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Dimachaer ChessA game information page
. Introducing the Dimachaer, a bifurcation piece that always lands on the diagonal second leg (zrf available).[All Comments] [Add Comment or Rating]
M Winther wrote on 2009-10-13 UTC
In Chinese Chess piece values are very flexible, but the Chinese Cannon and the Chariot (rook) seem generally to have the same value relation as knight and rook in chess. However, for different reasons I think it's a misfit on the the Western board, and the Korean Cannon even more so.

These are very interesting results for the Adjutant, which gladdens me. If it turns out that it's worth 6, then Zillions inbuilt piece evaluation is largely correct in this case. I'll have to correct my tweaking then. The reason why I didn't want to set it to higher than a rook is while it cannot give mate together with the king, something which the rook can. This is a very important plus for the rook. The Adjutant can only visit half the squares of the board. Despite this, if it holds a value greater than a rook, then it's a very interesting piece. The strategical tension becomes greater because the Adjutant cannot simply be exchanged with a rook. Two light pieces, or rook + pawn, is equivalent to an Adjutant. This creates a less trivial game. This is a piece that is easily graspable, very agile, but not trivial to exchange. Due to its great value it cannot simply block enemy pieces. It would be a good substitute for the mad queen, then.
/Mats

H. G. Muller wrote on 2009-10-13 UTC
Well, I tested the Xiangqi Cannon once in the Western context, and its opening value was also only slightly below that of a (Western) Knight. I guess the Secutor suffers a lot from the fact that it can be blocked, while the Cannon cannot. Another factor might be that captures contribute more to piece value than non-captures, while the Cannon has the more powerful orthogonal Slider move as capture.

As to the handling: You should not forget that I test with Fairy-Max, which also has no kowledge on specific handling of orthodox pieces. In a program with more knowledge (e.g. piece-square tables) the value of the Secutor might go up, but the value of the Knight likely would go up too, meaning that the relative value would be much less sensitive to this. Fairy-Max does distinguish pieces that should be centralized (P,N,B,K in orthodox Chess) and pieces that roam freely, though. I programmed the Secutor as a piece that could roam freely, because it is long-range and actually needs distance to its platform. For the Lion (FWADN) the value went up about 50cP by centralizing it! (It is short-range, so perhaps this should be expected, but before I tested it I had the belief that centralizing very valuable pieces was not good because it would make them easy chase targets. Of course the value increase by better handling should be seen relative to total value, and Lion > Q, so it is only ~5%.)

M Winther wrote on 2009-10-13 UTC
H.G., thanks for the investigation. It largely corroborates my intuitive evaluation with the aid of Zillions. It must be correct that an intelligent handling of the bifurcators, drawing on experience, must increase their value somewhat. Comparatively, chess programs are taught to place the knights in the centre rather than at the edge. A corresponding type of knowledge would have to be programmed if bifurcators are implemented in commercial programs. 

Bifurcators are not that strange if one compares them to the Asian cannons. The latter are also dependent on a screen for capture capability or movement. So bifurcators are related to the cannons. However, due to their low value, the Asian cannons won't work well in a Western piece context. But it is now proven beyond doubt that bifurcators work fine among bishops and knights. I have also invented some cannons proper, but their orthogonal jumping capability means that they are more difficult to harmonize with the dense board situation. So I have had to curtail their range beyond the enemy pawns. This is generally not necessary in case of bifurcators because of the oblique movement. However, Western chess players don't seem particularly fond of cannons and pieces dependent on screens. No doubt, they are more demanding. Korean Chess revolves much around the problem of activating the cannon.
/Mats

H. G. Muller wrote on 2009-10-12 UTC
I tested the Secutor, and (as expctd) it is stronger than the Dimachaer. Where a pair of Dimachaers clearly lost against a pair of Knights, (scoring only 44%) a pair of Secutors beat the Knights by 53% (over 1000 games). This was with the Secutor value set slightly below that of the Knight, so that the side playing the Secutors would trade them for Knights if he got the opportunity (and its opponent tried to avoid such trades).

I tried the same match with the value of the Secutor set higher than that of the Knight. In that case the Secutors won by 53.4%, i.e. not significantly better. (Statistical error is 1.3%..) So the opening value of the Secutor is marginaly stronger than the Knight (5-10 cP), but when you program a constant value for it during the game it doesn't hurt much if you choose that below the value of a Knight. This is likely caused by the fact that the value of the Secutor drops during the game, so that the average value during the game might very wel be below a Knight's value.

It is quite possible that the empirical value of the Dimachaer and Secutor would go up a little with better handling, e.g. when the value would be allowed to vary with the piece density. In that case the Secutor sid would initially avoid trading them for Bishops and Knights, in order to make use of their superior tactical possibilities on a densely populated board to create an advantage. But as the material thins out, it would start to seek trading it for Knight and Bishop, and conversely would try to avoid trading other material as long as it was still stuck with the Secutors (because it would devaluate them).

George Duke wrote on 2009-10-10 UTC
http://www.chessvariants.org/index/displaycomment.php?commentid=17227 I use Dimachaer here, where White cannot move in the theme. The opening set-up is plausible in these with both same-sided in full contingents.

H. G. Muller wrote on 2009-10-10 UTC
Well, I play them in pairs, so that doubles the advantage in the starting position to 1/3 of a Pawn. And Pawn odds is a pretty large advantage in normal Chess: it creates a score of about 68%. The white advantage has been estblished (from statistics of Grand Master games) to create an advantage of 53-54%, which is about 1/6 of a Pawn.

I guess the reason the Dimachaer is worth nearly a Knight is because in the early game phase when the board is still densely populated it has a very good forward forking ability, while every non-Pawn is a potential target. So it is almost always possible to force trading it for at least a Knight. I could run another experiment programming the value to between Knight and Bishop. This would already cause a major strategy change in the handling of the Dimachaer, as the program would not spontaneously rade it for a Knight anymore, but would wait for a chance to trade it for a Bishop. If this is good strategy, the value might go up to being close to a Bishop. OTOH, it might on average not be able to force such a trade, and then it would be stuck with a worthless Dimachaer in the end-game so often that it hurts more than it helps. The fact that it tests less valuable than a Knight now already suggest that it is already inferior to a Knight in the early middle game, as even when it could always be traded succesfully before th end-game, it is present for at last part of the game in every game, and does not seem to be able to creat an advantage during that time (e.g. by gaining Pawns).

I guess that pieces dependent on sceens and platforms only start to suffer when the population density starts to fall as low as about one piece per rank or line. Before that they will virtually always find a platform on their path, and it does not help much if there are two or three on their path: they can only use the first one. In fact the Colliding Bifurcation pieces, which do not only need a screen, but need a distant screen, might start to suffer again if the population density gets really high, because they can be blocked.

M Winther wrote on 2009-10-10 UTC
It amazes me that such small differences, the sixth of a pawn, can have such great impact on the end result. It would certainly be interesting to know the result for the Secutor. It differs from the Dimachaer in that it can always move (but not always capture), while the latter always needs a screen to be able to move or capture. Anyway, it is interesting that a piece which is so severly handicapped (i.e. that it is stranded when there are no screens) can be worth almost a knight. A Dimachaer is like a bifurcated Korean Cannon, but it is stronger than the Korean counterpart while it can go in many more directions, due to the bifurcation.

However, one must keep in mind that, from the standpoint of the human brain, a certain learnt ability to handle the Dimachaer could play a significant role. This could mean that a human understanding of the piece could very well create better results. With better theoretical understanding, it could equal a knight
/Mats

H. G. Muller wrote on 2009-10-10 UTC
After 404 games the Knights were leading against the Dimachaers by 56.2%. The statistical error should be about 2%, so I guess this should be taken as a significant advantage for the side with the Knights. IIRC Pawn odds results in a 68% score, though, so the difference is only about 1/3 of a Pawn. This is for a pair, so a single Dimachaer seems about 1/6 of a Pawn weakr than a Knight. With the Kaufman Knight value of 325cP, this would make the opening value of the Dimachaer equal to ~ 310cP (+/- 5cP). This value is consistent with the value I programmed for the test, which would encourage trading Dimachaers for Knights (and the Knights to avoid such trades).

For a really reliable value I should also test Dimachaers against a pair of Bishops. But I want to focus attention on the Secutor first, as this would not lose value in the end-game as steeply as the Dimachaer. Fairy-Max uses fixed piece values during the game, which might still not be perfect for the Secutor, but at least should be a better approximation than it is for the Dimachaer. The Secutor is Cannon-like, and I noticed that the Xiangqi Cannon only starts losing value if the board population gets really, really sparse. With just a few (friendy) pieces it is still quite useful. With orthodox Kings it retains its mating potential in combination even when assisted only by the weakest pieces (a single Ferz or Wazir).

M Winther wrote on 2009-10-09 UTC


This is a simple example of bifurcation tactics in Secutor Chess. Black
has played Bg4 and threatened the white queen, while the bishop is
guarded by the black Secutor at b8. White has played f3, and now black
can play Bh3! check. The Secutor on h6 checks the white king because
the black bishop placed himself as a collision screen. It's this types of
tactics that always occurs with bifurcation pieces, and it's a different kind
of tactics than with other pieces.
/Mats

M Winther wrote on 2009-10-09 UTC
I have not a clear judgement about these pieces yet. At least the colliding
bifurcators seem quite intuitive, and one soon gets a feel for them.
They feel much like bishops or rooks, which is not the case with the
super-knights. As yet, there is only one email preset, namely for the
Secutor: Secutor Chess

But some of these pieces might be non-intuitive and 'convoluted' attempts.
My interest in these pieces was initially of the mathematical kind. I
wanted to see if such a strange thing could work. But I soon found that
the ability to threaten around corners by distance invokes a new aspect. A
Retiar and a Murmillo, among others, can threaten the enemy king from
afar, despite the fact that it is hidden behind pawns on the first rank. A
collision against the rim, or a bounce can occur on the first rank, and
the king is thereby checked.

There are certain new interesting tactical
facets. The situation on the board can change significantly when one
moves *another* piece than the bifurcator, and new threats are
uncovered. These are cooperative pieces, so they introduce som new
concepts. One first thinks that they are significantly weaker in the endgame.
But I have found that they can still be powerful while it becomes easier
to position one's own pieces in such a way as to threaten the enemy.
Maneuvering becomes easier in the endgame. This, plus the free scope
of the pieces, makes the bifurcators keep much of their power.

I am not certain that all the simple pieces have already been invented. After
all, I recently invented the Adjutant, which a simple one, and it seems to
be very useful. One could easily replace a rook with an Adjutant:
Adjutant Chess
/Mats

H. G. Muller wrote on 2009-10-09 UTC
I wonder if Mats agrees with the 2s I replaced by 3s? Can the Bifurcation pieces really end on those squares with the same type of move as they are allowed to make to the 2s?

. 2 . 2 . . . .        2 . . . . . 2 . .
. . 3 . . . . .        3 . . . . . 3 2 2
. . X . . . . .        . X . . . X . . .
. . 1 . . . . .        . . 1 . 1 . . . .
. . 1 . . . . 2        4 4 4 # . . . . .
1 1 # 1 1 X 3 .        . . Z 4 1 . . . .
. 4 Z 4 . . . 2        . . . 4 . 1 . . .
4 . . . 4 . . .        . . . 4 . . X . .

I don't have Zillions, and perhaps I should learn how to read ZRF files. But the simplest way still seems to ask Mats! ;-) What was a bit ambiguous, and was also not explicit in Sam's diagrams, was what happens in the presence of the piece Z with a Colliding Bifurcator. Does it create moves to the 4s? Not for Mats' pieces, but you could make stronger versions that could.

Sam Trenholme wrote on 2009-10-09 UTC
My initial impression of these bifurcation pieces was that they are too complicated. As shown by Muller’s and my own confusion about the pieces, with both of us having years of experience looking at Chess variants, I think these pieces are too complicated to get widely played. And, indeed, I don’t think there have been any games played with these pieces on Game Courier.

What I see with pieces like this is that all of the simple pieces a Chess-like game can have are already invented, and that we’re having to come up with some pretty convoluted moves to come up with new piece types.

The simple Chess pieces seem to be:

It’s possible, of course, to combine leapers and sliders (Can you say “Capablanca Chess”?), but the only combined leapers + sliders in a national game are Shogi’s promoted rooks and bishops. There are also “riders”, sliders whose 1-move “atom” is not to an adjacent square; the knightrider is the most famous piece of this type.

Once we move past these simple pieces, things get complicated and the learning curve goes up. One relatively simple piece is a piece that captures differently than it moves; a piece that, say, moves like a knight or captures like a bishop.

Betza covered the “crooked rook”, “crooked bishop”, and “rose”—sliders which change their direction every square they slide.

Chinese Chess, of course, has the “Cannon”, which has inspired all kinds of pieces that leap before moving or capturing (or a combination thereof). Speaking of leaping pieces, I’m surprised no one has recently discussed having a checker’s king in Chess: A piece that moves like a Ferz, but captures by jumping over an adjacent piece, and can (optionally) capture multiple times in its move. We can, of course, have a wazir (horizontal and vertical) form of this piece, or combine it with any other chess piece.

So, yeah, it looks like pretty much any kind of piece chess can have with a simple move has been discussed here, so we’re moving on to complicated pieces that don’t seem very intuitive to me.


M Winther wrote on 2009-10-09 UTC
Sam, yes, but the bifurcation pieces have *four* different ways in which they make use of the screen: collision, bouncing, leaping on 1st leg, leaping on 2nd leg. See:
http://hem.passagen.se/melki9/bifurcation.htm
It is probably correct that other variants of bifurcation are possible.

One needn't have recourse to Zillions to see how the pieces work. I have created diagrams that describe them exactly. Follow the links to my homepage. FRom the above cross reference one can click in the rightmost column to see the actual bifurcator described with diagrams.

H.G., the second leg leapers can split their path in certain cases: either when capturing or when moving. This is like the Chinese versus the Korean Cannon (which, unlike the second leg leapers, are generally too weak in the Western context.) It's *not* necessary for a bifurcator that it moves and captures on the second leg. It can (1) capture on the first leg and move on the second leg, or it can (2) move on the first leg and capture on the second leg , or it can (3) both move and capture on the second leg.
/Mats

H. G. Muller wrote on 2009-10-09 UTC
I have no final verdict on the piece values yet (after 60 games the Knights are slightly leading), but  did form an opinion on the aesthetical value of these kind of pieces:

The square where the two legs are joined creates an ambiguity: does it belong to the first leg, the second, or both? This would involve a certain amount of arbitariness in the design of pieces that have different move rights on each leg.

Now for Bent Hoppers this ambiguity is naturally resolved, because the junction square is occupied by the platform, and not accessible anyway. For the jump-on-1st-leg Bifurcators it would be natural to count the junction square as part of the second leg, because, like the rest of this leg, it is behind the platform. The fact that the bend does not occur at the platform itself it aesthetically slightly displeasing. Note furthermore that jump-on-1st-leg hopping pieces only make sense if they have no capture rights before the hop; otherwise it would not be clear if an enemy piece is a victim or a platform.

The bouncing and hop-on-2nd-leg pieces reek of a non-causality that I strongly dislike. It is not logical that they are deflected by something that is not really in their primary path. Furthermore, they are not true bifurcators: their path does not split. (I guess you could allow them to continue on their first leg and take a later deflection; in that case their bifurction is asymmetrical, and could eventually split in more than two paths.) I like none of that.

The collission-type Bifurcator would be natural if the move rights where the same everywhere, as there is no natural break in the path. I would appreciate a piece that can do zero-or-more steps along the first leg better than one that could do only one-or-more. Of course 'the same everywhere' in practice means that it must be able to capture and non-capture everywhere, or it would be pretty useless. And this raises the concern about the ambiguity of enemy pieces serving as collision partners or capture victims. So I guess that the most logical piece of this type would only collide with friends, and capture foes.

Really interesting would be a colliding Bifurcator that would not lose its bifurcating ability on the second leg. I.e. it could make an unlimited number of collisions with friendly pieces, changing its direction by 45 degrees on each collision. This could be called a Billiards piece.

Apart from this, I like the jump-on-1st-leg piece without capture rights on the 1st leg best. And to not lose its value in the extreme in the end-game, it should have non-capture moves along the first leg. I would expect it to be able to capture directly after the hop, though. If not, I think a Bent Hopper would be more regular, and thus nicer.

Sam Trenholme wrote on 2009-10-09 UTC
Muller: Winther is very good about making Zillions rules files for his pieces, so, if you have Zillions (it’s only $25 and excellent for prototyping variants—buy it if you haven’t done so yet), you can see what moves a given piece of Winther can do if you have any questions.

In terms of bifurcators, I assume we’re taling about a piece that:

  • Moves in a straight line, either orthogonally (rook-like) or diagonally (bishop-line).
  • Hits another piece, either friendly or enemy
  • Changes direction, either 45 degrees or 90 degrees upon hitting the other piece
  • Finishes its move
Now, given these parameters, we have a number of new interesting pieces. To keep things simple, I will only look at pieces that turn 45 degrees on hitting the other piece. So, that gives us the following pieces (# is the bifurcator to move, X is another piece, either friendly or enemy, 1 is the first part of the piece’s move, 2 is the second part of the piece’s move, and . is an empty square)
. . . . . 2 . .        . . 2 . . 2 . . .
2 . . . 2 . . .        . . 2 . . 2 X . .
. 2 X 2 . . . .        . X 2 . . 1 2 2 2
. . 1 . . . . 2        2 2 1 . 1 . . . .
. . 1 . . . 2 .        . . . # . . . . .
1 1 # 1 1 1 X .        . . 1 . 1 . . . .
. . 1 . . . 2 .        . 1 . . . 1 2 2 2
. . 1 . . . . 2        1 . . . . 2 X . .

2 . . . 2 . . .        . 2 . . . . 2 . .
. 2 . 2 . . . .        . 2 . . . . X 2 2
. . X . . . . .        2 X . . . 1 . . .
. . 1 . . . . .        . . 1 . 1 . . . .
. . 1 . . . . 2        . . . # . . . . .
1 1 # 1 1 1 X .        . . 1 . 1 . . . .
. . 1 . . . . 2        . 1 . . . 1 . . .
. . 1 . . . . .        1 . . . . . X . .

. 2 . 2 . . . .        2 . . . . . 2 . .
. . 2 . . . . .        2 . . . . . 2 2 2
. . X . . . . .        . X . . . X . . .
. . 1 . . . . .        . . 1 . 1 . . . .
. . 1 . . . . 2        . . . # . . . . .
1 1 # 1 1 X 2 .        . . 1 . 1 . . . .
. . 1 . . . . 2        . 1 . . . 1 . . .
. . 1 . . . . .        1 . . . . . X . .
Here, we see three types of these bifurcators:
  • The piece bounces just before the other piece’s square (first row)
  • The piece bounces in the middle of the other piece’s square (second row)
  • The piece bounces just after the other piece’s square (third row)
Now given these six pieces, we can give these pieces four different powers:
  • The piece can move to an empty square on the first leg of its move (the second leg is not used)
  • The piece can capture on an enemy-occupied square on the first leg of its move (the second leg, again, is not used)
  • The piece can move to an empty square on the second leg of its move.
  • The piece can capture on an enemy-occupied square on the second leg of its move.
Pieces that can neither move or capture on the second leg of their move are nothing more than FIDE rooks and bishops, so are not interesting for our purposes. This leaves us with 12 types of powers for the pieces in question. With six types of movement for the bifurcators, this gives us 72 different types of pieces.

I’ll pull a Betza and create a notation so we can quickly describe a bifurcator. O means we start with an orthogonal move; D means we start with a diagonal move. B means we bounce just before the other piece, M means we bounce in the other piece’s square, and A means we bounce afterwords. 1m means we can end our move on the first leg, 1c means we can end our capture on the first leg, 2m means we can end our move on the second leg, and 2c means we can capture on the second leg.

OK, so where do Winther’s pieces fit in this Betza-esque scheme? Like this:

  • Gladiatrix OB2m2c
  • Crossrook DA1m2c
  • Crossbishop OA1m2c
  • Murmillo DB1m2c (also can bounce off the edge of the board)
  • Secutor OB1m2c
  • Provocator DB2m2c (also can bounce off the edge of the board)
  • Diamachaer OB2m2c
  • Sagittar DA2m2c
  • Venator OA2m2c
  • Laquear DB1c2m (also can bounce off the edge of the board)
  • Essedar OB1c2m
  • Gaul DA1c2m (not allowed to end its move in the square immediately after the second piece)
  • Thraex OA1c2m (not allowed to end its move in the square immediately after the second piece)

OK, there are some other pieces that don’t fit in this scheme, but this makes a reasonable introduction to these types of pieces.


H. G. Muller wrote on 2009-10-09 UTC
In my mind this makes these pieces less regular. A Cannon on c1 would not be blocked by Pawn on c2, despite the fact that it does not really move on the first (non-capture) leg of its move.

The most elementary Bifurcator, which does not seem to occur in your list, would be a Slider with a Y-shaped path triggered by a collision. That is, it would be able to make both captures and non-captures to squares on either leg. That means it might not get to the second leg of the path at all.

Anyway, even with your restriction the piece is easy to program. I have started a test now on 8x8, where I replace the Knights of on side by Dimachaers.I gave them a value slightly below that of the Knight, to stimulate trading it for light pieces. A piece like this should decrease enrmously in value in the late end-game, so trading it during the middle game should be a priority. (Similar to Grasshoppers.) This is much less true for the Cannon, which at least keeps its non-captures on an empty board.

After 30 games the Dimachaers-vs-Knights result is still pretty much even.

M Winther wrote on 2009-10-09 UTC
Thank you for testing this piece. All bifurcators must always move along *two* legs. So an orthogonally blocked Dimachaer cannot move. While this piece type always involves movement along a diagonal, I think they are capable of penetrating the position finely. But they are handicapped by the requirement of a screen. Due to this it'¨s possible that the piece power changes depending on piece density.
/Mats

H. G. Muller wrote on 2009-10-09 UTC
One thing is not completely clear to me: Does the Dimachaer actually have to move along the orthogonal leg, or is it enough that it is orthogonally blocked? I.e. if I have a Dimachaer on c1, and a Pawn on c2, is the Dimachaer allowed to move to b2,a3 and d2,e3,f4,...? My interpretation of your description would say this is allowed: IMO hitting something on the first orthogonal step is a case of 'colliding orthogonally', even though the second leg would originate from a point where the piece is currently standing.

Bifurcators such as the Dimachaer are actually very easy to implement in Fairy-Max, with a small change in the code. They are a slightly modified case of Hoppers. The way Fairy-Max implements Hoppers is that on hitting an obstacle, it replaces the primary move rights and vector by the secondary move rights and vector to continue the move. This would allow 'bent hoppers' which change direction at the platform. The 'true' Bifurcators, which split their move path, would do this one step earlier or later.

So what I did in Fairy-Max is just alter the Hopper code to take one step back along the primary ray before proceeding with the normal Hopper code. This then implements a Collider-type Bifurcator. The SMIRF symbol for Archbishop (two crossed swords), which is also used by WinBoard, seems to fit the Dimachaer perfectly!

I will report a value estimate soon!

M Winther wrote on 2009-10-07 UTC
I also believe that the bifurcators have fine qualities, and they aren't
as hard to understand as one would expect. There is a cross reference
list of the bifurcators here: Bifurcation pieces.

In fact, I have also created a wholly validated email preset here of
Secutor Chess.

Concerning the bifurcator piece values. Of course, I have rounded
them off and say that it's a 'preliminary estimate'. That's what they
say about the bishop and the knight, too, i.e. that they are both worth 3,
although this isn't quite correct. I have guessed the piece values by
watching play, how much they are utilized by the engine, compared
with other pieces. I have also removed the bifurcators for one party
and inserted light pieces instead. If both parties are roughly equal, then
I have concluded that the specific bifurcator is worth a light piece.

However, this method certainly doesn't give exact values. Muller
helped me to establish the values of the diverse Lama pieces.
He did this by playing very many games using his own program. So in
this case the piece values became much more reliable. The result was
also somewhat surprising. It's an interesting fact that humans tend to
overestimate pieces with long-range capacity, but that are lacking in
capture capability. So we tend to overestimate maneuverability in
relation to capturability.
/Mats

George Duke wrote on 2009-10-07 UTC
Virtually all of these have more exhibitable symmetry than Carrera BN Centaur, because the latter's internal relations depend on the board size. Whereas bifurcators' internal relations depend mostly on piece density, which is going to change anyway during actual play. We need to start getting ballpark piece-type values for the bifurcators. Winther frequently uses phrases and clauses ''same as the Knight'' and ''DoubleCannon's value is four'' and ''Gladiatrix's value is six'' and ''Samnis' value is five, same as Rook'' and ''Pegniar's value is three.'' But are they really? Where is the deserved research? I was thinking of the bifurcators, the hurlers, the catapalters, the paralyzers, the CrossRooks and CrossBishops, for well-monitored piece values every step of the way. It can become as precise as bonus Quiz question of ChessboardMath11 #(11), 3.88, 412, 324, the rudiments of perfection in relative valuation, by if necessary simple compounding-enhancement with Wazir, Ferz, Knight. If there's a piece worth 5.50, there can be one 5.51.

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