[ Help | Earliest Comments | Latest Comments ][ List All Subjects of Discussion | Create New Subject of Discussion ][ List Latest Comments Only For Pages | Games | Rated Pages | Rated Games | Subjects of Discussion ]Comments/Ratings for a Single Item Later ⇩Reverse Order⇧ Earlier Alibaba. Jumps two orthogonally or diagonally.[All Comments] [Add Comment or Rating]Matteo Perlini wrote on 2012-09-24 UTCI thanks to all of you for the intriguing posts. I keep following the discussion. ----H.G. Muller---- Empirical values of symmetric short-range leapers suggest that the the value is mainly determined by the number N of moves they have (when unobstructed), according to the approximate formula value (30+5/8*N)*N. Measurements on asymmetric and divergent pieces suggest that forward moves in this respect contribute roughly twice as much value as sideway or backward moves, and captures contribute roughly twice as much as non-captures. ------------- I need examples for determining the value of N. The pawn move in 1 square (forward move) and capture in 2 square (forward moves). The ferz move and capture in 4 square (2 forward moves and 2 backward moves) and it is color-bond. How the value of N is calculated for pawn and ferz? H. G. Muller wrote on 2012-09-21 UTCResults of tonights run (2 x 280 games, for a statistical error of ~2.5%): 2 N - P vs 2 AD: 62.5% 2 AD vs N + P: 55% Both results suggest that Alibaba + Pawn is worth just slightly more than a Knight. I expect little systematic error in this, because I compare Alibabas directly to Knights, which are very similar pieces. (Neither of them has mating potential, alone or in pairs.) Note there is a general effect that drives the value of nearly equal pieces (or to a lesser extent piece combinations) on opposing sides towards each other: A more valuable piece loses value through the presence of weaker opponent pieces, because it cannot go on squares covered by such a weaker piee even if protected. By this effect, if the difference is only small, the loss of value by treating it as stronger (i.e. avoiding trades) could reverse the difference. In this case better use of the stronger piece is made by treating it as exactly equal. Then you cannot avoid it will be traded for the weaker piece, but constantly having to protect it from attacks by the weaker piece would lose you even more. This effect is largely responsible for the fact that it is so very hard to say whether a Knight is better or worse than a lon Bishop in orthodox Chess. I think the Alibaba case is similar, although not as outspoken: its intrinsic value is just a bit better than N minus P. But avoiding AD + P vs N trades (by not daring to go on squares protected by P that are twice attacked) loses you more of the value than no yielding to such an attack whe the opponent tries to chase away the Alibaba. The value loss is not as bad as having to avoid 1-on-1 trades, but squares protected by Pawns and nothing else are very common, and just the sort of 'outposts' where you would like to put your Alibabas in the middle game to have any use of them at all. So when the opponents has Knights on an 8x8 board, the Alibaba should probably be treated as if it is exactly one Pawn below a Knight (or lone Bishop). Color-binding of the AD is of course very bad, and manifests itself in the end-game as that the reduction for 'unlike Bishops' now should even be applied in end-games of Alibaba versus a color-alternator like Knight (in presence of Pawns). In KBPPKN the two Pawns are usually enough to secure the win, because to stop a Pawn from crossing a square where the Bishop has no clout, the Knight has to stand on the color where it can be attacked by the Bishop (or, when blocking the Pawn, has to move because if zugzwang). So you cannot set up a defense purely on the color the Bishop cannot reach. But you ofen can set up a defense on a color the Alibaba cannot reach, even with a Knight, as the Knight keeps jumping between two 'meta-colors' unreachable by the Alibaba. H. G. Muller wrote on 2012-09-21 UTCSecond attempt. If they keep disappearing, I will have to give up at some point, but fortunately I had this one from yesterday still under the Crtl-V key... > And how will you know that there isn't some other potential flaw that you haven't thought of? Well, 'exclude' in this context should be taken to mean 'beyond reasonable doubt', and not in any mathematical sense. The whole concept of piece value has no mathematical meaning to begin with. It is just a heuristic for estimating your winning chances. The point is that the empirical method is in general quite insensitive to poor play, and that the effects one might not think of (and even those we do think of) are tiny to begin with. Joker will stupidly 'gain' a Knight for his two Pawns in KNPPKN, thinking that it ups his score from +2 to +3. Because I never told it KNK is a hard 0, rather than +3. But the number of games where it ends in KNK or KBK in practice is very small, less than 1% even when you play it against engines of similar strength that do have this knowledge, which is why I never bothered to correct this behavior. Even such a major blunder hardly affects the overall score. And, even more important, the method uses self-play. When both sides are unaware of KNK being a draw, it happens even less frequently, because the side that is behind will foolishly resist the opponent 'driving up' his advantage to as much as a full Knight. So somtimes the side with the Knight will discover to his surprise that he cannot win in a +3 position because it is KNK. But at other times he will be lucky in a Knight ending, because his opponent will fail to recognize he could draw by a Knight sac in KNPKN, and will win rather than draw because the opponent does not get a second opportunity by the time they can see the promotion, and count the P as Q. The lack of knowledge always works both ways. This is why I would be very surprised to see a large effect of including mate-potential knowledge. And that holds even more for more subtle effects. > Should we dismiss a bunch of experts that have made no mistake you can point out simply because we also can't point out a mistake that your engine has made when it gives a different result? I am a bit skeptical when I hear the word 'expert'. How many games would these experts have played? My computer does thousands in a few days. So one obvious mistake I think I can point out for the experts is: 'not enough games to get a representative sampling of possible situations'. Furthermore, in human games it is difficult to have players of equal strength. Even if their overall strength is equal, they might have different weaknesses. A player that is pretty adept with Commoners will use them to spring unpleasant surprises on his opponent who isn't,and doesn't see the danger coming. So that player could mistakingly get the impression that the Commoners are pretty strong, while in fact the only thing he is proving is that his opponent is 'Commoner-blind'. It is just very hard to get the cancellation of weaknesses that is automatic in self-play. In addition computers do not suffer from systematic oversight,as they rely on exhaustive search. People rely on ideas, and the ideas can easily be wrong or incomplete. > Alternately: there are significant differences between the ways humans and computers play chess, so theoretically some pieces could have a different effective value in human vs. human games than in computer vs. computer... Yes, this is a possibility. But if the play isn't very poor, the empirical results turn out to be surprisingly independent of the strength of the players. So I would be very surprised if it did depend very much on the strength of the strategic or tactical contributions to this overall strength. Which seems the major differencce between computer and human play. It is a case in point that computers tend to value the Queen higher than humans do. But even statistics from human games shows that the Queen is more valuable than the humans think (9.75 rather than 9.5). So apparently humans have a tendency to underestimate the value the Queen has to themselves! But that doesn't fully disguise the fact that it is stronger. Jeremy Lennert wrote on 2012-09-20 UTC> > Under what circumstances would you possibly be able to exclude it? > When I have run the test games with an engine that does take full account of the mating potential or lack thereof. And how will you know that there isn't some other potential flaw that you haven't thought of? Presumably a less thoughtful person in your position could have failed to consider the issue you are now addressing, and would therefore already be just as convinced as you expect to be after your next test. Should we dismiss a bunch of experts that have made no mistake you can point out simply because we also can't point out a mistake that your engine has made when it gives a different result? Alternately: there are significant differences between the ways humans and computers play chess, so theoretically some pieces could have a different effective value in human vs. human games than in computer vs. computer. I don't see any particular reason that Commoner should be such a piece. But if it were, a bunch of human experts agreeing on one value and computer tests reporting a different value is pretty much what we would expect to see, right? H. G. Muller wrote on 2012-09-20 UTC> Under what circumstances would you possibly be able to exclude it? When I have run the test games with an engine that does take full account of the mating potential or lack thereof. Then there would be no reason why pieces with mating potential would be under-estimated compared to pieces without it, as the engine would not miss any tactical shots that secure a draw by exploiting such lack of mating potential, and would also resist putting itself in a position where the opponent can pull such a trick (e.g. like trading from KNPPKNP into KNPKN, so the opponent gets handed a NxP sacrifice as an easy drawing strategy). Spartacus should be able to do this: it recognizes drawish material combinations, and will reduce the evaluation by a factor if the naive additive evaluation says the side with such material has the advantage. In particular this applies when the leading side has no Pawns, and is not more than a minor ahead, (reduced by a factor 8), and positions with a single Pawn where the opponent would not more than a minor behind when he sacrifices his weakest piece for that Pawn (reduced by a factor 4). Unlike color-binding of the last piece on both sides also gets a reduction (by 2). The concept of 'minor' deserves some attention here: strictly speaking a minor is defined as a piece without mating potential, but most Chess players think about it as a piece with the value of about a Knight, as in orthodox Chess that amounts to the same thing. But there are pieces that in my tests perform like a Knight or even slightly weaker, despite the fact they have mating potential. (E.g. Commoner (FW), and Woody Rook (WD); even Gold (WfF), a subset of the Commoner, has mating potential on 8x8!) This makes the rule-of-thumb of 1 minor ahead a bit tricky: WD + N vs WD is only a Knight ahead, incontestably a minor, but it is a general win, as the stronger side can use his superiority to force the trade of N vs WD, to be left with a winning WD. The mating potential has no special advantage for the defender. With R + N vs R the defending R is too strong for the N to get a grip on it, and the only trade that can be forced is R vs R, which makes it a draw. So the rule seems to be that the stronger side can force a trade of his choice of an approximately equal piece, but the weak side can make a sacrifice of his choice (so that 2B+N vs R is a win, but in B+2N vs R the R goes for the B, making it a draw). Jeremy Lennert wrote on 2012-09-20 UTC> Not 'probably'. I just cannot exclude it. Under what circumstances would you possibly be able to exclude it? H. G. Muller wrote on 2012-09-20 UTC> (Commoner) One situation where your empirical measurement differs from common wisdom, and you think it is probably due to a failure of your engine; Not 'probably'. I just cannot exclude it. But I don't really expect it to make much difference if the engine would take full account of mating potential, and that it is probably the common wisdom that got it wrong. But it obviously has to be checked. Anyway, first results of the Alibaba trial: I played a pair of AD (replacing the Knights in FIDE) against 2N - P and against N + P. Both for about 200 games, which makes the statistical error slightly under 3%. With Pawn-odds producing an excess score of 15% (i.e. 65-35), this translates to an error of 0.2 Pawn units (1 standard deviation). 2N - P proved superior, scoring 15.5% in excess of 50%. 2 + P proved inferior, with a score deficit of 6%. Translated to Pawns that would be 1.0 and 0.4 Pawns, respectively, suggesting that the difference between 2N - P and N + P, which is N - 2P, would be 1.4 P. So that would make N = 3.4P. That fits very well with the Kaufman value N=325, especially taking into account that the statistical error of the measurements combined is 28. It would be even better when recognizing a Pawn on f2/f7 is not a particularly strong Pawn, and correct its value to 95. It seems that an AD is almost exactly worth 1 Pawn less than a Knight (230 on the Kaufman scale). But as there no doubt will a Pair bonus, and this was measured on pairs of AD, a single AD will be worth slightly less, the second AD slightly more. The trials were performed with internal value AD=270, however, and are thus not entirely self-consistent. I will repeat them with AD=240 and P=90 now. Derek Nalls wrote on 2012-09-20 UTCIn some of HG Muller's earliest comments under different topics, he did his best to explain in detail why the bishop-knight compound (or archbishop) has a significantly higher piece value than the naive sum of its parts in Capablanca chess variants. Jeremy Lennert wrote on 2012-09-20 UTCI thought the Rook measurement was off by at least half a pawn, but perhaps my memory is in error. In any case, we have: - (Commoner) One situation where your empirical measurement differs from common wisdom, and you think it is probably due to a failure of your engine; - (Rook) One situation where your empirical measurement differs from common wisdom, but you think it's a failure of definition and you weren't really measuring the same thing; - (Bishop-Knight) One situation where there's a strong possibility that your empirical tests have shone light on an important lesson for theorists, but we still don't know WHY this particular piece would have such a high value, so we can't extrapolate from it (for example, we can't make reliable guesses about the value of a Bishop-Camel or a Bishop-Nightrider) I have great respect for your work and I think it's very valuable, but these still strike me as emblematic of how far we have yet to go. H. G. Muller wrote on 2012-09-19 UTC> Odd, I seem to remember reading about your Joker engine testing material values for FIDE and getting a Rook value that was unexpectedly low. By now I am convinced that this was just a discrepancy between the value of a Rook in a closed position and in a position with many open files. It is well known Chess lore that it is quite important to get your Rooks on open files. A positional bonus that automatically gets added to the Rook value as the board empties, and virtually all files become open. The (opening)value I found was only a quarter-Pawn low, and awarding a bonus of that magniture for being on an open file does not seem excessive at all. So I think this is not so much a problem with the empirical method, but more with the concept of piece values itself. Relative values of pieces are not constants of nature, but change as the game progresses from opening to end-game. The additive model for material value, where you add values of individual pieces to get the value of the army, is also far from perfect. The Bishop-pair bonus is a clear example of a cooperative effect, as is the dependence of the B-N difference as a function of the number of Pawns in the Kaufman model. How large these cooperative effects can grow is most convincingly shown by the fact that 7 Knights totally crush 3 Queens in the presence of Pawns, while the Kaufman values suggest that the Knight side is 'two minors down'. >I also seem to recall the same engine testing the Bishop-Knight compound and finding its value unexpectedly high compared to the Queen and Rook-Knight (closer than the values of their component pieces). And I do not recall anyone offering a predictive theory capable of explaining that. That the empirical method finds unexpected things by no means implies there is a problem with that method. It is much more likely there was a problem with the expectations. Not to mention the fact that 'predictive theories' are usually based on little else than the most simplistic analogies (like R>B, so RN must be > BN, and never mind B is color-bound and has no mating potential). I have intensely watched many dozens of long-TC 10x8 games between strong engines, and I have no doubt at all the empirical determination that RN - BN ~ 0.25P is correct. In all cases where an imbalance of BN vs separate R + N or R + B developed, the owner of the two lighter pieces was utterly and mercilessly slaughtered by the BN. (In the presence of several Pawns, of course; BN vs R is draw in itself, So BN + P vs R + B is already very drawish.) That piece is just so powerful... > Betza also performed computer tests and human playtests on the value of the Commoner (nonroyal King), and was convinced that the computer value was wrong. Well, I don't now what computer testing Betza did, knowing he rejected the use of commercial software like Zillions. I admit that my tests leave room for under-estimations of the Commoner value, as the engine with which it was done did not properly pay attention to mating potential. So it would fail to recognize the possibility to draw by sacrificing a piece for the opponent's last Pawn when the opponent's remaining piece lacked mating potential. Which might lead to unjust wins by the opponent, if it was too late to force such a trade when promotion came within the horizon. > Have your formulas for short-range leaper values been verified by anything other than your own chess engine? No, they have not. > Incidentally, did you ever finish that new chess engine you were working on that you said you wanted to complete before running more complicated tests? Spartacus, I think. Unfortunately also little progress there. I am just too busy with other Chess(variant)-related projects. Like setting up an internet server, adapting WinBoard to play large Shogi variants, and writing an engine for those. But I really should put some more effort in it, because it already is at a level where it heavily outplays Fairy-Max in the variants that it plays. It is not so generally configurable for all kinds of unorthodox piece types as Fairy-Max is, though. But it certainly should be suitable to do a precision determination of the value of the Commoner, as it does take account of lack of mating potential in its evaluation function (multiplying the naive evaluation towards zero in 'drawish end-games', where the opponent can afford to sarifice his lightest pieces for your remaining Pawns). Jeremy Lennert wrote on 2012-09-19 UTC> In my empirical piece-value determinations I never noticed any significant discrepancy with the orthodox values. Odd, I seem to remember reading about your Joker engine testing material values for FIDE and getting a Rook value that was unexpectedly low. I also seem to recall the same engine testing the Bishop-Knight compound and finding its value unexpectedly high compared to the Queen and Rook-Knight (closer than the values of their component pieces). And I do not recall anyone offering a predictive theory capable of explaining that. Betza also performed computer tests and human playtests on the value of the Commoner (nonroyal King), and was convinced that the computer value was wrong. I recall that you disagree with him on this point, but that Wikipedia article I linked a couple of posts back cites two sources that seem to at least vaguely support his conclusions (end-game fighting power of the king placed higher than knight or bishop). Have your formulas for short-range leaper values been verified by anything other than your own chess engine? It's certainly impressive, and it's plausible, but if it's based entirely on one source, then it's hard to tell how far we should trust it. Perhaps more importantly, symmetrical short-range leapers are a rather special subclass of pieces; I would be hard-pressed to name a single CV whose pieces all fit into that class. Incidentally, did you ever finish that new chess engine you were working on that you said you wanted to complete before running more complicated tests? Spartacus, I think. H. G. Muller wrote on 2012-09-19 UTCI don't think the situation is as bleak as you describe. I think it is pretty well accepted nowadays that the orthodox pieces have value 100, 325, 325, 500, 950 (centi-Pawn), the so-called Kaufman values. And that the bonus for a Bishop pair is 50 cP on top of this. There is some discussion as to whether in Human games a Queen should be 950 or 975. (Human GM's feel that 975, with the raw statistics on human games suggests, is too much. The strongest computers, however, usually value a Queen even higher.) Of course the value of a Pawn depends very much on the kind of Pawn (edge, backward, doubled, passer, protected passer, connected passer). But that doesn't mean there is much disagreement over how much a specific kind of Pawn would be worth. In my empirical piece-value determinations I never noticed any significant discrepancy with the orthodox values. Empirical values of symmetric short-range leapers suggest that the the value is mainly determined by the number N of moves they have (when unobstructed), according to the approximate formula value (30+5/8*N)*N. Measurements on asymmetric and divergent pieces suggest that forward moves in this respect contribute roughly twice as much value as sideway or backward moves, and captures contribute roughly twice as much as non-captures. The predicted values are only gross averages, though, around which the values of different pieces with a given number of moves cluster. Within such a group there can be significant differences, and all the factors you mention no doubt contribute to that. They don't seem to be worth very much, however. I did experiments by adding a single backward-step capture (providing mating potential) and non-capture (lifting color-binding), and the resulting piece was hardly worth more than an ordinary Bishop. These are tentative conclusions, however, and need to be repeated by an engine with more end-game knowledge. What seems more important is the way how the various moves cooperate in actions for supporting and attacking Pawn chains. But the Alibaba, with its scattered moves, also scores pretty poor in that respect. I have started a match now where one of the FIDE armies has its Knights replaced by Alibabas, and gets a Pawn advantage in compensation (deleting the f-Pawn). After a few dozen games the Knights are ahead, but it will take many hundreds of games to be sure this isn't just a statistical fluke. If the Knights prove superior, (i.e. 2 AD + P < 2 N), I will try to play 2 Alibabas against a single Knight, giving additional Pawn odds in favor of the Knight. Jeremy Lennert wrote on 2012-09-18 UTCThere have been many attempts to write mathematical formulas or create tables of piece values, but I don't think any have gained widespread acceptance. Authorities can't really even agree on the values of the orthodox pieces in FIDE Chess, so there's no accepted way to determine whether any given valuation is "correct" (or even precisely what it would mean if it were). Some people have tried computer tests to determine values empirically, which I think is a promising direction, but results don't always agree with other computer tests or with the accepted values (such as they are) of the orthodox pieces, so it's not always clear what they mean, and they're not much help in predicting the value of any piece that wasn't specifically tested. Most people agree that value is primarily related to a piece's "mobility", or how many different ways it can typically move or capture (after somehow accounting for the fact that certain moves are more likely to be possible than others; e.g. a Bishop can move 7 squares, but only in rare situations). But then there's a bunch of other factors that we're pretty sure are real but that no one really knows the true value of, like mating potential, development speed, colorbound pairs, stealth, how they cooperate with allied pieces, and so forth. Notice that many of these aren't even intrinsic to the piece, but relate to the other pieces on the board, which means that they change from variant to variant (and even over the course of a single game, as pieces get captured and removed from play). I found Ralph Betza's About the Values of Chess Pieces to be helpful when I started researching piece values for For the Crown, so that might be a good place to start reading if you want to know more. Matteo Perlini wrote on 2012-09-18 UTCThanks John for you exhaustive reply. The problem for having 4 alibabas on different "colors" is you should have 2 alibabas on the first row and 2 on the second row. But the majority of CVs has just pawn on the second row. Anyway I would like to know if there is a general formula to determine the piece's value on 8x8 board with FIDE rules. Or a table with the values of the most common pieces. John Lawson wrote on 2012-09-15 UTCIn http://www.chessvariants.org/piececlopedia.dir/ideal-and-practical-values-3.html , Betza gives no value, but explicitly states the AD [funny notation] is in practice worth "much less than a knight". Jeremy Lennert wrote on 2012-09-14 UTCThe alibaba has similar qualities to a knight (leaping piece, similar range, same number of moves), but its movement pattern confines it to 1/4 of the board (similar to how bishops can reach only 1/2 the board, but moreso). Betza suggested somewhere in this article on the Crooked Bishop that a non-colorbound piece would be 1.1 to 1.2 times more valuable than its colorbound equivalent, which means a colorbound knight ought to be worth ~87% of a knight. But the alibaba is colorbound "twice", which I would imagine warrants at least applying the penalty a second time, giving ~76% of a knight--a very good match for your estimate. (Though I wouldn't be surprised if "double-colorbound" turned out to warrant a greater penalty than that.) However, Betza appears to have been talking about pairs of colorbound pieces on opposite colors (like the bishops in FIDE Chess), which are generally believed to be more than the sum of their parts. So that estimate is probably only good if you start with 4 alibabas, one on each "color"--they will be less valuable individually, especially in the endgame (when it becomes harder to find targets on your own "color"). Finally, if you are playing with bishops, rooks, and queens, I would guess that the knight is also benefitting from a "stealth" bonus, due to its ability to threaten these pieces without being threatened in return. The alibaba can only do this if there is another piece in the way (that it can jump over but the other pieces can't), and so I would guess that its true value would be a little bit lower again than the estimate above. So, in summary, I would guess than 3/4 of a knight is close, but probably a little too high, and would expect the value to fall significantly in the endgame or if you don't get a complete set. Matteo Perlini wrote on 2012-09-14 UTCHow is the value of alibaba compared to other orthodox pieces in a 8x8 board? By intuition I would say 3/4 of a knight. Charles Gilman wrote on 2004-11-28 UTCThe Alibaba appears as a promotee in my Courier variants, where Pawns are promoted to compounds of the (simple) array pieces (http://www.chessvariants.org/other.dir/n_europe.html), and under the name Sail in Ken Franklin's Leap Chess (http://www.chessvariants.org/44.dir/leap-chess.html). Another place where it could occur is in a Fusion Timur's Chess along with the Bishop, Gnu, Marshal, Prince, Waffle, and many less familiar pieces. Picket+Elephant could be marked by the ability to leap two intermediate pieces, one for each component, and given a name along the same lines as Alibaba: Alitalia! Charles Gilman wrote on 2004-01-10 UTCExcellent ★★★★★The same-armies game with a total of 40 Thieves is a neat idea, but perhaps there should be at least one more colour-changing piece to increase interaction between the sets of squares - a Gnu (Knight+Camel) would fit in with the scale of other pieces. How about 12 files with TTTTTGKTTTTT at the back and TTTTTAATTTTT in front? I see the idea behind Pickpocket - a pocket-shaped set of moves. However, there is another use of of that name that MIGHT be more use. I have recently been considering extrapolations of the Picket of Timur's (aka Tamerlane) Chess, which is a sort of 'Bishop minus Ferz', and Pickpocket has an obvious commonality in its lettering. So how about Pocket for 'Rook minus Wazir', Pickpocket for forward moves of both (plus others if Shogi-prefixed), and Fagin for Picket+Pocket? Doug Chatham wrote on 2003-12-29 UTCHmmm...how about a 44-square Hecatomb-like variant where each side gets a King, an Alibaba, and 20 promotable Thieves? Call it something like 'The Game of Forty Thieves' :) L. Lynn Smith wrote on 2003-12-27 UTCMight an Alibaba without the forward orthogonal leap be called a Pickpocket? ;-) Charles Gilman wrote on 2003-12-26 UTCGood ★★★★Would a (presumably promotable) piece with just the three forward moves of the Alibaba be a Thief? Perhaps there could be a variant with an army of a King, an Alibaba, and a bunch of Thieves against the FIDE army, with the Thieves stealing the move of any FIDE piece jumped over except a Pawn and becoming Alibabas if they reach the enemy camp without jumping anything. Would 14 or 22 Thieves best match the FIDE army, and should the robbed pieces become Pawns or Thieves? Possibly more interesting intrinsically than the Thief would be a Goldthief with 6 Alibaba moves and a Silverthief with 5 - and anyone familiar with Shogi should be able to work out which 6 and which 5! Charles Gilman wrote on 2003-03-01 UTCGood ★★★★Alibaba is a very clever name for this piece. It is meaningful in its own right and also uss elements of both the pieces whose moves it combines. 23 comments displayedLater ⇩Reverse Order⇧ EarlierPermalink to the exact comments currently displayed.