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Comments by HGMuller

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Game Courier Logs. View the logs of games played on Game Courier.[All Comments] [Add Comment or Rating]
H. G. Muller wrote on Sat, Mar 2, 2024 09:14 AM UTC in reply to Jean-Louis Cazaux from Fri Mar 1 08:29 PM:

@HG: it is really a mess to recover this!

Yes, I am very sorry about this. I do know this site should not be used as a alpha-test facility. But the include file did contain a bug that had to be fixed, and I was dealing with some apparently illogical and therefore unpredictable behavior of GAME code itself.

@Fergus: wouldn't it be better to include the application of the PHP intval operator in the implementation of piececount itself, so that it would return the expected 0 for pieces that are not on the board?

It might also be useful to apply intval to the operands of arithmetic operators such as +. (Or is that very detrimental, performance-wise?) I was really baffled by that 1 + undefined = 0.


Play-test applet for chess variants. Applet you can play your own variant against.[All Comments] [Add Comment or Rating]
💡📝H. G. Muller wrote on Sat, Mar 2, 2024 03:33 PM UTC in reply to H. G. Muller from Sat Feb 24 10:07 PM:

I have now upgraded the content of this article to the new version, which also allows you to define morph and captureMatrix by filling the corresponding cells with pieces from the table. Other differences with the previous version is that under "Specify more rules" you can also define what are the royal types, and select a spell.

Note, however, that the generated GAME code does not support capture matrices or spells yet, and morphing only to other piece types. (No confinement or win squares.)


💡📝H. G. Muller wrote on Sat, Mar 2, 2024 09:49 PM UTC:

The spells do not work yet in the PTA itself, because this is still using betza.js instead of betzaNew.js, and spells are only supported in the latter. Bot they do occur in the spell description that appears when pressing the Show HTML button, so this could be used with betzaNew.js. I will upgrade the PTA to using betzaNew.js shortly.

What is worse is that the interface doesn't allow you to specify spellZone and blastZone yet.


Constabulary Chess. Chess on an 8x10 board with compound piece types added. (8x10, Cells: 80) [All Comments] [Add Comment or Rating]
H. G. Muller wrote on Sun, Mar 3, 2024 06:33 AM UTC:

If you already use the name Kirin for one of the pieces, it would be better to use Phoenix for the WA.

'Warmachinewazir' might be a good mnemonic name for an image, but it is a horrible name for a piece in a game. I think 'Woody Rook' would be better, despite its high Waffle content. Or something like (Seige-)Tower.


H. G. Muller wrote on Mon, Mar 4, 2024 06:43 AM UTC in reply to Kevin Pacey from Sun Mar 3 04:25 PM:

Yes, you are in a bad mood; people are only offering advice for how to improve your article. In English one doesn't form longer words by stringing together other words, as one does in German or Dutch, so warmachinewazir is awkward for that reason alone. If you have aversion to the word Dabbaba, it is inconsistent that you use that very same word in the next sentence. I might have done some editing on the fairy-piece wiki, but I am sure Modern Elephant is not my invention. (The history tab on the Wikipedia article shows that it was in fact added in June 2014, with Shako as reference, together with a host of other pieces, by an anonymous user logging in from an IP address that maps to Spain.) In my mind Elephant is synonymous to Alfil, and Alfaerie pieces strengthen that impression by using an Elephant as symbol for it. Just as much as Tower could be understood to mean Rook, because the Rook is depicted as a castle tower. (Which is why I proposed Seige Tower; I still think this would be an excellent name for the WD, because it is very Rook-like, and seige tower would be a valid translation of the Persian word Dabbaba).


H. G. Muller wrote on Mon, Mar 4, 2024 10:10 AM UTC in reply to adella hardy from 12:38 AM:

What doesn't work? It would be more productive to fix that than to revert to the previous version. The new version should not really be different from the previous one, as long as you don't click the 'morphing and confining' link under the piece table.


Home page of The Chess Variant Pages. Homepage of The Chess Variant Pages.[All Comments] [Add Comment or Rating]
H. G. Muller wrote on Mon, Mar 4, 2024 10:43 AM UTC in reply to Fergus Duniho from Sun Mar 3 09:54 PM:

The double undelining of links cause a larger line spacing with the next line, which is a bit ugly.


Accelerated Constable-Spiel. Chess on a 16x8 board with an assortment of pieces. (16x8, Cells: 128) [All Comments] [Add Comment or Rating]
H. G. Muller wrote on Mon, Mar 4, 2024 03:41 PM UTC in reply to Kevin Pacey from Sun Mar 3 03:46 AM:

Very nice. On such a wide board the Bishop will be significantly more valuable than a Knight or any of the other 8-target short-range leapers. And the Dragon Horse will benefit similarly. This gives it an almost ideal value spectrum: 16 Pawns, 8 Knight-class pieces, 4 Bishop/Rook-class pieces, 1 Queen and two pieces between Rook and Queen.


Betza notation (extended). The powerful XBetza extension to Betza's funny notation.[All Comments] [Add Comment or Rating]
💡📝H. G. Muller wrote on Mon, Mar 4, 2024 05:32 PM UTC in reply to Bob Greenwade from 05:05 PM:

would pO (with other appropriate modifiers, of course) allow castling with an intervening piece?

Sort of. I chose that as the notation of 'Fast Castling', which allows the King to jump over arbitrarily many pieces or attacked squares. The Rook would then move to the King's square of origin, though. So it is not normal castling that can hop.

The O atom is treated somewhat differently from other atoms. (E.g. the range indicator specifies the exact number of King steps, not the maximum as ranges normally do.) It had to, for making the combinations useful.


About Game Courier. Web-based system for playing many different variants by email or in real-time.[All Comments] [Add Comment or Rating]
H. G. Muller wrote on Tue, Mar 5, 2024 07:53 AM UTC in reply to Fergus Duniho from 01:51 AM:

Is this fixable?

It is easily fixable, and I think fixing it is long overdue, as it looks very amateurish.

The fix is to display the piece images not as content of the table cells, but as background. So instead of something like

<td style="width:50px;height:50px"><img src="xxx"/></td>

you would generate something like

<td style="width:50px;height:50px;background-repeat:no-repeat;background-position:center center;background-image=url(\"xxx\")"></td>

(where the extensive style specification can of course be achieved through a CSS class). You can then highlight through a border with negative margins. This is how I got rid of these annoying expansion/contraction effects in the Interactive Diagram. (But there I highlight by small symbol images displayed as content of the cell.)

If the cell backgrounds have transparent parts, you see either the specified cell background color there. Or, when no valid color is defined, the background of the surrounding (<table>) element. So you can still use full-board images as background.

The problem is that browsers refuse to fit 50x50 images in 50x50 cells, because they align the bottom of the image with the text line it is on, and text lines need extra space below them for the 'true descenders' on letters like j, p and q. So they always leave some space below the image, even if there is no text at all in the cell. How much depends on the font size that currently applies, but it is not possible to set that to 0.


Play Chess Variants with Jocly. Missing description[All Comments] [Add Comment or Rating]
H. G. Muller wrote on Tue, Mar 5, 2024 12:35 PM UTC in reply to François Houdebert from 08:25 AM:

How about using the Prince fairy sprite for the Jeweled General?

Sorry I had so little time to spend on Jocly; there was an emergency at the talkchess.com forum, for which the hosting was terminated per March 1, and for which we had to set up a new server. We managed to do that in time, but there still are some imperfections that have to be ironed out. But I will try to update the shogi picto sprites.


Hybrid Decimal Chess. Chess on a 10x10 board with unusual compound pieces included. (10x10, Cells: 100) [All Comments] [Add Comment or Rating]
H. G. Muller wrote on Tue, Mar 5, 2024 03:39 PM UTC in reply to Kevin Pacey from 03:03 PM:

The generated GAME code by default applies the 50-move rule. You can change that by appending a line

set rulemoves N;

to the Pre-Game code, where N is the number of half-moves. (So by default N=100.)

I believe at one time there was a rule that in cases where the theoretical distance to forced mate was known, you got 1.5 times that number of moves, if this was more than 50. They stopped doing that when solutions for larger end-games were calculated, and sometimes would require hundreds of moves to force the win.

Whether it pays to increase the number of moves where a draw can be claimed depends on how important the affected end-games are for the variant. In general drawishness is seen as a bad thing, so you don't want to increase the number of draws by making frequently occurring endings that could have easily been won in a larger number of moves draws instead. OTOH you don't want to force players to play exhaustingly long games when their opponent is too stubborn to accept he cannot win.

On a larger board games are naturally longer to begin with, so one can assume these are played by patient players, who do not mind to spend a little more time in the late end-game. In Team-Mate Chess I increased the limit to 64 moves, even though it is just 8x8, because the 3-vs-1 end-games you could get in were the crux of the design. So I did not want those to be spoiled by too tight a limit. In orthodox Chess most games would end in KQK, and that can be won way faster than 50 moves. I am pretty sure the choice for a 50-move rule was based on 1.5 times the number of moves needed foor KBNK.


Play Chess Variants with Jocly. Missing description[All Comments] [Add Comment or Rating]
H. G. Muller wrote on Tue, Mar 5, 2024 09:55 PM UTC in reply to François Houdebert from 08:25 AM:

OK, I pushed the reworked sprites to pullreq, under the name shogi-picto-sprites.png.


Janggi - 장기 - Korean Chess. The variant of chess played in Korea. (9x10, Cells: 90) (Recognized!)[All Comments] [Add Comment or Rating]
H. G. Muller wrote on Wed, Mar 6, 2024 07:28 AM UTC in reply to Kevin Pacey from 02:55 AM:

An N-move rule is in general a good thing to have, but so is a clock, and historic rules usually don't specify anything about that either. There do exist variants without irreversible moves, though, (e.g. Shogi), and there it becomes very difficult to formulate a sensible rule for terminating no-progress games. Even in Chess the criterion is not irreversibility, but progress. (Castling does not reset the count.) Pawn advance can be seen as progress towards promotion, but if a promoting piece does not move irreversibly, moving it back and forth obviously would not. Variants that forbid perpetual chasing can often have very long stretches of checks without repeating, (which would be allowed) with which they could delay the loss. In combination with an N-move rule with any reasonable N this would make the ban on perpetual checking ineffective. And games that are theoretical wins could take thousands of moves to reach it, because every move that progresses towards a promotion is followed by some 100 checks before the losing side runs out of non-repeating checks.

A rule that I proposed to solve this was to limit the number of consecutive checks to, say, 3, unless you end the sequence with a capture. Then you can still make such sequences when they serve a purpose, but the delay you can achieve with the checks is limited. And checks+evasions can then be discounted for the N-move rule, without the total number of moves getting unacceptably high.

An alternative would be that the player that refuses a draw after a reasonable number of moves will lose when he does not manage to win within some unreasonably large number of moves. Then being forced to play the unreasonably large number of moves by a stubborn opponent who does not want to recognize the game-theoretical outcome is at least rewarded.


@ Gerd Degens[All Comments] [Add Comment or Rating]
H. G. Muller wrote on Wed, Mar 6, 2024 02:08 PM UTC in reply to Gerd Degens from 01:46 PM:

Looking at the Diagram definition in the Page Source I notice this:

  • It says royal=K , but it really should say royal=KJLMOT .
  • The non-royal Queen is defined twice, in the same way
  • The promoChoice parameter isn't useful without zone or morphing to *, and could have been omitted.

The morph parameters seem to do what you describe in the text.


Play Chess Variants with Jocly. Missing description[All Comments] [Add Comment or Rating]
H. G. Muller wrote on Wed, Mar 6, 2024 02:56 PM UTC in reply to François Houdebert from 02:11 PM:

It is always a dilemma whether one should pick the representation based on the move or on the name. I tend to go for the move; names you can in principle do without. E.g. in Elven Chess the Warlock is represented by a Lion, not by a Wizard symbol.

A move-inspired pictogram representation for Tori Shogi would use the King for Phoenix, Leopard for Crane and Elephant for Falcon. The Goose, Pheasants, Eagle and Quails are unique to Tori Shogi. We happen to have a good Eagle pictogram, but for the others 'anything flies'.

Name-based pictograms offer the challenge to make many different distinguishable and recognizable birds. That is hard, as some of the birds would look very similar. The sprites you used are all very much out of style with the Eagle. We do have a Phoenix and a Falcon in the fairy-sprites, and as far as I am concerned the Stork we have there is indistinguishable from a Crane (other than by coloring, which the pictogram does not show). That also leaves Goose, Pheasant and Quails. For the Quails this one would be in style. (But is seems to be copyrighted...) For a Pheasant this one is the closest I could find. (But it is too wide; to match it in scale we would have to cut off the tail feathers. But a long tail is the main charcteristic for a pheasant; otherwise it looks like an ordinary cock. This one has the right style, but I don't recognize it as a pheasant.) This seems a good Goose. (But again a copyright issue.) Perhaps something based on the top-right drawing here (omitting the colors).

[Edit] This could also make a good Quail.

And the head of this one could be a good basis. I recognize it as a pheasant even without the tail.


H. G. Muller wrote on Wed, Mar 6, 2024 07:09 PM UTC in reply to François Houdebert from 04:43 PM:

The Goose is excellent. For the Pheasant and Quail, I made these:


H. G. Muller wrote on Wed, Mar 6, 2024 07:18 PM UTC in reply to François Houdebert from 07:16 PM:

I think so. The Stork, Elephant, Leopard and King are all in the fairy-sprites.

There only is the matter of diversifying the Quail. It is probably best to have the Left Quail look left for both colors. Then the pieces are flip-invariant. I does mean that in player A view the white Quails look inward, the black Quails outward.


H. G. Muller wrote on Wed, Mar 6, 2024 09:21 PM UTC in reply to H. G. Muller from 07:18 PM:

I made the Tori sprites, and pushed those to pullreq. I did demagnify the Goose somewhat, as it seemed unreasonably large for such a weak piece.

To have the white Quails look towards the center, the Right Quail had to look to the left. I don't know if that should be considered confusing.


Lumberjack. Pieces move depending on the column they are on. (8x8, Cells: 64) [All Comments] [Add Comment or Rating]
H. G. Muller wrote on Fri, Mar 8, 2024 06:40 AM UTC in reply to Bob Greenwade from Thu Mar 7 06:54 PM:

The recommended way for making Querquisite-like pieces is morphing. This game is in fact a special version of Avatar Chess, with a more regular morphing patters. Except that in Avatar the King was not morphing, because it was too difficult to checkmate it in thet case.

This seems to be solved here by making checking the King the goal here, rather than checkmating it? Wouldn't that be to easy?

The Interactive Diagram doesn't support checking as a winning condition. Currently the AI is not even aware that it is in check. (Except sometimes in nodes that would otherwise be leaves, when the last-moved piece delivers the check.) Perhaps I should change that, at least in the non-leaf nodes. (It is a rather large computational effort to detect every check and doing it in leave nodes would severely reduce the search speed.) Then for games where checking is the goal it could declare game end when it detects one.

In fact there could be two flavors of this: checking with a pseudo-legal move could be enough (as for King capture), the move would have to be legal (as for baring). In the variant 3-checks only the third check wins. I suppose the whole thing could be implemented by subtracting one royalty quantum from the royalty counter that the I.D. keeps track of incrementally whenever there is a check, which otherwise only happens when a royal is captured.


H. G. Muller wrote on Fri, Mar 8, 2024 08:20 AM UTC in reply to adella hardy from 08:07 AM:

(pa)mF would do it, where you can limit the number of jumps by putting a umber behind the parentheses. The jumps would not capture, though; to do that you would need (ca)mF.

The Diagram is designed for chess variants, though, not for checkers. Subtle checkers rules, such that you cannot jump over the same piece twice, cannot be implemented.


Including Piece Values on Rules Pages[Subject Thread] [Add Response]
H. G. Muller wrote on Fri, Mar 8, 2024 06:00 PM UTC in reply to Kevin Pacey from 04:33 PM:

I am in general against spreading misinformation; there already is so much of that on the internet, no one is longing for us to add more to it. The problem with posted piece values is that they are very often not rooted in reality, but spring purely from the imagination of the author. I don't see what value that would add to an article. Every reader should be able to make unfounded guesses without any help. And it is especially bad if people post values that contradict a large body of evidence.

If I were to edit the article on orthodox Chess, claiming that the piece values are P=1, minor=2, R=4 and Q=8, based on the theory that piece values are inversely proportional to the number of those pieces you have in a game... Should it be allowed to stand? Should it go accompanied by an explanation of how these were calculated? Or by a disclaimer like "virtually everyone in the world agrees that these values are very much off, but I present these here anyway for no other reason that I could calculate them through a method that did not require any experimental evidence"?

My brother always says:  "if you don't have anything to say, then don't do that here!". Information should flow from where there is knowledge to where there is none. It is only useful to publish information that is better / more reliable than what the reader already has. Sadly, for piece values that will usually not be the case.


H. G. Muller wrote on Sat, Mar 9, 2024 07:40 AM UTC in reply to Kevin Pacey from 01:26 AM:

The problem with intuition is that it is notoriously unreliable. Humans suffer from an effect called 'observational bias', because one tends to remember the exceptional better than the common. This is probably the reason that GMs/world champions have grossly overestimated the tactical value of a King (as ~4 Pawns): in games where the King plays an important role it can indeed be very strong, but there are plenty of cases where a King is of no use at all (because it cannot catch up with a passed Pawn). These tend to be dismissed, as "the King played no role here, so we could not see how stong it really is".  While in fact you could see how weak it was by its lack of ability to play a role. In practice two non-royal Kings are conclusively defeated by the Bishop pair, (in combination with balanced other material, and in particular sufficiently many Pawns). In games between computer programs that most humans could not beat at all. Of course none of these GMs ever played such a game even once.

Other forms of intuition often result from application of simplistic logic, rather than observation. It is 'intuitively obvious' that a BN is worth several Pawns less than RN, as B is worth several Pawns less than R, and it is their only difference. Alas, it is not true. They are almost equivalent. It ignores the effect that some moves can cooperate better than others, and in games BN + Pawn would score convincingly better than RN (and on average even beat Q).

We should also keep in mind that piece values are just an approximation. It is not a law of nature that the strength of an army can be obtained by adding a value of individual pieces, and that the win probability can be calculated from the difference between the thus obtained army strength. And indeed, closer study shows that it is not true at all. The win probability depends on how well pieces in the army cooperate, and complement each other, and how effective they are against what the opponent has.

For example, A=BN and C=RN are more effective against a Queen than against a combination of lighter material (say R+N+2P) that in itself would perfectly balance a Queen. Because all squares attacked by the latter, even though very similar to the number of squares attacked by a single Q, are no-go areas for a C or A, even when they are protected, while they would not have to shy away from a Q attack in similar situations. This causes Q+C+A < R+B+C+A, in Capablanca Chess, even though Q > R+B as usual. The extra C and A on the Q side are effectively weaker pieces than their counterparts on the R+B side, so much that it reverses the advantage. An extreme manifestation of this effect is that 7 Knights easily beat 3 Queens on an 8x8 board. Something that cannot be explained by any value for N/Q that would make sense in a context with more mixed FIDE material.

Your claim that B+2P ~ R and N+2P < R, which I don't doubt, cannot be used to conclude that B > N because of these subtleties. Piece values are not defined as how well the pieces do against a Rook, but by how well they do against a mix of opponent pieces such as these typically occur in end-games. And I have no doubt that the average performance of the Bishop suffers from the fact that there are many cases where B+2P ~ B+P, while N+2P would have done much better (namely when the Bishops are on unlike shades).

Note that the claim lone B ~ N was not based on what I would call a 'computer study'. I have no doubt a computer was used in the process, but just as an aid for quickly searching a huge database of human GM games. Not by playing computers against each other. The fact that a computer was used thus in no way had any effect on the conclusion. In the Kaufman study the claim was detailed further by stating that the B-N difference correlated with the number of Pawns, and exact equality only occurred when each player had about 5 Pawns; for fewer Pawns the Bishop performed better, for more Pawns the Knight. It is also common knowledge that Knights typically perform poorer in end-games where there are Pawns on different wings than when all Pawns are close together. This is of course also something that transcends piece values, which are defined as the best estimate for the chances without knowing the location of the pieces. Piece values are not the only terms that contribute to the heuristic evaluation of individual positions.

But to come back to the main topic: I don't think it would be a good idea to dismis any form of a quality standard on published piece values because "people should know that they should not believe what they read". That is an argument that could be used for publishing any form of fake news. It is already bad enough that this is the case, and we should not make it even more true by adding to the nonsense. There can also be piece values that have a more solid basis, and I think readers should have the right to distinguish the one from the other. So as far as I am concerned people can publish anything, as long as they clearly state how they arrived at those values. Like "personal experience based on N games I played with these pieces" or "based on counting their average number of moves on an NxN board" or whatever. If there is a non-trivial calculation scheme involved, it is fine to publish that as a separate article, and then refer to that.


H. G. Muller wrote on Sat, Mar 9, 2024 06:12 PM UTC in reply to Kevin Pacey from 03:29 PM:

Well, from what you say it appears that with 'computer study' you mean statistical data from games that computers played against each other (or themselves). As I would. But as I said, the B = N observation came from the Kaufman study, which was nothing of the sort. He just filtered positions with a B-vs-N imbalance from a huge database of human GM games, selecting those that were the imbalance was stable for some number of moves (to weed out tactics in progress), and counted the number of wins, draws and losses in which these games ended. Which apparently was a 50% score.

It doesn't sound like rocket science to me, but I suppose a complete idiot could bungle even the most simple tasks. And I have met other chess-engine programmers that have done similar things for themselves. (The Kaufman study did not publish more specific things, like how the N or B would do against Rooks, or whether the difference also correlates with the presence of other pieces than Pawns, and some programmers want to make their engines aware of that too, and put a complete table of every conceivable material compustion in their engine.) And they never told me they had proven Kaufman wrong.

The problem is that implying someone is a bungling idiot that even cannot do the simplest thing right, or a fraud who intentionally publishes falsehoods, is a pretty heavy accusation. Most people would hesitate to make such an accusation without having very solid evidence that the published results were indeed wrong. "It was not checked by anyone, so it must be wrong" is not really a valid line of reasoning.

You seem to have a wrong impression of the peer-review system. The 'peers' that are asked to referee a scientific publication will NOT redo the reported work. They only judge whether the described method according to which the results were obtained is a proper procedure. If the claims are in contradiction with earlier results the referees have a hard time. They would at the very least insist that the authors of the new manuscript give an explanation for why their method would be more reliable than what people previously did, and even then they stand a large probability of rejection if that doesn't convince the referees. In a sense everyone is a peer on the internet, and could have contested what others publish there, in particular the Kaufman results. But it didn't happen, and that means much more than when he would just had to fool one or two referees. And there isn't really any need for mathematicians, people that know how to count seem sufficient. You are aware that Larry Kaufman is a GM himself?

I don't really understand your third paragraph, but I am intrigued by the term "more knowledgeable". True knowledge should of course never be dismissed. But what knowledge are you talking about, here?

I agree the Amazon result is suspect; it was only based on a couple of hundred games where the Queen and a Knight where replaced by an Amazon, and the baseline pieces were shuffled to provide more game diversity. That is a very different story than GMs not being able to convert a B-N 'advantage' into a better result in a few thousand games. The remarkable thing about computer games is that it doesn't seem to matter much what the level of play is. Errors tend to cancel out, when both players make them. Even random movers systematically win more games when you give them stronger material. (Although quantitatively they don't make the most of that, as they too easily give the strong material away.)

Rather than describing the calculations in a large number of articles, which is likely to lead to a lot of duplication, you could make a separate article of it. That could lead to a more coherent presentation, and the other pages could then just refer to that.

 


Tenjiku Shogi. Fire Demons burn surrounding enemies, Generals capture jumping many pieces. (16x16, Cells: 256) [All Comments] [Add Comment or Rating]
📝H. G. Muller wrote on Sat, Mar 9, 2024 09:43 PM UTC in reply to A. M. DeWitt from Fri May 26 2023 05:53 PM:

I created a new version of the Interactive Diagram script, which currently is only used by the Tenjiku Shogi diagram to which this is a reply. It uses a different method for selecting Shogi promotions, and I would like some feedback on whether this is better than the old method ("Promote? yes/no" links above the board), or that it just is confusing. What happens now is that the destination and a square next to it get highlighted in red, and display the two pieces between which you must choose. Clicking one of those then completes the move. So double-clicking the destination is enough to promote.

The AI is now also searches deeper on lines with checks in them, with as a consequence that it even sees the mate-in-2-threat in Tenjiku Shogi after 1.j6 SEn11 2.BGi6 at a setting of 2 ply.

The I.D. (at 2.5 ply) beat Jocly (at 10 min) at the Modern Tenjiku Shogi correspondence championship!

[Edit] An alternative method for the promotions is to define special highlight markers for promotion and deferral, rather than highlighting by (red) color and displaying the pieces to choose between. That could be less confusing than just seeing pieces appear that don't belong there. The problem is that square-covering markers would have to be dependent on the square size that is in use. But using a red disk with a + on it for promotion, and a green one with an = for deferral, which just fit into a 33x33 cell might be usable.


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