[ Help | Earliest Comments | Latest Comments ][ List All Subjects of Discussion | Create New Subject of Discussion ][ List Earliest Comments Only For Pages | Games | Rated Pages | Rated Games | Subjects of Discussion ]Comments/Ratings for a Single Item ⇧Earliest ⇧Earlier ⇧Reverse Order⇩ Later Asymmetric Chess. Chess with alternative units but classical types and mechanics. (8x8, Cells: 64) [All Comments] [Add Comment or Rating]H. G. Muller wrote on 2016-11-29 UTCYou should not take the scores reported by Fairy-Max that seriously. For one, the positional part of its score is relative to the current position. Its own position can be far better, and it can still report a negative score (with equal material). Because the opponent can improve his position a bit more (or faster) than the side to move. This is for instance clearly to see directly after it castles, when the opponent has not castled yet. Castling gets a large bous (or it would never do it), and of course it is usually not possible to prevent the opponent from doing it. So even though this only equalizes, and the opponent could have other disadvantages too (like more-poorly developed pieces), Fairy-Max will still report a negative score, just because it can never match the castling bonus that it already collected. Dmitry Eskin wrote on 2016-11-30 UTCMaybe, it will be better, if the Fairy-Max will have not static, but dynamic evaluating of pieces, according the same criterias, that I use for balancing new units (you may read it at the end of main post). It is very experimental way, but if all weights are correct, an evaluation of positions will no depends on user's subjective values and mistakes. The basic idea is that the 60-80% of the piece's value is determined by its average attack/speed and can be pre-calculated at the beginning of game, but then add a positional part of values to each piece depending on it's current position, attacks and moves limiting by obstacles, activity (value of attacking squares), acceleration, agility and potential. There is no fact that it will work as universal, but if there is successful, it will be great. Now, if any chess variant (with different armies) is imbalanced, there is no assurance that it is objective not only because of poor openings but because the engine makes decisions depending on evaluations of positions, which depend on the user's subjective values of his own pieces.If the user is wrong by his values, then the engine is wrong by its evaluations. And I believe that adding right positional criterias make the openings much better without dependings on the openings book. Because any passive openings like c3 or d3 will impair the positional power of own pieces, blocking their current moves and abilities. For example, move c3 kills orthodox Knight's potential to attacking squares d5 and a4, limits its speed and agility (moves x threats) to attacking b5 and e4 and commonly is the very bad move, except cases that square of d4 is under opponent's attack and it is important to defend this square or to prepare active move of d4 (not d3). There is no need to calculate any deep variants to understand for the engine that this move is terrible. Maybe, it will be better to create position's evaluating special config, in which user can define values of many position criterias (weights) if he want to correct the engine's playing style. And the next idea is to improve endgames by learning the engine any type positions. For example, user edit the position, place some pieces and then launch the analysis and the program automately generate "Nalimov's tables" for the endgames with these peaces. I find that now the engine have several problems to checkmating a bare king with the Wyvern or even with the several Queen-type units (sometimes make stalemating instead checkmating), although this is a problem only if there is very fast timing control. But as for me, it's very interesting to explore the endgames of Dragon (Chancellor) versus Griffin (Rook) and many others with new units, because these endgames haven't any existing Nalimov's tables yet. Dmitry Eskin wrote on 2016-11-30 UTC It doesn't work (5.0b3). Update: I add the string /sameColorGames 100 to the end of file winboard.ini and it seems to work correctly, thanks. H. G. Muller wrote on 2016-11-30 UTCYou left out the leading hyphen when you typed the option in the startup dialog. All option names (as opposed to the option values) should start with a '-' or '/'. (These are treated as equivalent; one is Linux style, the other is Windows style. The '=' between an option name and value is also optional, and can be a space instead.) So in the ini file it worked because you did write the '/'. Dmitry Eskin wrote on 2016-11-30 UTCI had saw that pawns' moves puts to hash as "c3" but with elvish diagonal pawns there are some different pawns able to move here. Update: sorry, it was my mistake - I use the same file for different match-ups, they were not elves in that game. As I understand, I need to use different hash-file for each match-up. Dmitry Eskin wrote on 2016-11-30 UTCH. G. Muller: I'm interested in your statistical method of evaluating pieces with Fairy-Max. But I don't know details and need to some advices. For example, I want to evaluate Pegasus, then change 2 Rooks by 2 Pegasus and launch long series. If I get a result of winrate of 60% (for example), how to convert it to centipawns? And how long match need to played for accuracy of 0.25 pawn? 0.1 pawn? 0.05 pawn? Is it works if I set a fast timing limits of 1 sec per turn, or maybe better to use another timing limits for such tests? How is better to test pawns if they have different options to promote (different versions of queens)? First test Queens, get values, exchange them and then test pawns? Do you have results for famous peaces, like Archbishop, Chancellor and Elephant (jB2) on the 8x8 board? What about base (orthodox) pieces: Knight, Bishop, Rook, Queen, King (as Man)? H. G. Muller wrote on 2016-11-30 UTC"For example, I want to evaluate Pegasus, then change 2 Rooks by 2 Pegasus and launch long series. If I get a result of winrate of 60% (for example), how to convert it to centipawns?" You can then delete the f-Pawn of the side that scored 60%, and repeat the test. If it now scores 45%, you know the Pawn was worth 15%, so the 60% translates to an advantage of 2/3 Pawn, which would be 33cP for a sigle Rook-Pegasus difference. If you do this for many piece comparisons,and each time deleting a Pawn makes the score drop ~15%, you can expect that to be always the case, and skip the Pawn-odds test if the score was close to 50%. "And how long match need to played for accuracy of 0.25 pawn? 0.1 pawn? 0.05 pawn?" The standard deviation of the score percentage in a match of N games is approximately 40%/sqrt(N). For N=100 that would be 4%. If a Pawn is indeed worth ~16%, that would be 25cP. But in 32% of the cases the error would be larger than that. To be within error bars 95% of the time you should count on an error of 2 standard deviations. To get that to 25cP you would need 4 times as many game, i.e. 400. To get the standard deviation to 0.1 Pawn (2.5x smaller) you would need 6.25 times as many games, i.e. 2500. For 5cp it would be 10,000. But I doubt that piece values can be defined with this precision anyway; the whole idea that pieces have aconstant value is only an approximation, and in practice the value could differ dependent on what the opponent has, or where they start in the opening. (As is most dramatically demonstrated by the fact that in 'Charge of the Light Brigade' 7 Knights crush 3 Queens.) "Is it works if I set a fast timing limits of 1 sec per turn, or maybe better to use another timing limits for such tests?" It is better to use time cotrolslike 40 or 60 moves per minute, than a fixed time per move, so that the engine can allocate the time where its needs it, and is not forced to abort a depth iteration that took unexpectedly long, and let the effort goto waste. But 40 moves/min should be fine. I never tried faster games. More later. Dmitry Eskin wrote on 2016-11-30 UTCI think that I need 10k games per unit. My aim is the balance not far than 55%, such as 52 +/- 2% in each match-up. There are 5 different unit's types: queens, knights, rooks, bishops and pawns, so for each type accuracy must be +/- 0,8%. And it equals 0.05 pawn per type (if pawn = 16%). "Per type" means the full complect (for example, 2 bishops or 8 pawns). Have you tested this values for orthodox pieces? This is no important for balancing (because of relative difference), but good to publicating absolute values. And I know that this is only statistics, real values are dynamical and positional. I think that if the Orcs will be the imba, the best way to balance them is limiting promotion of their Guards, for example don't promote to a Dragon. But the next 2 candidates, a Wyvern and a Werewolf are very closed, so Fairy-Max "promoting to only Queen" (where the "Queen" is one of that pieces #7 or #9) will not be as good as for natural Queen. Aurelian Florea wrote on 2016-11-30 UTCHello Dmitry, As a fellow inventor I'd like to congratulate you for taking the daunting endevour of creating a different armies game. I'm, personally on the fence for that as balance is hard to obtain. But in the end "Nothing worth doing is easy" (I don't remember who said that first). H. G. Muller wrote on 2016-11-30 UTC" My aim is the balance not far than 55%, such as 52 +/- 2% in each match-up. There are 5 different unit's types: queens, knights, rooks, bishops and pawns, so for each type accuracy must be +/- 0,8%. And it equals 0.05 pawn per type (if pawn = 16%). "Per type" means the full complect (for example, 2 bishops or 8 pawns). " That is not an efficient approach. Whether the armies are balanced follows from having the complete armies play against each other. That is only a single measurement, which would need a much lower accuracy (2% ~ 1/8 Pawn) than each of the individual piece values would need to make the sum accurate enough. Adding piece values is an approximation anyway. Some pieces cooperate better than others, some pieces combat some opponent pieces better than others. You would never see that when you measure the pieces in isolation. Of course you need to approximately know the piece values to make realistic measurements, though. (E.g. whether a player should seek a certain trade or avoid it could influence the valuesof the involved pieces.) Dmitry Eskin wrote on 2016-11-30 UTCH. G. Muller Maybe, it's not an efficient approach, but changing and testing only the one type of pieces doesn't suffer from the openings as all different army. Also, I don't know how much evaluation of pieces affects to engine's decisions. Then it's better to get these values first. In any case, it's more effective to test it by parts, for the first iteration 100 games per unit. Aurelian Florea Hello! Thank you, but it's not hard for me, because there is only 10-15 units, not hundreds, which I usually work with. Hard are only limitations that I set: no any exotic peaces, no any strange extra moves and no any special rules. And now I need to understand, is it a perfect balance now, or not. The automatic tests are interesting type of balancing tools, and it's a very big privilege to have it for chess, but it is unclear how they can be trusted and what limitations they have. Because the engine have features that may affect to results, or may not, and this is provisional, not real statistics of games played by real people. And for statistics there are very small numbers yet. Aurelian Florea wrote on 2016-11-30 UTCWell, Dmitry I'm not sure about the elo of fairy-max, but it seems fairly respectable, anyway better than my 1800 or so! Also HG method is ELO free, so it depends only on material balance after all substractions and divisions have been made! Dmitry Eskin wrote on 2016-11-30 UTCAt this post I will publicate and update statistics: Pawn = 63,0% of 1000 games (+/- 2%) Dragon (RN, Chancellor) + pawn f = Angel (Q), 51,9% of 500 games (+/- 3%) Phoenix (BN, Archbishop) + pawn f < Angel (Q), 46,0% of 500 games (+/- 3%) Exchanging the Queens also affects to pawns, so the advantage of Angel may be only 75% of the full, because other 25% difference pawns gains for the better promotion. 2 Wyverns (R>1) + pawn f > 2 Griffins (R), 54,1% of 500 games (+/- 3%) 2 Pegasus (jR3) > 2 Griffins (R), 53,6% of 500 games (+/- 3%) 2 Werewolves (N+) = 2 Knights (N) + pawn f, 52,5% of 500 games (+/- 3%) 2 Unicorns (Nx) > 2 Knights (N) + pawn f, 58% of 500 games (+/- 3%) 2 Hunters (B>1) + 2 pawns f/c = 2 Monks (B), 48,1% of 500 games (+/- 3%) 2 Centaurs (jB2) < 2 Monks (B), 39,9% of 500 games (+/- 3%) 8 Fairies (px) + Phoenix (BN, Archbishop) > 8 Footmen (p) + Angel (Q), 60,2% of 100 games (+/- 3%) 7f Guards (p+) + Dragon (RN, Chancellor) < 8 Footmen (p) + Angel (Q), 42,4% of 500 games (+/- 3%) Current equalities: Dragon = Angel - 0.6 Phoenix = Angel - 1.0 Wyvern = Griffin - 0.3 Pegasus = Griffin + 0.2 Werewolf = Knight + 0.6 Unicorn = Knight + 0.8 Hunter = Monk - 1.1 Centaur = Monk - 0.4 Fairy = Footman + 0.2 Guard = Footman + 0.15 I think that auto statistics can't be very exact, because of experience of other strategies shows that a balance depends on player's skills and styles. The balance will be different for grandmasters and novices, for humans and engines, with openings' books and endgames' tables and without it. But it is important to ensure that the basic balance of the matchups is within acceptable limits, 40-60% rather than 70% or higher. And the automatic tests can be useful for adjusting units evaluation as provide an alternative, machine evaluation as one of the approximate marks. So I will test separate units by 3 iterations: 100, 500 and 2500 games (200, 1000 and 5000 for basic pawn evaluating), that gives an accuracy of 0.25, 0.1 and 0.05 per unit (0.05, 0.02 and 0.01 for pawns). Then I will launch 1000 games for each match-up and 500 for mirrors. Mirrors are important to get statistics about white and black balance. All games will be launched at the minimal time limits, 1 sec per turn, because it provides greater accuracy per spent time. Overall (by the 1 iteration of test): Hero = 3.0 Footman = 1.0 Knight = 3.25 Monk = 3.5 Griffin = 5.0 Angel = 9.5 Guard = 1.15 Centaur = 3.1 Werewolf = 3.85 Wyvern = 4.7 Dragon = 8.9 Fairy = 1.2 Hunter = 2.4 Unicorn = 4.05 Pegasus = 5.2 Phoenix = 8.5 Dmitry Eskin wrote on 2016-12-01 UTC1) This is surprising that Centaurs are much stronger than Hunters 2) These values can't be used as average powers because they show only unit's potential from the very beginning of game; the side which have better potential, have better chances commonly 3) The starting power of Queen-type units is really high, much more than 2 Rooks 4) These values (starting powers) don't correlate with average power, neither of orthodox pieces, nor of new peaces with my calculations, that's why better accuracy is not important, and I think I will stop on the 2nd iteration, 500 games per unit. I have an idea for weakening Orcs, to deprive them of the right to castling, but at first, I need to collect their massive statistics before it and after it to ensure that this measure is really necessary. Jörg Knappen wrote on 2016-12-01 UTCInteresting game and worth trying out. I also love your pieces, specially the Werewolf and the Unicorn that are new to me. Here are a few remarks: The "jumping rook" and "jumping bishop" pieces are known as "ski rook" and "ski bishop" (think of ski jumping!) for a long time, for a reference see, e.g., here: http://www.mayhematics.com/q/mccs.htm Since your Chess Variant is a themed or Humans, Elves and Orcs, some artistic freedom in piece nameing is generally granted, But I think you are going overboard in renaming the Human pieces (the standard Chess pieces) only to create unnecessary confusion. Also, the name Phoenix is given traditionally to another piece (WA) and should not be reused. A Centaur is usually understood as a KN compound piece (also known as knighted King or crowned Knight). The piece you name Centaur is usually known as Ferfil (Fearful being a wordplay on that) or as Modern Elephant. For list of piece names, you may consult these references: http://www.chessvariants.com/piececlopedia.dir/whos-who-on-8x8.html (My favorite reference list, because you can find a piece when you know its approximate strength) http://www.chessvariants.com/index/mainquery.php?type=Piececlopedia&category=&startswithletter=&language=English&daysyoung=0&daysold=0&minyearinvented=&maxyearinvented=&boardrows=0&boardcols=0&boardlevels=0&boardcells=0&authorid=&inventorid=&orderby=LinkText&usethisheading=Search+Results&displayauthor=on&displayinventor=on®expurl=®explinktext= (The long link above gives a list of Variant Chess piece article in the piecoclopedia on this site) And an external link: https://en.wikipedia.org/wiki/Fairy_chess_piece Jörg Knappen wrote on 2016-12-01 UTC[Duplicate posting removed. --JKn] Dmitry Eskin wrote on 2016-12-01 UTCJörg Knappen I also love your pieces, specially the Werewolf and the Unicorn that are new to me. Thanks, they are very basic units, because the move of orthodox Knight often explained the same way as moves of Werewolf and Unicorn are. But the Knight is jumping over the first square, while the Unicorn/Werewolf can stay there. Werewolf/Unicorn were the first invented units but later Pegasus became my favourite unit, because its movement is very easy but its abilities are wonderful and I was very surprised why there are many such units as R2, R3, R4, R5 which are only weaker versions of Rook, but no one is jumping R3, although the Pegasus is the strongest basic orthogonal leaper which is balanced (jumping R4 is imba on 8x8 boards). Pegasus is a real antipode of Rook (Griffin), playing on the same lines but by the opposite way, preferring closed rather than open lines. The "jumping rook" and "jumping bishop" pieces are known as "ski rook" and "ski bishop" (think of ski jumping!) for a long time, for a reference see, e.g., here: http://www.mayhematics.com/q/mccs.htm I was very surprised when I didn't find the same pieces in fairy lists, because in my opinion they have very simple, almost basic, mechanics of move, much easier and more obvious than most fairy pieces. Someone had to invent something like them. Since your Chess Variant is a themed or Humans, Elves and Orcs, some artistic freedom in piece nameing is generally granted, But I think you are going overboard in renaming the Human pieces (the standard Chess pieces) only to create unnecessary confusion. I like orthodox pieces but don't imagine what the Bishop, Rook and Queen do at the battlefield and why they are battle units? I understand that the orthodox pieces are likely to retain their old names, but still suggest alternative names. On the other hand, orthodox names may be used and reserved as type names, for example Griffin, Pegasus and Wyvern are Rooks. Also, the name Phoenix is given traditionally to another piece (WA) and should not be reused. Yes, but I can't imagine any other unit as elvish "champion". Maybe the Dragon, but the Dragon is already used by Orcs. The Pegasus is almost ideal alternative, but it makes the Angel and the Dragon incomparable with them and need to find extra new names (and for the Elvish Rook too). A Centaur is usually understood as a KN compound piece (also known as knighted King or crowned Knight). The piece you name Centaur is usually known as Ferfil (Fearful being a wordplay on that) or as Modern Elephant. I'm thinking to rename Centaur to Harpy, but doubt because the Harpy is the 3rd flyer unit (Wyvern, Dragon). The Elephant is bad because it associated with clumsy, but this unit is much more agile. The main problem of "usually known" names is the same as orthodox names - they are not thematic (if we include several pieces to the same army) and often not logic. There is a huge number of different names of fairy pieces, and it is a problem for any developer. Because if you want to use, for example, the image of the Dragon or the Griffin to thematic army, suddenly you find that these pieces are already exist but they do not at all what you need for the game and for the balance. The names are controversial, but at least now they are 100% relevant to the theme and how these units work (for all new people who don't know any peaces besides orthodox). I can't find another good names, to achieve two goals at once, that's why I had chose only thematic names. I'm opened to new suggestions for names, but thematic and logic are the most important aspects. Maybe, I should rename Fairy to Sprite, and Pegasus to Valkyrie. Valkyrie is something that can be Elvish Queen instead of Phoenix, but which name give to Elvish Rooks? Dmitry Eskin wrote on 2016-12-02 UTCH. G. Muller If I'll want to make Elvish Pawns have leaping push like spartan pawns: h:70 15,E4 17,E4 16,5 15,6 17,6 E4 means this leaping push? If I'll want to make Orcish Pawns have promotion only to Nightriders (unique unit, which is absent in starting army), then I would put the Nightrider to #7 (or #9 black), moving the Orcish Queen to the end of list, it will work correctly? I think that promotion to Nightriders is "positive" nerf making Orcish games more interesting than standard promotion (to Dragon usually). And this is better than a ban of castling or a single ban of promoting to Dragon. I like Nightriders but as a rare option, not at starting army (because it is upgraded Knight and have difficult move). It is interesting that although Nightrider is stronger than a Rook/Wyvern, it can't checkmate a bare King (need the second Nightrider). The same story with the other Orcish Knights - Werewolves. Jörg Knappen wrote on 2016-12-02 UTCWhile there are lots of evil creatures (to be associated with the Orcs) in Tolkiens legendarium, the number of good or ambivalent races is rather limited. There are Goblins, Hobgoblins, Uruk-Hai, Trolls, Balrogs, Dragons and Worms, Wargs, and the Nazgul with their (unnamed) flying animals. For the airforce of the "good ones", there are the Eagles (taking part in the Battle of Five Armies). Than, there are the Ents, and maybe an Ent is a good picture for a rookish piece. Of course you can look up other mythologies for suitable names. Dmitry Eskin wrote on 2016-12-02 UTCJörg Knappen Eagles are bad because there are Griffins (Human Rooks), which are hybridize eagles. Ents are good, but require redefining all the Rooks from flyers to something siege. There is no problems for Orcs because of Cyclops instead of Wyverns. But there is a problem for Human, because the closest analogue is Elephant (instead of Griffin). As an alternative there is a (sieged) Tower, usually associated with the Rook (by image), but I'm not sure that it is a good unit. In general, there is a very interesting idea to use exactly Tolkien's setting, but I'm afraid that I haven't enough imagination to find matching units for all 3 races. H. G. Muller wrote on 2016-12-02 UTC"E4 means this leaping push?" Indeed. Fairy-Max uses the primary rights-code '4', which would mean a move without any rights and without continuation (which thus would be totally useless) as an 'escape' for indicating moves that only 'virgin' pieces have. The preceding hex digit is then used to indicate what the move can actually do: a slider step creating e.p. rights on the visited square, (which is interpreted as castling on a royal piece), a sliding step creating no e.p. rights, or a jump over the square. The lowest two bits would specify the rights for the next (terminating) step. For E = 1110 this would be move-but-not-capture "If I'll want to make Orcish Pawns have promotion only to Nightriders (unique unit, which is absent in starting army), then I would put the Nightrider to #7 (or #9 black), moving the Orcish Queen to the end of list, it will work correctly?" That might not work. IIRC Fairy-Max will always use #7 for both sides, unless the initial setup would contain a #7 for white, and a #9 but not #7 for black. So if black does not initially have #9... Of course the code can be easily changed; the rule that each side promotes to #7 if it has a #7 in the initial setup, and to #9 otherwise, irrespective if they have it or not would have the same effect in all currently supported variants. And Fairy-Max 5.0 takes the promotion piece from a board-size table ayway (to make Grant Acedrex possible, where the piece is determined by the promotion square). Before version 5.0 Fairy-Max did have problems with Nightriders, because these make mutual perpetual chack possible, Sooner or later the search would stumble on a position where this occurred, and because each check evasion extends the search by 1 ply (i.e. the check evasions are not counted in the search depth), this wouldleed to infinitely deep recursion, and a crash due to stack overflow. But Fairy-Max 5.0 doeshave in-search recognition of repetitions, and I think this should cure thisproblem. (Otherwise Griffons (in theGrant Acedrex sense) would also have been a problem, as these can do this too. Dmitry Eskin wrote on 2016-12-07 UTCCurrent statistics with Fairy-Max: Human-Orc: 40,1% (human) by 500 games Orc-Human: 43,6% (human) by 500 games Human < Orc, 41,85% by 1000 games (+/- 2,5%) Elf-Orc: 42,3% (elf) by 500 games Orc-Elf: 40,8% (elf) by 500 games Elf < Orc, 41,55% by 1000 games (+/- 2,5%) Human-Elf: 49,2% (human) by 500 games Elf-Human: 48,1% (human) by 500 games Human = Elf, 48,65% by 1000 games (+/- 2,5%) Unit's power by statistics (500 games per unit): Hero = 3.0 Footman = 1.0 Knight = 3.25 Monk = 3.5 Griffin = 5.0 Angel = 9.5 Guard = 1.15 Centaur = 3.1 Werewolf = 3.85 Wyvern = 4.7 Dragon = 8.7 Fairy = 1.2 Hunter = 2.4 Unicorn = 4.05 Pegasus = 5.2 Phoenix = 8.2 Dmitry Eskin wrote on 2016-12-08 UTCOverall: Human = Elf Orc > Human/Elf by 0.6 pawn I think that there are two good ways to balance Orcs: 1) Orcish pawns promote only to Nightriders (special unit) but it seems as too big nerf, or 2) Orcish pawns promote only by simple turn (don't promote automatically) In both cases I need to configure promotion at Fairy-Max. Dmitry Eskin wrote on 2016-12-15 UTCUpdate rules: Guards (Orcish Pawns) can promote only to minor pieces: Werewolves (Orcish Knights) and Harpies (Orcish Bishops). Update statistics: Human-Elf: 49,2% (human) by 500 games (+/- 3,5%) Elf-Human: 48,1% (human) by 500 games (+/- 3,5%) Human = Elf, 48,65% (human) by 1000 games (+/- 2,5%) Human-Orc: 48,2% (human) by 500 games (+/- 3,5%) Orc-Human: 47,7% (human) by 500 games (+/- 3,5%) Human = Orc, 47,95% (human) by 1000 games (+/- 2,5%) Elf-Orc: 50,3% (elf) by 500 games (+/- 3,5%) Orc-Elf: 48,1 (elf) by 500 games (+/- 3,5%) Elf = Orc, 49,2% (elf) by 1000 games (+/- 2,5%) Update units' power: King: Hero = 3.0 Queen: Angel = 9.8 Rook: Griffin = 5.0 Knight: Knight = 3.3 Bishop: Monk = 3.3 Pawn: Footman = 1.0 Queen: Dragon = 9.0 Rook: Wyvern = 4.8 Knight: Werewolf = 3.9 Bishop: Harpy = 2.9 Pawn: Guard = 1.1 Queen: Phoenix = 8.6 Rook: Pegasus = 5.1 Bishop: Ranger = 2.2 Knight: Unicorn = 4.1 Pawn: Sprite = 1.2 Dmitry Eskin wrote on 2016-12-16 UTCIt was surprising for me, but as it has appeared, that a pair of Unicorns can ckeckmate a bare King without the help of the own King, like a pair of Werewolves. The demonstration of this: Unicorns 25 comments displayed⇧Earliest ⇧Earlier ⇧Reverse Order⇩ LaterPermalink to the exact comments currently displayed.