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Comment by infogulch | Hacker News Reader

For me the most surprising observation from this thread is this tweet [1] showing that the knight's moves are exactly the squares that the queen can't get to within a 2 square manhattan distance.

I.e. the knight is an anti-queen.

[1]: https://twitter.com/skidbladnirr_/status/1750285122769957129

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Some non-orthodox chess variants have a piece which combines Queen+Knight: https://en.wikipedia.org/wiki/Amazon_(chess)

very old variants often had such compound pieces like for example: https://en.wikipedia.org/wiki/Grant_Acedrex

There's also the knook that combines the Knight and the Rook, which was hilariously invented as a meme on /r/anarchychess: https://anarchychess.fandom.com/wiki/Knook

Here's a whole page of chess variants, many of which include such invented pieces: https://en.wikipedia.org/wiki/List_of_chess_variants

Wherein the hex-knight is the anti-hex-queen.

https://en.wikipedia.org/wiki/Hexagonal_chess#/media/File:Gl...

You mean a 2-square Chebyshev distance, I think (the Linf norm). A 2-square Manhattan distance (the L1 norm) wouldn't include any of the squares that a knight can get to.

For those that want a visual description there's a blog I serendipitous recently ran into[0]. If you're too lazy to open:

- n-square Chebyshev distance forms concentric squares around your starting point

- n-square Manhattan Distance forms diamonds (meaning up/down/left/right = 1 space away but all diagonals are 2 spaces away)

We could also say the Knight can go everywhere a queen can't in the 3-square Manhattan distance and this might be more useful since it is all points along that boundary except the corners! (We got an in the wild "2 problems in computer science" with an "off by 1" error :)

[0] https://chris3606.github.io/GoRogue/articles/grid_components...

That's very much a picture that you learn on the usual mathematics route of learning about these things, yes. More generally, there's a fair amount to be said about how you can derive shapes of unit balls from distance functions and vice versa (in functional analysis in particular).

I think, unfortunately, not many get visual descriptions when learning about these topics. To be fair, these visualizations can harm one's intuitions about how the metrics work in higher dimensions, so I can understand arguments against visualization. But in another example, I think even many people can come out of a Linear Algebra class and not understand that a matrix operating on a vector always performs an affine transformation on that vector (yes, I know we can abuse and get other forms but let's keep general due to context).

I think this is something math education can improve on since we are highly visual creatures. But it is always a tough balance since that same tool to help learning can decrease generalization if internalized improperly. But that can also be said about anything and is why I dislike the common test focused paradigm of education. As I think we've been doing it long enough that students have found studying for the test -- as opposed to study for learning -- is optimal. Goodhart always wins lol

> a matrix operating on a vector always performs an affine transformation on that vector

A simple matrix multiplication only performs a linear transformation. To get a nonlinear affine transformation you need to augment the matrix and vector with extra dimensions containing carefully selected extra ones and zeros in addition to the translation. The end result is still a linear transformation of the supplied vector, but looks like an affine transformation if you ignore the final dimension.

But yes, I did not pick that up in linear algebra. It wasn't even mentioned there. All the fun stuff was taught in computer graphics.

Specifically,

https://math.mit.edu/~gs/linearalgebra/ila5/linearalgebra5_1...

For more on the underlying geometry, see, e.g.,

https://archive.org/details/elementarymathem0000klei/page/70...

or, for a more modern, abstract formulation,

https://archive.org/details/geometry0001berg

This is why in some rare endgames you might want to promote to a knight instead of a queen to avoid a stalemate!

Chess would be a more elegant game without the stalemate rule.

(And apparently it would be a more _interesting_ game without castling. At least according to the game that the AIs at Deep Mind played when they trained them on this variant on a whim. Castling was supposed to speed up development of chess, but it favours the defense a lot.

So what the game's appeal wins from castling in cutting out some busy work moves, it loses in attacking spirit.)

Extremely strong disagree with this[1], and the obvious question that arises is what would you do if a side is in stalemate and there isn't a stalemate rule? Would the other side win? Would the stalemated side pass? Neither seems at all satisfactory.

The non-castling variant was invented by Vladimir Kramnik[2] explored by Deep Mind not on a whim but specifically as part of Vlad's exploration of whether this chess variant would help to revitalise the game given how computer evaluation has affected theory.[3] There were also a couple of no-castling tournaments played, but like chess960[4], opinions were very much divided about whether or not it was actually any kind of improvement.

[1] Even though I pretty much always fall into stalemate traps in blitz.

[2] Former world champion and lately old guy shouting at clouds about cheating

[3] https://www.chess.com/article/view/no-castling-chess-kramnik...

[4] https://lichess.org/variant/chess960

> what would you do if a side is in stalemate and there isn't a stalemate rule?

The game keeps going until one side gets too tired and gives up (or makes a mistake that breaks the stalemate).

Another idea would be a "Fortnite rule": after $X turns, the outermost edge of the board kills any pieces left in it (or becomes "the storm" in Fortnite parlance); repeat every $X turns until someone's forced into checkmate because there's only a 2×2 square left and nowhere else for the kings to go except within each others' attack range.

What do you mean the game keeps on going? One side has to move but has no legal moves.

That's what I was asking - do they get to "pass" and the other side gets to move? This is how it works in the game of go for instance. In chess that would be more or less equivalent to the non-stalemated player winning, because in many stalemate positions if you had an extra move you could checkmate.

The stalemate rule adds something that the attacking player has to keep in mind and gives the defending player something to shoot for - lots of creative and tricky play arises from this, with IM Eric Rosen being the most famous exponent[1] so much so that he now has a particular type of stalemate trap named after him.[2]

I'm not seriously entertaining your Fortnite rule. That is such a radical departure from the game of chess as not to be the same at all, so you could have that as a chess variant or whatever but it's fundamentally a different game.

[1] https://youtu.be/YB_LLivPlY8?si=dWl2bEAEeRmzhBiC

[2] Defending king in the corner with the opposing queen a knight's move away and no other legal moves for the defender. https://www.chess.com/forum/view/game-analysis/the-rosen-tra...

> What do you mean the game keeps on going? One side has to move but has no legal moves.

As said elsewhere, I would suggest removing rules against putting yourself in check.

> The stalemate rule adds something that the attacking player has to keep in mind and gives the defending player something to shoot for - lots of creative and tricky play arises from this, with IM Eric Rosen being the most famous exponent[1] so much so that he now has a particular type of stalemate trap named after him.[2]

That's why I was careful in saying that the proposed change would make the rules more elegant, less clunky. I did not say anything about this making chess a better game. Sometimes adding weird corner cases to your rules can make the game better.

Just look at all the weird rules that you can find in Skat or Doppelkopf. (Though Mü is an example of a game with much cleaner rules that captures a similar spirit.)

But who cares if the rules themselves are more elegant if the game they describe is different and worse because of it?

It's like you introduced a bug in refactoring and are insisting on the change because the code looks nicer now.

Yes, you are right that this is only an intellectual exercise. (Btw, it's quite popular in the Go world to reformulate the ancient clunky rules of Go into a more modern, mathematical version that might or might not have small differences in gameplay.)

However, I don't actually think that removing check rules make chess worse. But that's subject to experiment.

See https://arxiv.org/pdf/2009.04374.pdf for exactly such experiments.

I do not get this Rosen Trap example you linked to... It's an easy win for blacks, just move the rook to 8th row instead of moving the pawn and it's checkmate. The pawn move seems kind of random, as if it was played by a child.

Generally the attacking side can avoid the stalemate trap but often it happens under time pressure etc so the attacker doesn’t have time to calculate whether the defender has any valid moves.

> One side has to move but has no legal moves.

If you run out of legal moves then that seems like an outright loss condition, no? My understanding of "stalemate" is when you can move but there's no way to actually force a win (e.g. kings chasing each other around the board). I ain't a chess expert, so maybe I'm using the wrong terminology.

No. In ordinary usage "stalemate" means when people are kind of stuck and not making progress, but in chess stalemate specifically means when one side has no legal moves but is not in check. In chess that is a draw. The type of situations you're describing about are covered by four rules.

1. The threefold repetition rule: if a board position repeats 3 times either side may claim a draw. In online chess this is automatic but in over the board chess people may not always realise the position has repeated especially if there is a gap between repetitions.

2. The 50 move rule: if 50 moves pass without a pawn moving or a capture being made then it is a draw.

3. Insufficient material: if neither side has sufficient material to checkmate then it is a draw. In some rule systems [1] it has to be sufficient material to force checkmate in some it's just sufficient material to mate[2] (ie the defending side can blunder into it). So if there are no pawns and both sides just have a lone king, or it is king and knight vs king or king and bishop vs king then it's always a draw. In USCF and chess.com king and 2 knights is a draw in FIDE tournaments or on lichess it's not. King and 2 bishops on up is not a draw because you can force checkmate.

4. Draw by agreement: the two sides can agree a draw. In some tournament situations this is only allowed after a certain move but there are ways the players can deliberately go into some sort of situation which is a known theoretical draw and just repeat 3 times by tacit agreement. Including amusingly/notoriously (depending on your pov) the Carlsen/Nakamura "double bongcloud" game [3]

[1] US Chess federation and chess.com in particular

[2] FIDE and I think lichess

[3] https://www.theguardian.com/sport/2021/mar/18/bongcloud-meme...

The important thing to mention: in many respects, you can pretend that in chess the objective is to capture the opposing king. But for historical reasons, the game actually tops one move before, and the king is never captured.

But they also have weird rules that both (A) forbid putting your king in check, and (B) hand you a draw when you are out of legal moves.

A simpler version of chess would actually play until the king is captured, and allow people to move into check.

https://arxiv.org/pdf/2009.04374.pdf reports on some experiments with variants like this.

Another interesting thing to try out is to allow players to capture their own pieces. Or to abolish castling.

In that case the stalemated player has no moves. So you can't keep playing

Well then whomever's forced into check would be the loser, then.

> forced into checkmate because there's only a 2×2 square left and nowhere else for the kings to go except within each others' attack range.

If the only two pieces left are kings, that is a stalemate. It is not even a legal move for two kings to come within one square of each other.

Sorry, when I was talking about the stalemate rule, I meant the rule that declares that the game is a draw, if the only move you can make is putting yourself into check.

I was not talking about the rule that handles games that do not progress. (Though I would perhaps replace that rule with one that just forbids repeating board states, but that was besides the point here.)

So to make chess more elegant in how the rules are formulated (but probably not a better game, mind you), I would suggest removing all special rules around check and checkmate. Just replace it with the rule that you lose when your king gets captured (and obviously, keep Zugzwang).

Sorry, I was not clear that I also wanted to remove special handling of check and checkmate.

(Btw, this feels very similar to the rules against self-sacrifice in Go.)

This would make the game less elegant, in my view. There are too many endgames that are just trivially winning if you make stalemate a win for the stalemating player.

Stalemate mostly just makes the game more interesting when otherwise the position would be completely uninteresting.

That would just result in earlier resignations so that both players could get back to a more interesting game.

No, you're fundamentally changing the game, not when people resign.

If stalemate is a win, almost any king and pawn vs king position is a win, which means almost every rook or bishop ending up a pawn is a win, which means it's incredibly easy to trade into a winning endgame from the middle game. Now the middle game is less interesting, because taking any kind of risk is never worth it.

Which in turn makes only the most boring openings viable.

Now every game is boring.

Also, in tournament play, finishing your game early just means you have to wait longer until the next round, even in a round robin event. Because chess tournaments pretty much never start a round until the previous round is completely over. Not since adjournments stopped being a thing.

When people complain about chess being uninteresting, they usually complain about too many draws. Adding more opportunities to turn draws into wins/losses might seem like a good thing?

> Now the middle game is less interesting, because taking any kind of risk is never worth it.

Chess is a zero sum game. Taking the right risk can also increase your chances of winning (instead of just stalemating).

(I mention the zero sum game, because eg real life war is not a zero sum game. So it's perfectly possible for certain actions, like starting to shoot medics, to be bad for both parties. But in chess, what's bad for one party is automatically good for the other.)

> [..] which means it's incredibly easy to trade into a winning endgame from the middle game.

You know that the other guy gets to have a vote on this, too?

You can also have a look at the Eastern chess variations (Shogi etc) that https://news.ycombinator.com/item?id=39141452 mentions that don't have check rules, and see whether for them every game is boring?

Btw, see https://arxiv.org/pdf/2009.04374.pdf for an investigation of the impact of no-check-rules on western chess. It doesn't make everything boring.

Addendum from the linked paper:

> Stalemate=win chess has little effect on the opening and middlegame play, mostly affecting the evaluation of certain endgames. As such, it does not increase decisiveness of the game by much, as it seems to almost always be possible to defend without relying on stalemate as a drawing resource. Therefore, this chess variant is not likely to be useful for sidestepping known theory or for making the game substantially more decisive at the high level. The overall effect of the change seems to be minor.

This does not seem to support your suggestion that eliminating check rules and ending the game with the capture of the king, would make all games boring.

Personally I find it very entertaining when a player pulls a rabbit out of their hat by turning an obviously worse position into a stalemate by some crazy trick. I think it adds some skill expression to an otherwise boring end game, and gives the losing player something to fight for at the end. Admittedly this is more interesting in a format that results in a time crunch at the end.

According to the paper (https://arxiv.org/pdf/2009.04374.pdf) removing the special handling of stalemates did not really change the draw ratio much. So the weaker party can still gun for the draw in most situations.

(However, they had really good computers play, who do not get stressed out, albeit they do make worse moves on shorter time controls.)

> Sorry, when I was talking about the stalemate rule, I meant the rule that declares that the game is a draw, if the only move you can make is putting yourself into check.

There is a Chess game that does this. It's Chinese Chess (which is just as old as Western Chess, and has the same origin in India). In Chinese Chess, the objective is to capture (or as the Chinese say, "eat") the king. There is no prohibition against putting your King in check. It works just fine, unless the only pieces left are Kings. The main problem is players doing checks forever or chasing pieces around forever: the rule is that if you do that you lose.

https://en.wikipedia.org/wiki/Xiangqi

Now stalemate by repeated back-and-forth is a different matter. I would favor some variant of Go's Ko rule, which is that you may not make a move if the result would be a board that looked exactly the same as it had in the past.

Thanks for bringing up the Chinese chess example!

Btw, see https://arxiv.org/pdf/2009.04374.pdf for an investigation.

> Now stalemate by repeated back-and-forth is a different matter. I would favor some variant of Go's Ko rule, which is that you may not make a move if the result would be a board that looked exactly the same as it had in the past.

I would favour that rule as well, but keep in mind that even in Go that is a very modern formulation / variation of this rule. Past formulations were about as clunky as chess's stalemate rules, and even left some situation (that were extremely rare in practice) without a satisfactory resolution.

Btw, in Go the 'board' isn't the only state the game has: you also count prisoners. After a complicated ko exchange, the board might look the same, but the prisoner count has gone up for both players. I assume the rules probably only care about the difference in prisoner counts (and board state) to determine whether a state has been repeated; not the absolute prisoner count.

I don't know about Japanese rules, which are borderline insane. I use the only rational ones: Chinese rules! As far as I know, in Chinese rules, Ko only considers the board configuration.

Well, I have the same reaction (though a bit muted) the Western Chess's refusal to capture the king.

> what would you do if a side is in stalemate

Then that side loses. If they can't move, it's because any move would put them in check and therefore captured next turn. If they're in that position, they've obviously lost.

Passing makes no sense since it contradicts the rule that says chess players must always make a move during their turns. There is no contradiction with stalemates: since they must make a move and any move puts the king in check, it is hopeless and they lose.

> Neither seems at all satisfactory.

Can't be worse than drawing a game because the clearly beaten opponent can't legally move into a check.

This is an actual strategy. Refuse to resign and try to get the opponent to blunder his pieces into a stalemate in the end game thereby drawing the game. It's asinine but sometimes it actually works, especially at my noobish level.

I dont get it. Queen can do everything a knight can, and then some. How can electing a knight instead of queen can be of any usefulness?

Or do you mean it avoids stalemate by making it harder for the elector to defend/attack, and loose? I dont see any advantage there, isnt stalemate always preffered to loosing?

A knight can attack squares the queen cannot. There are positions that promoting to a knight is checkmate, but a queen would lose.

Sometimes it's tactically crucial that the promotion is a check, and in these cases you might need a knight.

Making a knight to avoid stalemate is actually extremely rare, because very often a knight isn't enough to win. Making a rook is more common there, but even that's rare.

Promotion generally happens when there are very few pieces on the board, and often the opponent's king is the only piece they can move, and it often is close to the promotion square since it was likely trying to block it.

If the promoting piece can't check but immediately blocks the last movement squares for that king, as a queen may easily accidentally do, the player who has only the king gets a stalemate draw out of a technically lost position.

In Chinese chess, the horse behaves like the knight in western chess except that (a) it cannot jump over pieces, and (b) the definition of jumping over is that it first move horizontally/vertically 1 step and then diagonally 1 step. It cannot first move diagonally.

This also leads to situations where two knights can be positioned amongst other pieces such that one knight is able to capture the other but not vice versa.

Most people think of the knight move as one straight and then one diagonal. However, this is still ambiguous. Imagine one up, then one down left for example, resulting in the square left to the original square. A better description is moving 2 squares straight, not diagonal, into one direction, then moving 1 square into a 90° turned direction.

Forward two steps and the knight falls off his horse.

Most people think of the knight moving 2 and then 1.

Chess players do, yet experience of basically any non-player I have ever heard describing it as "one straight, one diagonal" tells me otherwise. Also "moving two and then one" is ambiguous and would allow moving 3 squares into the same direction.

I've always thought of it as an L

This probably _heavily_ depends on the country where you live.

I don't know people that think or thought that horses move "one straight, one diagonal".

I was taught that a knight moves by going to the opposite corner of a 3x2 rectangle

Both western and Chinese chess knights move to the opposite corner of a 3x2 rectangle.

However in western chess, it doesn't matter what pieces are in the other 4 locations of the 3x2 rectangle.

In Chinese chess, you can only move from A to B if X is unoccupied. The locations marked "-" don't matter if they are occupied or not.

    A X -
    - - B

    A -
    X -
    - B

    - - B
    A X -

    - A
    - X
    B -

etc.

Of course, all the other pieces have their own rules as well, so it's a significantly different game. There is no queen, the elephant is sort of like a bishop but must move 2 squares at a time, the cannon moves horizontally and vertically but cannot kill without having exactly one (not zero) piece in-between, the "guards" and king all cannot leave the king's palace which is a 3x3 box, the pawns can move sideways after they cross the river in the middle of the board, etc. Come to think of it the only piece that has the exact same rules are the 2 "chariots" which are positioned in the same place as western rooks and function exactly like western rooks.

But in Chinese chess, if the knight can't pass other pieces, thinking of it that way would be a rule change.

But you can do the same unambiguously if you are going to involve angles.

"Move one straight and then move one diagonal into a 135° turned direction."

To add to that, this also makes a knight and queen a very strong attacking combo, because they can uniquely cover the most squares.

Given... they stay... in the same... square?

In the image above, imagine a knight on H3, which covers g1, g5. Neither a bishop, a pawn or rook can help that way.

After one move, it can go to f4 where it can add on as a second attacker to the queen on 4 squares. After another move it can go to d3, where it can again cover squares the queen cant (e2,e5).

So I guess its both because it has a lot of alignment in doubling as an attack, and covering nearby squares the queen cannot.

There was a big discussion of this topic on chess stackexchange[1]. It used to be just universally believed to be true but some modern analysts challenge the idea.

[1] https://chess.stackexchange.com/questions/27815/why-does-a-k...

It also enables some powerful, well-known attacking patterns, like a smothered checkmate with Queen sacrifice (or Philidor's mate).

Yup that's how they designed it in hexagonal chess: https://en.wikipedia.org/wiki/Hexagonal_chess#Rules

"the knight may move to any nearest cell not on an orthogonal or diagonal line on which it stands"

I thought that was the main deal about knights...

A very interesting property:

Knights have odd-even parity, i.e., they attack only opposite colored squares.

Pop quiz: How many knights can you place on a board so that they don't attack each other?

Well, since a knight on a white square can only attack knights on black squares, and half of the 64 squares are white...

I'm not a master or anything, but I've always found it interesting that the queen is usually thought of as the most powerful piece while I think it's the knight.

Yeah the queen can move everywhere, but she's "predictable". The way a knight moves makes it more difficult to defend against IMO.

A huge difference is that a queen, like a rook, can attack a whole row and column, making it possible to box in an opponent's king. A knight has limited reach and can never attack a square the same colour as the one it is occupying.

Another is that a queen, like a rook, can reach any unblocked square in at most two moves. A Knight can require up to six moves!

Add to this the ability to move and attack diagonally like a bishop, but without the restriction of staying on the same colour of squares...

The queen is immensely powerful.

Knights are useful but are very much not as difficult to defend against as a queen.

Knights are objectively nowhere near as powerful as queens. However they're underrated in blitz chess, where their visually hard to see moves often catch people unaware.

Even grandmasters fall for knight forks in blitz games quite often.

Try playing a game where your opponents knights are replaced with queens. I suspect you will change your mind very quickly.

Knights tend to be overrated at lower levels of playing strength, where their irregular moving pattern is indeed challenging for the players, and as a result, Knight moves often get overlooked.

However, once a player becomes more proficient, not only the obvious superiority of the Queen, but even of the Bishop (other things being equal, Bishops are thought to be actually somewhat more valuable, especially as a pair) starts becoming apparent.

Knights are prone to getting "trapped". There are limited places a knight can go, so the opponent only has to maintain pressure on those squares. This is the reason why it's generally a bad idea to put a knight near a corner of the board.

>>> I.e. the knight is an anti-queen.

The complement-queen I would say

I have no source for this but that is the definition for a knight's valid moves, it goes where the queen cannot within that distance

The official definition, as per the FIDE laws of chess [1], article 3.6 is

"The knight may move to one of the squares nearest to that on which it stands but not on the same rank, file or diagonal."

[1] https://www.fide.com/FIDE/handbook/LawsOfChess.pdf

That's a horrendously hard to parse definition. Couldn't they say "3 squares away but not on the same rank/file/diagonal" or something a bit easier?

"3 squares away" is not entirely obvious though. The definition as it stands is mathematically unambiguous. Obviously, it's not how you teach the rule to a beginner, but that's irrelevant.

If "nearest" is unambiguous then "3 squares away" is also unambiguous. The nearest squares are 1 square away, the ones nearest to those are 2 squares away, etc.

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