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My family has just moved to china, and my son and I are learning Xiangqi.
We have a question about a move his classmates are using regarding the
kings on open file.  If there is a peice between the two kings, and that
piece is moved, the other player is immediatly capturing the other king
to
win the game.  My interpetation is that this is an illegal move, like
putting yourself into check.  Are the kids (high school level) just
playing with house rules or is this a special scenario?

In XiangQi, it is suppose to be illegal to expose your King to capture. 
But these students might merely be playing these games through to capture
in order to learn its nuances.  That way they can learn to visualize all
these potential lines of attack on the King and realize a 'checkmate'
position.

Then again, it might be a local form of play. ;-)

World Chess Champion Emmanual Lasker (and some other chess writers) put the fighting value of a king at 4 pawns. I have been wondering how he arrived at that estimate, and today came up with a theory, however unlikely that it's true. If we take his (also commonly used) values of P=1;N=B=3;R=5 and Q=9, we can reach an average fighting value for the king if we look at 4 basic endgame situations:

1) K normally stops P+N, so in this case let's say the K's stopping power (or fighting value)=P+N=4;

2) K normally stops P+B, so in this case the K's fighting value =4 again;

3) Let's say K+P draws vs. R considerably more often than not, so in this case the K's fighting value =4 again;

4) Let's say K at least restrains 3 passed pawns more often than not, so in this case the K's fighting value =3.

The average fighting value of a king based on the above would be (4+4+4+3)/4 =3.75, or rounded to the nearest 0.5 pawns would be =4 pawns, Lasker's given figure.

Another matter I've wondered about is, if we assume a Guard's value is at least that of a king's fighting value, in chess variants that include Guard(s), what values should a King and a Guard have respectively in such variants (assuming for a start that the values will be close)?

Well, if a chess variant was 8x8 and had all the piece types in chess, plus one or more Guards, then the above 4 endgame situations would still be relevant to a Guard or King's value. In fact, the above four would lead to the final value for a Guard in such variants, the way they did for a king in chess. A king's value must be recomputed for such variants, however, by averaging in a fifth endgame situation:

5) Let's say a K+P and Guard+P(on the same file) situation favours the Guard about 1 time out of 4, in that the Guard drives away the K and wins the enemy P - in this case the fighting value of a K is only worth about Guard-P=3. All guesswork, I suppose, but I'm only demonstrating the sort of calculation I'd use if I knew more about this case.

Thus, the average fighting value for a king in such 8x8 variants as I described earlier would be (4+4+4+3+3)/5=3.6 pawns, or 3.5 if rounding to the nearest 0.5 pawns.

For the record I would use P=1; N=3.49 (or 3.5 when rounded); B=3.5; R=5.5; Q=10 if doing my own calculations for the above values of the K and Guard. Also, computer studies put the value of a Guard at 3.2 pawns, I recall.

This is what is known as 'observational bias': you select favorable situations, and ignore those where the performance of the King did not impress you so much. No doubt Lasker did the same.

But a fact is that a King loses more often than not against a single passed Pawn. (Let alone against N+P or B+P.) So K < P. Now take that in your average...

So you noticed that when a King is in a position to fight, it fights pretty well. Unfortunately it quite often is in a position where it isn't able to put up any sort of resistance, and you conveniently ignore that. But it does strongly suppress the value a King has in real life.

A Knight has better chances to stop B+P than a King.

For what it's worth, I once tried to estimate how often a K was randomly able to stop a lone pawn, if the enemy king was out of the picture or otherwise preoccupied. A pawn is on rank 4 or 5 in the average case, I suppose, so there's a lot of cases where the king is in the square of the pawn, depending on the files involved. Moreover, a king might more often be in its own half of the board based on previous play from the start position, but I didn't know how to quantify this.

I did this sort of guesswork in trying to estimate how effective my own primitive formula for estimating what the value of a king might be, when it came to computing it for large boards. For a 12x12 board I came up with a very low value for a king/guard, and someone questioned it. All I could say is that a king takes a lot more moves to cross from one side to the opposite, compared to a knight. I could have added that in an average case, a king on a 12x12 board might have an impossible task on average in stopping two isolated passed pawns x files apart. That's when I began to try to estimate how often a king might be in the square of both, and it seemed surprisingly often, though as I wrote above I could only do guesswork.

On a 10x8 board with all other normal chess pieces present, and using HGM's Fairy-Max software I once did conclude (verify HGMs earlier work) that the guard is worth almost exactly the same as a bishop, and slightly superior to a knight.

I've always heard (but never confirmed) that bishops have slightly higher value on larger boards, because they can slide quickly across whereas a guard cannot.

If bishops are worth more on a 10x8 board (compared to 8x8), then a guard (non-royal king) is equal to this superior value (because they were equal even when played on a 10x8 board).

This makes me think that a guard might be worth slightly more than a bishop on an 8x8 board, although I've never confirmed it.

A King can never stop two Pawns that are separated by more than 1 empty file.No matter whether it is in the square of both. Not even on a 4x4 board. This is related to the point whether a King would be more frequently on its own half. Often it will be behind a single Pawn, because it needed to go there to capture another Pawn when there still were two.

As to larger boards: Kings will be weaker, but other leapers will be as well. It has not been investigated whether their relative values change. They will certainly go down in value compared to sliders. It would be very interesting to measure values of the orthodox Chess pieces K, N, B, R and all their compounds on 10x10 and 12x12 boards. Perhaps one of these days I will be able to find time for that.

I think this discussion relates a bit to my zebra vs knight with increased board size question, as has the same element of a longer leaped piece versus a shorted leaped piece (the man is actually a stepper rather that a leaper). But there is a key difference. The man move is much more fluid that that of the knight which helps a lot with things like pawn defense. In a way the knight has a more fluid move that the zebra but there are probably rather rarer situations when this comes in front :)! On the other hand the man speed is roughly half of the knights speed, where the zebra has 3/2 of the speed of the knight :)!

@ H.G.:

Fwiw, a king can stop two isolated pawns more than a file apart at times, at least when one pawn is on a sufficently different rank from the other. How common such cases are I cannot be sure.

Otherwise, yes, there are many endgames where a king is diverted to the enemy half of the board to capture pawn(s). Is this the average case? Again, I am unsure..

This is why I have no faith in this type of 'numerology'. If you play actual games, this will provide a representative sampling of what kind of situations will occur in the end-game. So just play a thousand games or so, starting from a material imbalance involving the piece under test (pitting it against various more or less equivalent combinations of material), and you will discover how often the piece will survive into the end-game, and what problems it will on average have to face there.

What's a bit of a mystery to me is how, long ago, strong players more or less reached agreement on what piece values should be for chess piece types, if we set aside the K's fighting value and the question of fine tuning the fractional part of, say, the exact value of a bishop. I suppose that it was a combination of experience (say based on personal collection of many strong player tournament games, or on individual players trying to apply a piece value system they thought up, until it often seemed to work for them) and intuition (e.g. thinking about various possible endgames).

As far as king stopping a passed pawn possibly being the average case goes, I might be able to take some small comfort from knowing that statistically the average and theoretical case for the result of a game of chess is a draw. Whether that means endgames involving a passed pawn and a chasing K are also normally drawn is an open question. I also wonder whether just a thousand games is anywhere near enough to be an adequate sample in regard to many questions about chess [variants], since the number of possible chess [variant] games is so vast, and the ways of playing a position can at times depend on the players involved (though I've seen somewhere that at the top level in chess, the number of moves in an average game where style can be applied, i.e. in case of more or less equal choices, is shockingly small).

I don't think this is so mysterious. It will become apparent pretty quickly that a Rook is stronger than a minor, and that with equal number of Pawns on each site, the Rook almost always wins. And that one extra Pawn is still not enough compensation, and the minor has to fight very hard to achieve a draw. A couple of hundred such end-games, possibly with deep analysis to see if they were played optimally, would already lead you to such conclusions.

@Kevin & @HG

I hope you 2 guys are not bothered by me inserting myself into your little talk. For Kevin, I'd like to help answer the last question about the sample size of 1000%. One problem, as always in statistics is if the sample is representative (in principle I think that is your main concern), if so HG has a neat formula for the error taken from physics I guess. The point to be taken for there was that the error decreases by the square root of 2 function meaning you have to quadruple sample size in order to half the error. Still there is the concern of representative sample. At first glance, and it could be sufficient ,the distribution of reasonable situations is rather constant. Even if situations are not constant through the course of games (meaning in average you exchange rooks rather later than minor pieces in an very orthodox like variant) , they are probably very alike in between games. There are probably effects of what Ralph Betza calls sociability (an example would be Fergus's observation about his own Gross Chess that knights cooperate better with vaos) meaning so me pieces have better relationships with each other. Unless very weird cases these effects are likely to be small! Point of whole of this is not to make an truthfully scientific enquirement because that is very difficult but rather to argue that such samples are mostly representative. If they are not you are basically screwed.

I think there is some small friction between the two of you, and maybe I'll post myself somewhere in between, although probably a bit close to HG's take :)! I see HG experimental method more as a tool rather than the answer and I'm confident you do, too HG, as you enquirement about short range leapers on and 8x8 board ended up in a formula :)! On the other hand the mathematician in me does feel the need for explanations of the empiric data. There principles like the ones postulated by Ralph Betza in his essays on piece values (I'm sure you know what I'm talking about so I won't insist) come into play. In my knight vs zebra context on increasingly larger boards the knight has better mobility (due to less of the board jumps), and the zebra has better speed (due to longer jumps). When thinking about a basic formula I don't think it should be that difficult to come up with. It probably is enough to count how many moves a knight and respectively a zebra needs to get from A to B on boards of various sizes starting with 8x8 and ending with say 20x20. That it is not hard to do, maybe tomorrow I'll get the courage. I certainly hope so. But in real variants there is more to take into account than simple triangulation like relations with pawns, checkmating power with another specific minor piece . Here is where HG's statistical method comes into play. But even so for each variant, interpretations may be given.

PS Actually practical questions are allways more difficult the ZvsN theorethical one is not that hard eventually. But the practical problem on what to use in a computer program (hopefully not in a long time I'll go back to Greg's chessV) for the value of the joker in each of the 2 apothecary games is a pretty nasty one :)! Cheers everybody, I hope I have not bored you :(!

Hi Aurelian

As chess author Cyrus Lakdawala wrote regarding a position in his Slav book where Black just played ...Qxb2-pawn, after White offered it, "Anger flashes when our values are questioned". For what it's worth I have somewhat less invested in such discussions. My chess play does depend a bit on believing bishops are normally worth a tiny bit more than knights, though, so I do have some real skin in the game. In general I still wonder how much is up for debate on questions of piece values, at least on this website. Otherwise, H.G. has indirectly done me at least one big favour on the web in the past, in regard to my Sac Chess variant, so I try not to overdo things too much.

On another topic, I'm having a problem finding a suitable new minor piece type for a possible 10x8 variant that's close to being Capablanca Chess. In fact it's an impossible problem to solve, given my desires for such a piece. An elephant-ferz doesn't quite seem right, as though I do want the piece type to be symmetrically moving, I want it to be worth about a N or B on a 10x8 board, which I'm doubting in the case of the elephant-ferz, and I also ideally wish to place bishops on certain squares in the setup, yet have all pawns protected, and avoid possible common early exchanges of pieces from the start.. So far an elephant-wazir would be perfect, except it's hard to develop in the setup unless it takes a good square away from a knight, and in the setup I have in mind it would also guard the rook beside it, which somehow seems unattractive. The next best piece would be the guard, but it's a bit boring and also tedious to develop from the setup. The moral seems to be that sometimes you cannot have everything you want, even in chess variants, where you might even invent a piece.

@Kevin

Why not try something like Elephant+Silver General and Dababah+GoldGeneral? They seem like nice pieces :)! You may even try but I think is worse Gold elephant and silver dababah.

Thanks for the suggestion, but Jose already has used a version of a silver general in a 10x8 variant of his. Plus I was hoping the mythical perfect piece type I want (one of it on the kingside & one on the queenside) would be symmetrically moving, i.e. the same in all directions.

There is a difference between 'minor', in the strict sense of not having mating potential, and in the sense of 'not worth much more than N or B'. There exist enormously valuable pieces that have no mating potential, and less-than-N-valued pieces with mating potential (R2). To get a leaper on par with N or B it needs to have 8 targets; 12 is too much (about Rook-valued), 4 is too little, and for totally symmetric pieces there is nothing in between. One oblique step already causes 8 targets, the Knight is already taken, and the Camel and Zebra are nasty pieces on an 8-rank board with Pawns on 2nd rank, because of their distant forking power. That leaves pieces with two different steps on Queen rays. Perhaps FH (1 diagonal 3 orthogonal) would be an option. Although I am not sure that would be stronger than a Knight.

If you want to consider less 'pure' pieces, rather than breaking symmetry, you could consider divergence. E.g. move as Zebra, capture as King could be pretty strong. (But it might have mating potential; I did not check that.) Or slam extra non-capture Wazir moves on the Elephant-Ferz. (These would then break its color binding.) Or you could resort to 'lameness'. E.g. a piece that would jump as Dababba, and, when that is possible, can continue with one outward diagonal step. It would have 12 targets, but 8 of those would be blockable. The 8 targets of the Xiangqi Horse, which are similarly blockable, are only worth half as much as those of the FIDE Knight. Such a piece is easy to develop, not colliding with anything else, and would not be able to fork trapped back-rank pieces because of its lameness preventing it to reach beyond the Pawn line.

Thanks H.G.

It's plain I don't have the experience/imagination yet to go much beyond piece types that are precidented that I'm aware of. I also thought symmetry (and relative simplicity) in a piece type might lure more variant players to try, and perhaps stick with, such a 10x8 variant in the first place, but I may be wrong about what players on Game Courier find attractive - plenty of really weird stuff is played there, a lot. At any rate, I'll ponder your suggestions at my leisure.