Comments/Ratings for a Single Item

Using magic number = 0.7, I calculate the mobility of the Ferz->Bishop bent
rider as 8.8, placing it midway between the Crooked Bishop (mobility 9.7)
and Cylindrical Bishop (mobility 7.9)--so, roughly Rook-value, or a smidge
less.

I think only its first 3 moves are the same as the Zigzag Bishop, though,
not the first 4.

The value of Griffon and Aanca

Synergy - Betza's calculation does not take into account that in combining sliders to a compound, there usually is a synergy. Applying his method to the orthodox Queen, for example, would lead him to conclude its value was that of Rook (5) + Bishop (3-3.5), i.e. 8-8.5, while every Chess player of course knows it is more like 9.5-10. The reason, no doubt, is that the B moves of the Queen help you to position her such that the R moves can do damage, and vice versa.

Griffon - The true value of the Griffon in a FIDE context is slightly under that of Queen minus Pawn, around 8.3 (if we take the Kaufman value 9.5 for Q). That is, a Queen lightly beats a Griffon + Pawn (like 53%), if that imbalance is present in the opening position. (I.e. replace the Queen of one side by a Griffon, and delete the f-Pawn of the other side.) That is 1.66 times the Rook value, 0.2 larger than Betza's 'naive' adding of the values of the left and right Griffon. (Which was not that naive, because it had to correct for moves they had in common.) So there seems to be a synergy of about 100 cP.

Aanca - Similarly, the difference of the Queen and Aanca turns out to be significantly less than that of Knight and Pawn: Aanca + Knight beat Queen + Pawn, from the FIDE opening position (i.e. delete a Knight for one side, replace the Queen by Aanca for the other and delete his f-Pawn, and the Aanca scores ~59%, which correspond to about half a Pawn. As Knight minus Pawn is about 2-2.5, this means Queen minus Aanca ~ 1.5-2. That makes Aanca something like 7.5-8, say 7.8. Which is 2.4 times the Bishop value if we set the latter at 3.25 (the Kaufman value of a lone Bishop) and 2.22 x B is we set it at 3.5 (to incorporate half the B-pair bonus of 0.5 Pawn). In any case way larger than what Betza calculates.

The Chiral Bent Riders

Chiral Aancas - I was of course curious why the Aanca value would be so much above Betza's guesstimate. So I decided to check the individual steps. The first step is to get the value of the 'left' or 'right' Aanca, which Betza values as "8/7 of a Bishop plus some bonus for not being color bound". So I set up a match where one side has the Bishops replaced by a Left and a Right Aanca. I put them such that they bend 'inward', i.e. white's Left Aanca starts on f1, so that it can move f1-f2-e3-d4-c5-g6 as soon as the f-Pawn is pushed. When black has the Chiral Aancas, he starts with the Right Aanca on the f-file, as FIDE is mirror symmetric, and the chirality flips on reflection.

Turns out the Chiral Aancas win by about 60%, which is over half a Pawn, even if you start them with Pawn odds. (In which case I delete the g-Pawn in stead of the regular f-Pawn, because the latter would develop the Aanca like normally a missing g-Pawn would develop the Bishop.) This means a pair of Chiral Aancas is worth about 150cP more than a pair of Bishops, meaning that a single Chiral Aanca is a full Pawn more valuable than a lone Bishop (or Knight). With the Kaufman values B = N = 325, so that a Chiral Aanca would be 425cP.

That makes the high value of the complete Aanca understandable: adding the values of Left and Right would produce 850, but you have to subtract ~150 for the overlapping W moves, which would leave you with 700. Now I found about 780, and a synergy of 80 cP sounds reasonable, although a bit on the low side when we realize a Queen, which is not that more valuable in total, has ~150cp synergy between R and B moves.

Chiral Griffons - I did a similar test on the Griffon, where I started one side with Chiral Griffons in stead of Rooks (such that the one on a1 would bend left: a1-b2-b3-b4-...). Because it was too cumbersome to set it up such that Fairy-Max could castle with the Griffons if I gave them alternately to black and white, I also forbade castling with the Rooks.

The Rooks won this match very lightly, some 54%, translating to ~1/8 Pawn value loss per Chiral Griffon compared to Rook. Now Rooks and other Wazir-like pieces are known to test about 25cP low from opening positions compared to other pieces (i.e. they test as 475cP rather than the classical 500). They only reach their full potential once there are open files (or, in the case of Wazir, managed to get in front of the Pawns). Now I am not sure if the Chiral Griffons suffer a similar devaluation in the opening. According to the test the opening value would be ~465cP. If they do their value on a non-crowded board might rise to 490cP. For comparison, Betza's 7/8 of a Rook would amount to 440cP even when you set R=500, and seems a slight under-estimate.

Combining the two Chiral Griffons into a complete Griffon would then naively give 2 x 465 = 930cP, of which after subtracting a penealty of ~150 (The Ferz value) for the moves they had in common 780 would be left. The empirical value of 830 then includes 50cP worth of synergy. If we assumed 490cP for the Chiral Griffons, there would not be any synergy at all. (Which does seem unlikely. So Chiral Griffons are probably not really dependent on open files for getting them into play. Which fits with the observation that you can develop them through Pg2-g3, RGh1-g2, after which the Right Griffon already covers the f-file and the back rank.)

Mating potential - As a side remark: the Chiral Griffons have the capability to force checkmate on a bare King.

Just a thought - By playing with these pieces by hand on WinBoard, which highlights destination squares, I noticed that the Aanca has the same move pattern as two Bishops standing left and right of it (or, alternatively, in front of and behind it). The moves of these 'virtual Bishops' intersect each other in two squares, but the Aanca compensates this by also covering the two squares these virtual Bishops occupy, which the Bishops would not. So it does seem an Aanca should be the equivalent of two complete Bishops. The situation with the Griffon is different, though: most of the move pattern of a Griffon on e4 would be reproduced by a pair of Rooks on d3 and f5 (or d5 and f3). These would also intersect in two squares, and have the Griffon attack the squares they are on. But all W squares would not be covered by the Griffons, while they would be covered by the Rooks. So the naive Griffon value should be appreciably smaller than double the Rook's, unlike the Aanca.

For other newcomers than Florea, here are Gryphon and Aanca whose values Muller and Aurelian Florea have pinpointed through fascinating Fairy-Max discussion. It makes sense to player that preferred-emphasis-orthogonal Gryphon edges out Aanca. But are not G. and A. better implemented on 10x10 or even up to 12x12? Then on large boards they might equal Queen. In any case on 8x8 and 8x10 Gryphon and Aanca are higher value than 5.0-range Rook and Falcon. Of the four fundamental Chess pieces Bishop and Knight as 3.0 and Rook and Falcon as 5.0 are pairwise very close in values given the standard boards 8-deep and reasonably popular piece mixes. Betza had just become aware of Falcon when writing up "Bent Riders" in 2002, and Falcon's changing direction has the B.R. piece genre's mode but that move just beyond Knight of Falcon is better described as fixed-length plural-path.

The Florea-Muller exchanges on Fairy-Max are reminiscent of those of Reinhard Schamagl (Capablanca Random) ten years ago on Carrera-Capa RN and BN.

Another idea that I have would be knight then camelrider

@Kelvin that thought has crossed my mind too. But it had been any camelrider move after the knight move. Works with zebras, too I guess.

The opposite version and the zebra pieces. I will edit them later with the correct piece picture

One can also make many other Rook/Bishop-based bent sliders than Griffon and Aanca. The latter two bend their trajectories by 45 degrees after one step, but one could also bend by 90, 135 or even 180 degrees, and after another number of steps. Last year someone presented me a variant he designed, which featured a piece that moved orthogobally for maximally 10 steps, turning at right angles after exactly 5 steps.

@HG

Remember your point for an upcomming game of mine!...

I came on this page and I am horrified to read what I read.

"In H.J.R. Murray's History of Chess, page 181 states that the Alfonso manuscript was published in about the year 1211" >> no, not 1211! Murray wrote it right: 1283. 

"which on page 346 is said to have used algebraic notation, and to have described a chess variant that included the modern B and Q": not at all!
That chess variant wich used algerbric notation and modern move is another one, from India, written in Persian and dated 1796-8. It is reported quite clear in Murray page 181 for who has eyes to read!

Page 348, Murray gave a short description of Grant Acedrex from King Alfonso X.Today this is better known thanks to the PhD work of Sonja Musser. I worked a bit with her on this, this is reported in my book A World of Chess (McFarland, 2017). In few words: what was called Unicornio in medieval spanish was clearly a Rhinoceros. So the Rhino was a piece first jumping like a Knight, then going away like a Bishop.

It was the counterpart of another piece moving one step diagonal then moving away on rows and columns. That later piece is called Aanca in the manuscript. It's an Arabic word, not Spanish, designating a giant Eagle or prey bird, from oriental legends (able to carry elephants). This was mistakenly traduced by Gryphon by Murray. This is unfortunate as the Gryphon was a very different legendary animal. This is why I prefer to use the name of Eagle in Metamachy and not Gryphon to avoid replicating that mistake.
 

"Not described there is a piece which makes a one step Rook move and then continues outwards as a Bishop. For lack of a name, I'll call it the Aanca." 
No no no please! Aanca is the Giant Eagle, or the Gryphon if you want. Do not give that name of Aanca to a piece which is different and is more like the Unicornio / Rhinoceros. This is a very very bad idea. Stop adding confusion, I wish one is more careful when reading the work of Murray.

We have to blame Ralph Betza for hijacking the name Aanca, and assign it to the bent slider that first makes a W step and then continues outward like B. Unfortunately this new meaning of Griffon / Aanca became so well settled that it will be difficult to eradicate. There was a discussion about this in the comments on Team-Mate Chess (which also features a Betza Aanca).

Note that such a confusion is not unique: the Spanish word for a Bishop is Alfil, but in English Alfil is used in the original sense of the Shatranj piece. It will be even harder to make the Spanish Chess community see the error of their ways, and make them drop their erroneous use of the word Alfil.

My conclusion was that the simplest solution is to just accept that chess men have different names in different languages, which are not always translation of each other (e.g. Bishop - Runner - Elephant - Fool - Counselor), and that this can also be the case for unorthodox pieces. That way Griffon would be the English name for the piece that the Spanish call Aanca, while Aanca would be abused in English to dscribe the Betza piece, similar to how the Spanish abuse the word Alfil to describe a Bishop.

Dear HGM, I'm sorry to disagree with your demonstration.

It is very unfair to say that the "Spanish abuse the word Alfil". This is denying chess history. First we largely owe the Spanish to have transmitted to the rest of Europe chess from the Arabs in the years 950-1000. Alfil in Arabic is/was "al fil" which means "the elephant". It stayed in Spanish as Alfil, and was adopted in France as "fol" in Occitan, then "fou" in French, meaning "fol". Due to the shape of the piece, also borrowed from the Arabs, with 2 protuberances for the tusks, it was assimilated to a bishop's miter by Englishmen. So the Spanish just kept the name. The move was only modified 500 years later, also first in Spain! It is not because some American and English chessvariant lovers decided to revive the world Alfil in the 20th century that Spanish have abused whatever. 
Consider that Russians are calling the bishop a "slon" which also means elephant in Russia! Russia for chess is something no? Spanish are not so wrong  after all. But OK, I will not say that English are abusing :=)

I would agree with you that it would be difficult to forget the Gryphon or Griffon for CV lovers, but Aanca no. Aanca is Anka in Arabic and is what I said, a legendary giant Eagle, found in The 1001 Night Tales. In context of chess, it is found in Alfonso X's codex where it has clearly the move of what you call the Griffon. And shall I need to remind that Alfonso X was not writing in modern English but in medieval Castilian. (tired to read that the world griffon is found in the Libro de los juegos from king Alfonso >> it is not!).

Actually, the same Alfonso described a piece, in the same Grant Acedrex, almost moving like W followed by diagonals. Actually, it was just skiping the W squares, first jumping like a knight to the diagonal. That was an Unicornio, that it is demonstrated that it meant a Rhinoceros. Alfonso didn't have access to Wikipedia to check what a Rhino is. So, rather than Aanca, Rhinoceros would be a much better name. This is what I selected for Zanzibar chess. 

Calling that piece an Aanca, is like calling the a Bishop a Rook. Absurd.

I can confirm that the Russians do call the bishop an elephant "slon" (although their physical boards have pieces that look the same.)  So I would agree that the Spanish are in the mainstream by calling it an alfil.

I also don't think the use of "aanca" is that well established and, given this information, I think we should tend toward something else.

What to do with this page is a bigger question.  This page is more of an opinion piece by Betza than an encyclopediac entry.  (Despite the icon, this page isn't in the Piececyclopedia.)  That said, the page does present some things as facts that appear to be incorrect.  I am in favor of - at a minimum - correcting obvious errors with footnotes indicating we have updated Betza's text for accuracy.  Even better would be to retire this page and replace it with a new, more encyclopedia-worthy page on bent riders.  For example - I care not for the discussion about whether space aliens would use our chess pieces.

Betza wrote:

Not described there is a piece which makes a one step Rook move and then continues outwards as a Bishop. For lack of a name, I'll call it the Aanca (13th century Spanish for "Gryphon"). Although the Aanca is not described, one can suppose that the same mind who conceived the Gryphon and the Unicorn probably also considered the Aanca.

This is really irresponsible. Lacking a name for the Gryphon's orthogonal/diagonal counterpart, he just borrowed another name used for the piece called a Gryphon. The Aanca was described. It was described as being the same piece as the Gryphon. What wasn't described was a piece that Betza should have found a more appropriate name for. I don't have sufficient knowledge of Arabic or Spanish, but Gryphon seems like a fitting name on the basis that this piece is sort of a hybrid of Rook and Bishop, and the Gryphon is a mythological hybrid animal. With that in mind, it would be appropriate to use the name of another mythological hybrid for the corresponding piece, such as Chimera or Manticore.

Actually, I think Manticore is an excellent name.

Since the name Unicorn is already in use for other pieces, the piece he describes as moving like a Knight, then a Rook, could be called a Hippogriff, which I just learned is the offspring of a Gryphon and a mare. The hippo- root in the name means horse, which fits with a Knight move.

I used the name Aanca for apothecary chess modern and griffin too (I thiught that is the correct spelling at the time). So I don't see any problem with these. Anyway nobody can expect total consistency as one name for 1 piece over all chess variants.

Previous time this subject came up I proposed to rename the W-then-B to Ancaa.

I don't really like the name Eagle for the Griffon. And Eagle is an ortdinary bird, not a mythical monster. In that sense Griffon is much closer: a large mythical monster that can fly. OK, it doesn't really prey on Elephants, but who does? Arakis Sandworms, I suppose, but these cannot fly. If we want to be purist, we should keep the Arab name Roc.

Renaming the Grant Acedrex Unicorn is just as bad as renaming the Aanca. Unless we would rename it to Rhino, which was what the Alfonso Codex really meant. Using Unicorn or Rhino for other pieces than the N-then-B is exactly the same as hijacking the name Aanca for W-then-B.

I like the name Hippogryph, but I think it should be used for the W-then-B.

As to the Alfil: the move was changed without changing the name. Betza changed the move of the Aanca, but kept the name. Sounds like it is the same thing.

Using Unicorn or Rhino for other pieces than the N-then-B is exactly the same as hijacking the name Aanca for W-then-B.

No, it isn't, because people have used the name Unicorn for other pieces without borrowing it from another Chess variant where it was used for a different piece. Unlike Aanca, which is a word Betza came across only in Murray's description of a particular Chess variant, people far and wide know of unicorns and rhinos. This is one of the reasons I put a Unicorn piece in the logo. I knew people would recognize it even if they weren't familiar with games that use it.

I like the name Hippogryph, but I think it should be used for the W-then-B.

The name seems more appropriate for a piece with some kind of hippogonal move.

As to the Alfil: the move was changed without changing the name.

You have that backwards.  As Chess evolved the elephant was enhanced from leaping diagonally to sliding diagonally.  It was at a later point that the English name (and maybe other languages) was changed to bishop while other languages, such as Spanish and Russian retained the existing name.