds Norse or Scandanavian (I think that might be where gnomes came from).

I also like the simplicity of the game. Chess can already be very complicated, so only a minor rule change is needed to make a good variant. Your variant sounds intriguing. When I finish one or two other games (I'm usually playing about 5-7 games concurrently, one move per day), I might want to play Enep.:)

I also submitted a variant (Chess on an Infinite Plane using classical chess pieces) on 2/16, but haven't seen or heard anything yet.

I'm interested in seeing yours:)

Hi Aurelian,I'm not sure why my name shows up with question marks. I see it that way too.

Btw, did you finish any more work with Enap? Are you testing it by playing for real, or by testing it some other way?

:)

The Huygens Chess Piece

Btw, here is an example game with a joker (with images for the full game): Bulldog chess-angel joker game .

OK, thanks for answering. In a few weeks I might try ChessV.

I noticed the package includes Fairy-Max. Does ChessV and Fairy-Max share any code, or are they distinct programs?

Regards, :)

Thanks Fergus,

It looks like there's a big set of Alfaerie graphics. I was looking for a war machine icon that I saw somewhere, and i was able to find it. In fact, there's a few versions. I appreciate it.

Thanks! :)

Infinite Chess on YouTube

http://www.chessvariants.com/d.betza/chessvar/pieces/index.html

That's funny. Pancake numbers can come from two ways: stacking them and cutting them! It's making me hungry.

I'll eat and then enjoy web-surfing to learn more about some of the other number sequences you listed!

At one chess-playing forum, there are at least two games of my version "Chess on an Infinite Plane" being played. One of them is a team competition (3 players vs. 3 players). Another game in-progress is called "Chess on an Infinite Plane with Huygens Option" which is the same as Trappist-1 (a game described here at CVP).

Also, the huygens has received attention among the math community. An example is at the StackExchange Talk Forum, and also at the Talk Page of Joel Hamkins "A position in infinite chess with game value ω^4".

Please let me know if you'd like me to show any specific links.

Thanks as always Fergus,

Hi Fergus, here are links to the four examples:

Chess on an Infinite Plane with Huygens Option

Chess on an Infinite Plane - Team Competition

Question at Stack Exchange about the Huygens

A position in infinite chess with game value omega to the ω^4 (huygens mentioned in discussion)

Thanks as always for your support,

Hi Aurelian and Greg, just to throw in my opinion, I don't think draws are a problem in chess. It's just a 3rd possible outcome.

Games can still be intense and filled with interesting and spectacular play. Here's a good example of an interesting game that ended in draw:

Magnus Carlsen (age 13) vs. Kasparov

See Kasparov shaking his head.

Greg: Btw, I've been enjoying your ChessV program. Excellent work!

Thanks for the update. Also, I think it's usually better to work on one game, and release it, rather than two at once.

Also (btw), I think games are fine even if they aren't in ChessV.

ChessV is awesome because you can play games against it, as well as see your moves analyzed. On the other-hand, if you play chess on-line with long time-control, you may always be worried your opponent might be using ChessV to help.

So some games not being in ChessV is good.

Also, we have to keep Greg busy with programming!!

One of the last places where he left comments is here:

<< Mullers Short-Range Leaper Law >>

It's a short but useful thread to explain how leaping piece values can be estimated with a formula. I also did some work to try to "disprove or prove" the formula, and I found it to be very accurate (comparing the value of a guard and knight to that of a bishop for example). The formula now takes the name "Muller's Short-Range Leaper Law".

Btw, I like your idea of playing cyborg chess. It is currently being used in one game of "Chess on an Infinite Plane" but unfortunatelly, the computers don't help much because the game has not been well simulated by any code.

Thank Fergus,

(Q1) Playing eight games concurently with time-control (4 moves/week) is playing an average of 4.6 moves per day. Obviously it would be good to learn all the games rules before-hand. That's a bit much to learn "on-the-fly".

(Q2) Thanks for the answer.

(Q3) Thanks. Someone on one of these forums once mentioned cyborg chess, and I wasn't sure if that was the format. A format (human + computer) vs (human + computer) could be fun too. I just wanted to know for sure (better to know the rules before the game starts rather than in the middle of the game).

Playing eight games concurently is a bit much for me with my other commitments, but sounds fun. Good luck!!

Thanks Greg for the clarifications. I'll sit this tournament out, but it does look like fun.

Some time I would like to play an engine-assisted game. Might be interesting for a game on a large board, like chess on a 12x12 board. I've always been curious if a flank attack (attack from behind the king) can be well-executed. It takes extra moves to get there, but the defense from that angle might be weaker.

Have fun in the tournament.:)

Thanks for the info. I did suspect that this game for a few reasons is pressing the limits of what variant chess engines can play. (Games will always be ahead of engines, if for no other reason that nobody makes an engine before the game. Plus, I'm sure programming is not a fast easy task, especially for chess).

HGMuller: Keep us up-to-date when HaChu is released!

Greg: I didn't know that a link to ChessV is in the "Play" menu. In an internal (CVP) search for ChessV, results for both pages come up (plus less related pages). Users will have a 50/50 chance of finding a useful page on the first try.

About "Chess and a Half". This board is 1.5x as wide as a normal chessboard - is that why it's "Chess and Half"? At this size it has 144 squares, so has 144/64 = 2.25 times as many squares. Not counting the new pieces, I think this is massivelly more complex than ordinary chess. Btw: I like how you kept the tradition "queen starts on her own color", and I also like how each of the two knight face in opposite directions. Even late in the game, when there's just one knight of a color, you can know which one survived.

Even the minutiae appears to be well-thought out. Great work!

Maybe rules 5, 7, and 8 should just be eliminated. Then the game would have the quality that the graphics are clean, crisp, and interesting, and the rules are simple and concise.

Only the game is mathematically complex (due to large size and extra pieces).

I believe it just means that that the Cat and Star-Cat can capture pieces within its intermediate move pattern. So for example a Star Cat on d4 can capture pieces on e5, f6, and g7 (in one move).

But being optional is interesting. I think it would be uncommon for one of the cats to not want to capture a piece while jumping over it. But there may be situations, for example to leave an opponent's pawn in place if the pawn is blocking a slider from attacking a more valuable piece.

I just realized it also probably means that Fairy-Max can't play this. And my estimate for the value of a Star-Cat may be low, due to multi-capture ability.

But since it's not a long distance mover, I don't think it has overwhelming power, and would still play perfectly in this game.

Kevin, thanks for your information about piece values. Your comment about a queen being more powerful on larger boards is interesting, and a good point.

A few months ago I used HGMuller's Fairy-Max to play a bunch of games to estimate the value of a guard (to confirm or dispute HGMuller's earlier work). I did it on a 10x8 board, and used the 4 extra squares for pieces to add guards and/or minor pieces. From these tests, I found that a guard is very nearly equal to a bishop, and slightly superior to a knight. (let me know if anyone would like more info about test details).

Nicolino, I hope you don't get rid of any of the Star Cats just because of their power.

This is a big board, so there's room for a few powerful pieces. Opponents have plenty of room to maneuver, create defenses, avoid attacks, and to create counter-attacks.

Btw, even though HGMuller's formula applies only to short-range leapers, I suspect it still might be good for pieces which jump up to 3 squares away when played on a large board.

Long range jumpers on a small board have the problem that long jumps go "off the board". But on a large board, attack points are more likely to useful, possibly helping the formula to remain accurate.

Maybe within a week or two, I'll calculate the "power density" of this game. It's a method I've used to determine the relative power of all pieces as a ratio to board size, and compare it with other games. I believe it's a useful measure of how "dynamic" the play can be expected to be. I'll update here if/when I finish the calculation.

Joe Joyce (and others interested):

Here's the details of my tests to estimate the value of a guard. Using Fairy-Max, I set up games on a 10x8 board. Pieces on each side placed as: RXNBQKBNXR.

X represents a variable piece, which was always different between black and white.
For example, black might have X = two knights and white has X = two guards (or vice-versa). Then I ran games (engine vs. engine with long time control) and kept track of scores. In all cases the armies were switched (W/B) so that half the games were each way. (scores are based on win = 1, draw = 0.5, loss = 0).

First, one problem in setting up a test is that Fairy-Max requires all pieces to have an assigned value, and going into a test the assigned value of a guard is unknown.

The first approximatelly 200 games were to "hunt" for the guard's best assigned value. I found guards play best with an assigned value between 300 and 375. Within this range games were not sensitive to their assigned value. But if the assigned value is lower (tested 250) or higher (tested 400) then guards don't help their side to play well (and these results are discarded from the final summary).

An Overall Summary of only games where guards have this "optimal" assigned value (300, 350, or 375):

asymmetry: [2 guards vs. 2 bishops]
guards win (score) = 40/80 = 50.0%

asymmetry: [2 guards vs. 2 knights]
guards win (score) = 46/80 = 57.5%

asymmetry: [2 guards vs. 1 bishop and 1 knight]
guards win (score) = 101/200 = 50.5%

Conclusion (on a 10x8 board, with other FIDE chess pieces):
A guard's value is:
1) equal to a bishop.
2) slightly superior to a knight.

If any questions or comments feel free to leave a message.

HGMuller's formula is interesting, and it's good to see there's a way to expand its scope by using ELC. Muller presented the formula as:

  value = 33*ELC + (33*ELC)*(33*ELC)/1584)

I prefer it a little more as:

  value = 33*ELC + 0.6875*(ELC)^2

In this form the variable occurs once for its linear component (33xELC) and once for its polynomial component (0.6875*(ELC)^2).

But this is just a minor stylistic preference. More generally, it's very interesting that a rather simple formula can be quite accurate for a wide range of leapers. Not sure if there's any future possibility (by Muller or others) to ammend it for longer range leapers. Of course, work like this always requires a lot of engine analysis, and follow-up evaluation of the data.

Good work on the formula!

Btw, do we know that Lasker's estimate of a king's value in an endgame (4) might not be too far off? The study that I did (which basically just confirmed previous work by Muller) was to estimate the value of a guard/commoner for the entirety of a chess-game (10x8 board).

From my study alone, I cannot dispute Lasker's estimate. As far as I know, it might be possible that a non-royal king might be worth a little more on an 8x8 board, and yet a little more in an end-game only situation.

As mentioned before, I completed a calculation of the "Material Power Density" for "Chess and a Half".

First, to explain what this ratio is - it is simply the sum of the value of all chess pieces on the board, divided by the playing area (number of squares).

This paramater may give some prediction of the style of play that can be expected from a game. Higher density usually means pieces have more attacking possibilities, and games can enter into dynamic attacks quickly - requiring fast responses from the other side. A lower density means that the opening development may last longer, with a stronger focus (for example) on the placement of pawns.

From lowest to highest is the "Material Power Density" of several games I've analyzed, with "Chess and a Half" now added in the list.

                     Board         Number   Piece    Power    Ratio to
                     Size (sqs)    of Pcs.  Density  Density  Classical Chess
———————————————————— ————————————  ———————  ———————  ———————  ———————————————
Classical Chess      8 x 8 =  64   32       50%      1.34     1.00
Infinite Plane(1)            360   76       21%      0.56     0.42
A Chess Endgame(2)   8 x 8 =  64    7      (11%)     0.64     0.48
Chess and a Half     12 x 12= 144  48       33%      1.04     0.78
Bulldog Legacy Chess 10 x 8 = 80   36       45%      1.10     0.82
Bulldog Chess        10 x 8 = 80   40       50%      1.31     0.98
Bulldog Chess(Witch) 10 x 8 = 80   40       50%      1.35     1.01
Janus Chess          10 x 8 = 80   40       50%      1.53     1.14
Capablanca Chess     10 x 8 = 80   40       50%      1.55     1.16
Seirawan Chess(3)    8 x 8 =  64   36      [56%]     1.88     1.40
Musketeer Chess(4)   8 x 8 =  64   36      [56%]     1.88     1.40
Chu Shogi            12 x 12= 144  92      [64%]     1.91     1.43
Amsterdam Chess      11 x 8 = 88   44       50%      1.95     1.46
Waterloo Chess       10 x 10= 100  60      [60%]     2.72     2.03

(1) - Chess on an Infinite Plane: Play is assumed to be in an 18 x 20 range. This is the horizontal span and 2 ranks less than the vertical span of the outermost pieces (starting position). Little play is presumed to take place in the outermost ranks except for pieces moving inward.
(2) - A chess endgame: A sample 7-piece ending with KQRR vs. KQR.
(3) - Seirawan and Musketeer Chess: Data assumes all pieces are on board. Some pieces are introduced early or later in the game.
(4) - Musketeer Chess: Assumes game with archbishop and chancellor played as the new pieces.

Chess and Half (4th in list) is interesting in that is has a very low piece density (48 pieces on 144 squares), and the material density is 78% compared to that of chess (100%). This is in the range of games that have become my favorite to play. They usually feature an opening with time for tactical development, rather than the players always reacting only to exact threats from the opponent.

I've seen plenty of variants where the dynamics suffer from too much power concentrated into a normal size chess board.  I would really like to try this game sometime, and as mentioned, if it's ever withing the scope of a chess engine, see if a human can win, or how two engines do against each other.

I agree that the concept of power density involves some assumptions that might cause the value to be an approximation. As you mentioned, it does assume that pieces have fixed values, even with a different mix of pieces, and different board sizes.

I do believe that if every game has a mix of pieces (as they do), such errors would tend to cancel out. For example, as board size changes, some pieces might gain slightly in value, while others lose value.

The only way to overcome such possible errors is if there was an accurate way to identify a piece's value based on the specific board size. I'm not aware of any work that has been completed to do this for a range of board sizes. At best, maybe we know the rough difference in value of a few pieces when they go from an 8x8 to 10x8 board. To my knowledge, there is no piece which has its value altered by such a large amount that it would render power density as grossly innacurate.

I believe the biggest error currently found in the power density table is the data for Chess on an Infinite Plane . Here a board size of 18 x 20 was assumed because it's the approximate span of pieces in the starting position. But the bulk of the dynamics in actual play is usually found in a much smaller area.

In fact, the tendency of pieces to try to "fight for the center" might be a phenomenon seen in all games, so the stated "board sizes" themselves might be an opportunity for refinement. But I'm reluctant to complicate the formula based only on conjecture. As we learn more about piece valuations for variant chess, I certainly can plan to refine the formula when there is merit to do so. For now, it's based on the theory that "Simple and approximate" is better than "Complex with speculation".

There may have been some differences in the test setup, for example HGM's test may have been on an 8x8 board where one side had the bishops removed altogether, and replaced by guards (commoners).

Maybe an army likes to have bishops and guards working together, but if the bishops are removed and replaced with guards, the army is slightly worse than the one that still has the bishops? Maybe HGM will shed some light on his specific test set-up, or the scope of his conclusion.

Note all tests by me were done with HGM's Fairy-Max engine, which is very well-designed for playing these types of pieces.

Oh Thanks. Sorry I didn't notice. Congratulations to all the players!

I'll probably check back occasionally - this is interesting!