Friday, November 20, 2009

Sometimes tournament chess is like getting hit upside the head with a huge fish

Those people who don’t play our royal game often imagine chess to be this detached, cerebral exercise. (Cue fiendish laughter here) For the last couple months, I’ve been getting settled in at my new job with Wolfram Research which has been great but has also left me with very few spare brain cells for appreciating chess. Which has meant that – through absolutely no fault of my own – in the few spare moments when I’ve tried to get this blog going again, my taste in games has gravitated towards games that are the Caissic analog of a roadrunner cartoon.

Case in point:
Victor Chubakov-Leon Shernoff Maroon Kings Tournament, University of Chicago, 2002

1. d4 d5 2. c4 c6 3. Nf3 Nf6 4. Nc3 dxc4 5. e4

The Geller Gambit (or Tolush-Geller Gambit), a sharp continuation that is considered unsound at the GM level, but not below. On the one hand, I’ve seen games where the young Kasparov lost as White; on the other hand, I was present at a tournament where perpetual “could be IM if I felt like it” David Gliksman used it (as white) to crush Igor Ivanov, who then immediately used it himself (even though of course it’s completely not in Ivanov’s style) to crush another IM in the last round. (Gliksman, if I remember correctly, drew with Michael Brooks in about 7 moves to share first place between them)

I used to play a gimmicky continuation invented by Smyslov, but gave it up because of bad results, even though I always got good positions. Also because I realized it was just unsound in one game (confirmed by John Donaldson in the post mortem), even though my opponent didn’t see it in the game.

5... b5 6. e5 Nd5 7. a4 e6

The main line. The other major continuation is 7...h6. Also possible is 7...Qa5 (according to my book, anyway) and 7...Be6 (Smyslov's continuation), with the idea of Nxc3, Bd5, and e6.

8. axb5 Nxc3 9. bxc3 cxb5 10. Ng5 Bb7 11. Qh5 Qd7

Okay, this is also a legitimate way to reach the main line (11...g6), but also a gimmicky sideline in its own right. White can't take 12.Nxh7 because after 12...Nc6 Black threatens sacrifices on d4. You’ll get the idea of what happens then from what happens here.

12. Be2 Bxg2 ??

Based on the results of this game, I have to say that this move is just losing, even though it had been played at the GM level before (and possibly since).

Black should just play 12...g6 to enter the main line. He then plays Be7 and goes for the setup with Bd5 and Nc6. I, however, had memories of Black grabbing this pawn thus keeping White's king in the center (see Kievelitz-Crouch, Decin 1996 at the end of this post), so I got over-ambitious.

Though, to be honest, I had another problem with the main line, in that book (at this point in time) had White able to force a repetition with stuff like 12...g6 13.Qh4 Be7 14.Qh6 Bf8 (if Black can’t castle, he’s doomed) 14.Qh4 Be7 (see previous comment) 15.Qh6, etc. I don’t know whether this assessment has changed in the meantime. But it seems like an attractive line for a lower-rated player, as Black either has to allow this draw or play one of these cheesy sidelines that subject him to a ferocious attack.

13. Rg1


Afterwards, I thought that better would be 13...Bd5, as in Kievelitz-Crouch. However, I was still hoping to get my Nxd4 sac in, so I left the d-file open. And in fact, Couch should have been vulnerable to the exact sort of continuation that I was here, but he wouldn’t have had the open d-file for counterplay (such as it is).

14. Nxh7 Nc6 15. Nf6+ gxf6 16. Qxh8 Nxd4!?

Quiz question #1: What is White’s best response to this move? Hint: there is a very, very, very clear answer. To help you avoid peeking, I will babble on for a little while to fill up the page.

My thought at the time was that objectively Black's just losing here, so he has to start the complications right away. 16...0-0-0 just hands Victor the correct line: 17.Rg8, and now if Black starts the same sort of fun with 17...Nxd4 Qxd4, White is always a capture ahead of him after, say, 19.Rxf8 Qxa1 20.Rxd8+ Kc7 21.Rd1

After the game, I said to Victor that this was similar to last year's game, in that after a certain point we entered a tactical flurry, and I prevailed in the period of mutual oversights. The interesting thing about this year's game is that neither of us realized how early that period began.

Okay, if you’re still looking for a hint, I’ll tell you that the most important feature of Black’s last move is that it gives up control of the b4 square, which means that black can’t push his b-pawn if he needs to.

Here, Victor can win with 17.Qxf8! Kxf8 18.Ba3+, giving Black the lovely choice between 18...Ke8 19.Rg8# and 18...Qe7 19.Bxe7+ Kxe7 20.cxd4, capturing the knight under, ummm, superior conditions.

Is the roadrunner theme becoming clear to you yet?

So objectively, Black should have castled last move, even though it sucks. I think you’ll agree with me that the pawn grab Bxg2 was bad, in that Black’s two moves ...Bxg2 and ...B retreats are worth a lot less than White’s corresponding moves Rg1 and Rg8. So better, actually, to castle on move 12 instead of grabbing the g-pawn. But then White‘s king wouldn‘t be trapped in the center and Black would have little play. Best not to enter this line at all.

Victor was also (unbeknownst to me) starting to hallucinate, in that he rejected 17.Rg8 because of 17...Nc2+ 18.Kf1 Nxa1 19.Rxf8+ Ke7

because he "didn't see a mate" and didn't want to be "only up a piece" after taking twice on a8. After the game, we agreed that just being up the piece was good, but there are also at least three quick mates in the position! See if you can find them! The answer is at the end of the post, as “Mate cluster #2”.

I, of course, was going to castle on move 17, entering the "correct line" that I didn't want him to find (above).

The game continued:

17. cxd4?? Qxd4 18. Rb1?

Again, there's another long mate after the thematic 18.Rg8 Qxa1 19.Rxf8+ Ke7

and 20.Rxf7+. I’ll let you find it on your own, and I’ll put the answer at the end of the post. It’s not easy to visualize all the way to the end. Objectively, it looks like the passive move in the game blows White's advantage. Which means that if it takes so many ?? moves for White to not be winning then ...Bxg2 probably deserves something like four or five question marks instead of two.

18... O-O-O

19. Rg8??

At this point, White has to come back and defend against Qc3+ with Qh3 or Rg3. Fritz’s reasoning for declaring the position almost equal then seems to have been something like 19.Qh3 Qxe5, but I tend to distrust grabbing pawns as compensation for a whole rook. Ramming the c-(and b-?)pawn(s) down the board looks more appropriate to me. Another line is 19.Kf1 Bf3! (threat Qd1+; 19...Bc5? 20.Qxf6 defends f2) 20.Bxf3 Bc5! with threats against f2 and h8.

19... Bb4+??
Black wins after 19...Bg2!! The threat of Qc3+ compels 20.Rg3 or Rxg2, and Black then wins White's queen with 20...Bb4+
20. Rxb4??
It is especially embarrassing to admit that I was planning on 20.Kf1 Qd1+ 21.Bxd1 Rxd1+ and Re1#, not noticing that this is illegal because my rook is pinned on the back rank (by two major pieces!). But at least this is a mate. I'd been playing for it in some other lines where the Re1 is supported by a Nc2 instead of the bishop (after 17.Rb1, for example). There, it isn't even mate, as the king can go to d2!
20... Qc3+
Victor had overlooked that his Bc1 would be hanging here.

Here are a few of the games I had studied, with light notes.
(this is well-documented getting hit upside the head with a fish)

Kievelitz-Crouch, Decin 1996
1. Nf3 d5 2. d4 Nf6 3. c4 c6 4. Nc3 dxc4 5. e4 b5 6. e5 Nd5 7. a4 e6 8. axb5 Nxc3 9. bxc3 cxb5 10. Ng5 Bb7 11. Qh5 Qd7 12. Be2 Bxg2 13. Rg1 Bd5 14. Nxh7 Nc6 15. Nf6+ gxf6 16. Qxh8 O-O-O 17. Qxf6? (Rg8, of course, but White is still winning.) 17...b4 18. Bb2 a5 19. Bh5 a4 20. Rxa4?
(20. Qxf7 Qxf7 21.Bxf7 a3 22.Bc1 and Black's would-be pawn roller is already pre-undermined.)
20... Nxd4 21. Rxb4?!
(21. Ra2 Nc2+ 22.Kf1 (22.Ke2?? Bg2) 23.Bc6 and Black will have to play for an attack)
21... Bxb4 22. cxb4 Be4! 23. Qxf7 (23.Rg7 Nf5 0-1) Nc2+ 24. Kf1 Qd3+ (25...Qh3+ is also good) 0-1

Jiretorn-Chmielinska, European Women’s Ch, Warsaw 2001
This game shows what happens if White takes the exchange without playing 12.Be2 first.

1. d4 d5 2. c4 c6 3. Nf3 Nf6 4. Nc3 dxc4 5. e4 b5 6. e5 Nd5 7. a4 e6 8. axb5 Nxc3 9. bxc3 cxb5 10. Ng5 Bb7 11. Qh5 Qd7 12. Nxh7 Nc6 13. Nf6+ gxf6 14. Qxh8 O-O-O 15. Qxf6 Nxd4 16. f3 Nc2+ 17. Kf2 Bc5+ 18. Kg3 Rg8+ 19. Kh3 Nxa1 20. Be2 Nc2 21. Rd1 Bd5 22. g4 (and as you might imagine, Black won handily) Ne3 0-1

O'Cinneide-Hurley, Bunratty (Ireland) Open, 2000
And this game shows what happens if White tries to get fancy in the corner.

1. Nf3 d5 2. c4 c6 3. d4 Nf6 4. Nc3 dxc4 5. e4 b5 6. e5 Nd5 7. a4 e6 8. axb5 Nxc3 9. bxc3 cxb5 10. Ng5 Bb7 11. Qh5 Qd7 12. Nxh7 Nc6 13. Nxf8 Qxd4 (!!) 14. cxd4 Rxh5 15. Be3 a5 16. Be2 Rh4 17. Nxe6 fxe6 18. O-O-O Nb4 19. Bg5 Rh8 20. Bd2 Nd5 21. Kb2 Rf8 22. f3 Nf4 23. Bf1 Bd5 24. Rc1 Kd7 25. Rc2 Kc6 26. Rg1 Kb6 27. g3 Ng6 28. f4 Rac8 29. Be2 b4 30. Rgc1 Kb5 31. h4 Rc6 32. h5 Ne7 33. g4 Rfc8 34. Be1 Kb6 35. Bh4 c3+ 36. Kb1 R8c7 37. Bxe7 Rxe7 38. Bd3 Rec7 39. f5 b3 40. Rf2 a4 41. f6 gxf6 42. exf6 a3 43. h6 a2+ 44. Ka1 0-1

Moral of the story: all the sidelines in the 11...Qd7 line are good for Black; it’s only the main line that’s are completely lost!

Mate cluster #2

Mate 1) 20. Rxf7+ with Bh5+ and Qxf6#
Mate 2) 20. Qxf6+ with Qh8+ & Bg5+, with similar play; and
Mate 3) 20.Ba3+ which I consider emotionally equivalent to mate, even if it doesn't lead to a quick mate on the board

Here’s that longer mate at move 18:
18.Rg8 Qxa1 19.Rxf8+ Ke7

20.Rxf7+ 21.Bh5+ Now it's a mate in 8: 21...Ke7 22.Qxf6+ Kd7 23.Qf7+ Kd8 24.Qf8+ Kd7 25.Qd6+ Kc8 26.Qxe6+ Kc7/d8 27.Qd6+ Kc8 28.Bg4#

I have to say that after going over this game with a computer, I felt like it was a totally wretched game. However, after trying to find this particular mate myself, I have to admit it’s tough (or at least stressful) to do with (from the diagram position) mate hanging on c1 and the rook hanging on f8.
Likewise, a move before, it’s easy for the computer to see that 16...Nxd4 should be answered with 17.Qxf8, but it’s hard to notice that you should steel yourself and “ignore” the “loud” move Nxd4 and just play a sacrifice of your own. Another mental block in that position is it can be difficult to realize that you’re just winning so early in the game. On the other hand, Victor’s comments afterwards showed that he did realize he was winning, so

I suppose I can’t fault either of us much for missing the computer tactics on move 19; however, it was sheer luck that his ?? oversight outranked mine on move 20. Certainly not the most embarrassing game I’ve ever played, but it makes me think that perhaps one of the reasons chess is superior to roadrunner cartoons is that in chess both sides can get multiple anvils dropped on their head in the very same game.

Comments? I've added a poll.

Monday, September 7, 2009

Elizabeth and Lars have corrupted me

Okay, I tried to keep this all chess, all the time, but (a) Elizabeth has been blogging about all sorts of random things for a while now and why should she have all the fun? and (b) my soon-to-be former workmate Lars has shown me one of the few golden opportunities I've ever seen to use the power of the internet to further the cause of world civilization.

So here it is: which of the following re-dubbings of the same rant from the movie Downfall are the funniest?

Hitler Learns that the Vikings Have Signed Brett Favre
Sample line: "Everyone who wears a Helga hat to bed, please leave. You must not hear what I have to say." (3/4 of the Nazis leave the room)

Hitler Learns That Michael Jackson Has Died and so Will Not Be Performing at his Birthday Party
Sample line: "And the worst part is, I never even got a chance to go to Neverland."

Hitler Finds Out He Has Been Banned from XBox Live
Sample line: "I had over 2000 Microsoft points!! I bought a Call of Duty theme pack, too!"

Note that I stayed away from less skillful ones, like "Hitler Learns there is no Santa Claus" or political ones like "Hitler Learns that Sarah Palin Resigned", but you can find them easily through the YouTube related links. And for the connoisseur, I understand that there is also a version of this up where Hitler is ranting about all the different versions of this that are up...

Wednesday, July 22, 2009

Two – no, three – Utterly Ridiculous Games

Well, I was so bored the other day that I succumbed to the madness of reading Geurt Gijssen’s arbiter column on ChessCafe, and – as unfortunately so often happens with bad behavior – I was richly rewarded with the most ridiculous game I’ve ever seen, which someone had submitted for an official opinion.

I won’t have time to make anything like a real chess post for a week at least, so I thought I’d throw this out there – especially for any of you whose brain may have been fried by the 8 x 8 queens post.

The question was whether, in a game where White starts with his king and queen in each other’s positions, is the checkmate after
1.e4 e5
2.Bc4 d6
3.Kh5 g6#

legal? Or – more to the point – does the result stand in a tournament game?

The questioner quoted a whole bunch of rules relating to making an illegal move, which I didn’t read because I don’t really give a damn. Gijssen said that the more relevant rules were those governing pieces being set up wrong in the starting position – if it’s discovered during the game, the game must be restarted from the beginning (with, one hopes, the pieces set up correctly).

However, he added, since giving checkmate and pressing one’s clock ends the game, then the result of the game must stand, even though it would really pain him to allow it. He proposed several technical rule changes to allow such a game to be disqualified, which (of course) I didn’t read either, since if I’d been that bored I would have been banging my forehead against the computer, not reading stuff on it.

My main point in sharing is that this is just a hilariously ridiculous game, and it’s even more hilariously ridiculous to see an international arbiter approaching it seriously. I mean, this is a game where White (a) attempts the Scholar’s Mate while (b) not noticing that he’s moving his king to h5. What are the chances that this is going to occur in an event governed by the laws of FIDE?

Furthermore, I think that Gijssen is just dead wrong for two reasons:
First of all, what the hell are the standards for “discovering” that the pieces were set up wrong – or that an illegal move was made earlier in the game? It’s hard to imagine that Black delivered checkmate without “discovering” that it was his opponent’s king he was checkmating. I mean, unless he played 4...g6 and then went “Wait a minute! That’s your king!”

For Gijssen’s logic to be correct, both players have to have realized that the piece on h5 was a king immediately after (and only after) Black played 4...g6. After all, if White still thought it was a queen, he would have just retreated it to f3 and the game would have continued. To constitute “discovering” that the pieces were set up wrong, is Black somehow required to blurt out “Oh my God! That’s your king on h5! It must have come from d1, where it was set up incorrectly in the initial position and it was an illegal move to bring it out to h5!”? Give me a break. If Black delivered checkmate intentionally, it’s hard to imagine that he didn’t also realize that the king’s appearance on h5 was not quite kosher.

An alternative, of course, is that perhaps the helpful arbiter who submitted the question pointed out that a checkmate had occurred, when neither player had noticed it. (In this case, the arbiter should simply be shot, because he should have let the kids play) I have no idea what the rules are on continuing the game after checkmate, because (like I said) I’m not that bored. I’m just a bit intrigued by the ridiculousness of it all.

Okay: the second reason that Gijssen is dead wrong is that White should be forfeited on principle for attempting to mate on f7. Quite honestly, this should overrule all other considerations. Especially with children. Granted, perhaps this is a Nakamura game (blitz playoff?) where he decided to revert to his Qh5 repertoire, but in that case he should lose even more so, because it might influence young people.

I once had an eight year old student who would not accept that this strategy could possibly be bad. I lectured him sternly about the need to get all your pieces out, but he would not listen. “Okay,” I said, “we’re going to play a practice game.” This went:
Howard – Leon
CalArts, ~1990
1.e4 e6
2.Bc4 Nc6
You see, I have prepared for my opponent’s repertoire.
3.Qf3 Nd4
and here he joyfully picked up his queen to deliver checkmate at f7, only to discover that his bishop was blocked. And that he had a problem on c2. Consternation ensued.

(Here, actually, you see why this mindless checkmate goal is such a destructive meme: here’s a kid who – when he looks freshly at a position – can notice that not only his queen is attacked but there’s also a costly fork on c2. And yet, when planning his mate, he was so deeply on automatic that he didn’t notice that my e6 pawn blocks his bishop from f7)

The game continued:
4.Qc3 Bc5 I considered 4...c5, but thought it wouldn’t teach him as much about the importance of development if I won while behind in development. Luckily, the game finished very “instructively”:
5.d3?? Bb4
6.Qxb4 Nxc2+
because of 7...Nxb4 coming

A much nicer game, which actually accomplished its instructional purpose and got me the best parental response that I’ve ever had or heard of. Howard was a bit sulky after the game, so I thought it best to give his mom a little heads-up on what had happened.
So I drew her aside when she came to pick him up and said (quietly, out of the side of my mouth, like a secret agent) “We played a training game today, and I, uh, sort of kicked his ass.”
Mom: Well, I should hope so! We are paying you after all!

She also reported the following week that in his games with his father, Howard had taken to admonishing him over and over to get all his pieces out. It is good to have one’s instructions taken to heart.

If my math is right, Howard is almost 30 years old now. I wonder if he still plays chess.

The Third Ridiculous Game
When I started this post, I called the first game the most ridiculous one I’ve ever seen, but then I realized that wasn’t true. The most ridiculous chess game I know of goes
1.e4 e5
2.Qh5 Ke7

But that’s just so offensively ridiculous that I won’t even give it a diagram. It does seem to show, though, that there’s a theme of an early Qh5 being associated with ridiculousness. Elizabeth Viccary just made a post where she mentions an argument among some players about what move is most often a good one (or bad one) – wherever it occurs in whatever game. I’d make a case for Qh5 to most often turn out to be a silly move.

Sunday, July 5, 2009

8 x 8 Queens

The famous Eight Queens problem has always fascinated me, perhaps because I’m so bad at it. The problem is to place eight queens on the chessboard in such a way that none of them attack one another (diagram shamelessly copied from Wikipedia).

I actually think that the reason it’s so fascinating to me is because I can’t see any pattern for the placement of the queens. I mean, a lot of the queens are a knight’s move apart, which only makes sense, since it’s the closest way to pack them without them attacking one another. But obviously this also gets broken quite a bit, and there turns out to be no simple heuristic as to where to start over once you’ve run your string of knight-related queens to the edge of the board (as happens on the left-hand side of the diagram).

This would also be why I’m so bad at it – because I can’t see any overall pattern. So ideally, to “subdue” this problem, I’d want to find a way to see patterns in it – by which I mean human-recognizable ones, not paper algorithms that can generate a solution but can’t be perceived over the board.

Another thing I’ve wondered about off and on is how many eight-queen solutions can be placed on a single board, where one is not allowed to put more than one queen on a square. In other words, can I put one set of eight yellow queens on the board such that none of them attack one another, then put a set of blue queens on vacant squares such that none of them attack one another, and so on for eight sets of eight queens?

Packing: the Rotational Approach
Now, these patterns are much too random-looking for me to just try mixing and matching them. One needs a heuristic. The first one involved packing a whole 8-queen pattern into one quarter of the board via rotation.

For instance, if I give the board three quarter-turns, and on each turn I “trap” whatever queen is in the lower-right, then I’ve packed all eight queens into one quarter of the board. If I can do this without any of them landing on top of one another, I’ll have proved that the initial position can be rotated four times to give us four solutions that fit on the same board. The diagram below shows how our initial position fails this test:

Here I’ve colored the home square of each queen, and colored the square that it packs to (in the lower right) the same color. Six of the queens rotate without interference, but the two on the magenta squares map onto one another, ruining the situation.

Well, let’s not try to break down an existing pattern – let’s try to generate one of our own. And let’s not bother with the four rotations, let’s immediately test for eight.

Generating our own solutions
This quarter of the board can be split in half down the long diagonal. If I can take each eighth of the board and spin it out into a valid eight-queens pattern, then the way I’ve generated it will prove that (a) it can be spun three times to give us four non-interfering eight-queens patterns, and (b) it can then be flipped across the long diagonal to give us four more – our ideal solution of eight non-interfering patterns.

It’s not quite as simple as that, because the long diagonal squares can’t each be divided in half. But one can split them two and two: for instance in the diagram above, one can take the set of the six yellow squares plus the two deep blue ones on the long diagonal, and then take the set of the six light blue squares plus the two black ones on the long diagonal. If both sets can be spun out into a valid eight-queens pattern, it will show that it is possible to fit eight sets of eight queens on a chess board.

It is fairly easy to show that this is impossible. Let’s start with the central dark blue square and work our way down the column of yellow squares beneath it:

Already, ominously, there is only one rotational slot possible for the second square (possible rotations shown in green). And there is only one for the third square (shown in magenta). By the time we get to the fourth square, there is no space available for it that doesn’t attack an already placed queen.

The reason for this is easy to see: we’re starting out with four squares on the same column, and there are only four rotations available. Therefore, we have to distribute our rotations with optimum efficiency, one queen each on the two central columns and two each on the two central rows. But queen number one is already on both a central column and a central row, taking up two slots. Thus, we can only add two more queens before running out of slots. The exact same problem applies with the corner square.

I believe that there are similar issues with light-blue-centered eighth of the board, but those would be more complicated to demonstrate. In the meantime, it’s clear – there can be no fully 8x8 queen solution generated in this way.

Is this a packing problem or a tiling problem?
Well, this got me thinking about approaching the problem in a related way: these rotational rings of queens don’t attack one another; so if I can find a pair of rings that cover the whole board without attacking one another (a) it’ll be pretty cool and (b) it might open the way to seeing other patterns.

Well, that doesn’t work. In fact, if we make a ring of queens that are on rotations of the square e3:

We find that just two more queens (in the lower left and right corners) completely cover the board. Now we’re starting to get more human-grokkable patterns, but we’re also straying a bit from the task. So I started to wonder: is this a packing problem or a tiling problem?

It’s sort of a mixture of each: if it’s a packing problem, then it’s to find the most efficient packing given a constraint (queens can’t attack each other) that makes the packing very inefficient; or, if it’s a tiling problem then it’s to find the least efficient tiling given a constraint that tends to make the tilings maximally efficient. This is an issue especially with our 8 x 8 queens task, since it's actually a packing of a packing (or a packing of a tiling?).

Okay, so let’s take a look at whether this ring of queens idea can give us patterns that need less than six queens to control (or occupy) the whole board. One does not have far to look for a generalized solution!

For each non-mutually-attacking rotational ring of queens outside the central sixteen squares, the four queens in the ring leave exactly four squares uncovered. These squares are symmetrically distributed on the long diagonals, so that one more queen on any of these four squares covers them all and completely controls the board.

In other words, now that we’ve shown that e3 needs six queens to cover the whole board, the five remaining yellow squares from our original eighth-board formation all have five-queen solutions to the problem of covering the whole board. Here are those formations:

e2 (I’ve just colored the appropriate squares rather than drawing actual queens)





A few points here:

1) I think this exhausts the solutions possible with five queens. I don’t have a proof, but these five patterns strongly exploit both the rotational symmetry of the board and that of the queen herself. As soon as I mess up the symmetry, the solutions start requiring more queens.

2) I will hold out the possibility that there may be a solution with four queens, randomly and brilliantly placed (sort of like the famous problem of how to connect nine dots with only four straight lines). But I think the probability of this is low.

3) Among these five solutions, each uses a different long diagonal square except that two use the corner. However, you can see that I have cunningly notated one corner in the lower left and one in the upper left (all other long diagonal squares are in their lower left rotation). This means that (a) we can place each of these five solutions on the chessboard at the same time without any queens landing on top of one another, and (b) we can then flip this composite solution horizontally to give us a total of ten solutions packed onto the same chessboard.

I like this.

Update: after writing all this, I did some online research and found out that there already exists a proof that five queens is the minimum number to completely cover an 8x8 chessboard, and that moreover there are 638 “basic” solutions that can be rotated and reflected to produce a total of 4860 distinct positions satisfying the condition. But intriguingly, the position given as an example follows the same form as my five solutions: a ring of four queens plus one “cleaning up” on the long diagonal. I wonder how many of the solutions follow this form.

Back to the original problem
Can this shed any light on our original problem of the eight queens, and how many of them can be packed onto a chessboard? It does. Let’s take a look at Wikipedia’s solution #1 again:

Although the presence of queens at d8 and h5 mean that this solution can’t be rotated a quarter-turn in either direction, it can be rotated 180 degrees. That gives us the following pair of solutions, notated here with red squares for the original queens, and blue squares for the rotation:

Now I call that an amazingly regular pattern – one that is very human-understandable! First of all, this is a pair of solutions packed onto the same board. But one can also easily see by looking at the pattern that it can be flipped either horizontally or vertically without any of the queens landing on top of one another. But then it can’t be flipped again, because either of these flips gives us the same composite pattern of four solutions packed onto the same board. (This is actually a result of the operations we’ve done, since flipping horizontally followed by flipping vertically is the same as rotating 180 degrees)

Looking at the existing “basic” eight queens solutions (there are only twelve), there aren’t any whose composite patterns of four magically compliment one another, giving us the ideal eight solutions. However, there is at least one pair of eight-queen patterns where the solution and one reflection can fit into the space left by the full composite of another.

Each of the twelve basic solutions must also be a distinct eight-queen pattern in either of its reflections, since for a reflection to have a queen land on top of another means that those queens must have started out on the same row or column – an impossibility giving the starting restrictions. Each of the twelve basic solutions also possesses exactly one non-interfering rotational version. All twelve patterns contain at least one pair of queens that interfere with one another rotationally, but those that contain two pairs interfere with each other at the same number of rotations.

So each basic eight-queen pattern can be packed onto the board with exactly three other versions of itself. And once this is accomplished, there is at least one pair of basic patterns where two permutations of the second pattern can be packed onto the board with the full four permutations of the first.

Let’s take a look. Here is Wikipedia’s pattern #2:

In passing: I like the way this pattern contains two permutations of my e3 tiling covering the chessboard with six queens:

Okay, here is Wikipedia’s #2 pattern rotated 180 degrees and then flipped horizontally (or vertically):

So far, so good.
All right, here is Wikipedia’s pattern #12:

You can see that it fits in the blank spaces left in the diagram just above it. Let’s flip this one horizontally:

This one fits, too, so let’s put them all together:

This gives us six eight-queen patterns packed onto a single chessboard.

Okay, summing up:

1) I think that this is a maximum packing for eight queens. Just looking at the twelve basic solutions, they all interfere with themselves or one another if you try to do more than six on a board. This is still a very concrete result, though, achieved by spot-checking the twelve basic solutions for rotational interference. I have no idea how to give the result (generalized or specific) in theoretical terms – for starters because I have no idea if there exists any way to represent the basic solutions themselves in theoretical terms.

2) I find it interesting that the five-queen solution can be packed onto a chessboard ten times, and the eight-queen solution six times – as close to being inverses of one another as the number of queens being used allows. One occupies 50 total squares on the chessboard and the other 48. I wonder how this number will behave for other numbers of queens on other sizes of chessboard. I imagine that as the board gets larger, it will become possible to fill a larger and larger proportion of the available squares with queens – by an analogy with knot theory, where as you increase the number of dimensions, it becomes easier and easier to slip out of a knot, and also because as the board increases in size the queen becomes proportionally less powerful.

Sunday, June 28, 2009

My First Sicilian

There are first things we all remember. This is my first Sicilian. After years of being a Réti player, I was getting good positions out of the opening, but it was taking me too long to win them. I was tired of winning a game in the morning that took six hours and then having to play the second game of the day after a 20 minute break and (often) no food.

I tried switching to the English, but that wasn’t enough, so eventually I went whole hog and decided to switch to 1.e4 (a decision from which my rating has still not recovered). Anyway, my main fears about this shift were assuaged when I bought Nunn & Gallagher’s Beating the Sicilian 3. I have never been able to understand why Black doesn’t just always get mated in the Sicilian, but the statistics are very clear that there must be some reason. So I figured I’d by buy the book, use it in an email tournament (with the IECG) and by the time that was done I’d have enough quality experience to play it over the board.

Shernoff-Dunn, IECG Cup 1997
1.e4 c5 2.Nf3 d6 3.d4 cxd4 4.Nxd4 Nf6 5.Nc3 a6 6.f4

N & G’s recommendation against the Najdorf.

I was happy with it because I figured it wouldn’t have as much theory as the Bg5 lines, and because I’d seen a game with it where Korchnoi beat Geller in a style similar to that which I was hoping to achieve by switching to 1.e4.


Okay, now we’ve got all the knights developed on normal squares, just like I was taught as a wee lad when I was first learning about chess. Let’s be good and look in the book...

What? This is not a normal move? In fact, it’s so much of a sideline that it gets kissed off with “7.Nxc6 bxc6 8.e5 Nd7 9.Bc4 dxe5 10.0-0 e6 11.f5 Bc5+ 12.Kh1 and White has good attacking chances.”

What? I paid $23.95 for one little line, and to be told that I have good attacking chances?

7.Nxc6 bxc6 8.e5 Nd7 9.Bc4 dxe5 10.0-0 e6 11.f5 Bc5+ 12.Kh1

Son of a gun, he played into it. And now, as so often happens, that little untested line turns out to have a completely incorrect evaluation. On my very first use of that expensive book. Thanks, N & G!
12...exf5? 13.Rxf5 f6
My opponent, in New Zealand, always played very quickly (on a couple of occasions we exchanged three move pairs in one day) and was always happy to play “anti-positional” moves like this in order to hang on to material. Characterizing his play as quick, materialistic, and anti-positional, I decided he must be using a computer. (This is how they played, back in the day.)

In retrospect, I’m not sure why I thought that, since Black (from the diagram) has the much better and more materialistic move 12...Nb6!, after which White will either have to exchange queens or sac unsoundly in order to remain only one pawn down – for example 13.Bd3?! exf5 14.Bxf5?? Qxd1. I also don’t think much of 13.Qf3 Nxc4 14.Qxc6+ Bd7 15.Qxc5 Rc8 and exf5. This is not why White opens with the e-pawn.

Okay, so I’ve dodged a bullet here, but I still have to address the threat of Nb6, and I have to find a constructive place to develop my queen’s bishop, otherwise my lead in development will just melt away.

You might want to take a moment or two to contemplate how you would solve these problems...


My first Sicilian, and I get to sac a knight at d5. Sweet!
This was what I had expected, but 15...Rb8 is a much more constructive way to threaten to take the knight.

On 14...Nb6, White does not play the humorous line 15.Nxb6? Qxd1+, but instead 15.Nxf6+ Ke7 (15...gxf6 17.Qh5+ with widespread devastation) 16.Bg5! with that nice bishop development that I’d been aiming at – for instance, White has a nice checkmate after 16...Qxd1+ 17.Rxd1

You might want to stop here and try to spot the various mates.

The easiest is 17...gxf6 18.Bxf6+ Ke8 19.Rd8#

Then there's 17...Bxf5 (covering d8) 18.Nh5+ Ke8 19.Nxg7#

And the prettiest one is 17...Nxc4 18.Ne8+! Ke6 19.Nxg7#. It would have been difficult to spot this over the board...

Back to the game
(after 14...Rf8)

15.Qh5+ g6
16.Qxh7 cxd5
17.Qxg6+ Ke7
18.Bxd5 Rb8

And there’s that nice bishop development that was the whole point of the combination. I was glad I was playing this game via correspondence. It would have been tough and stressful to find all of this over the board.

At the time, coming off all those Rétis of mine, this seemed like a completely crazy position to me, but now as an experienced 1.e4 player, it seems completely normal! Well, not. But I certainly feel quite confident in saying that White has good compensation for the piece.

Again, I felt that this was an inappropriately materialistic move. On the other hand, 19...Nb6 20.Qg7+ Kd6 21.Qxf8+ (21.Be4 Bxf5 22.Bxf5 Rg8 23.Rd1+ Bd4 and I don’t think I have quite enough) 21...Qxf8 22.Bxf8+ Kxd5 23.Rxf6 and I should be able to win this endgame.

20.Bxf8+ Qxf8
21.Bb3 Bd4
22.Rd1 Nc5

The rook will be very active on the 7th rank. Meanwhile, my Qg6 is active enough; and it defends c2, keeping the Rb2 confined.
24.axb3 Be6
25.Rh7+ Kd6

See my comment above about this position now looking normal. In Rétis, the whole point is not to give your opponent any counterplay whatsoever. Umm, not that Black really has any here. But one never has to calculate as many tactical lines in the Réti as I’ve had to do here, unless you’ve screwed up badly and let Black off the hook. So although I kept telling myself that things were fine, my positional alarm bells kept giving off danger signals. And there were certainly a lot of tactics here that I might have missed over the board.

on 26.c3 Bd5, the weakness of g2 may cause me some problems. Better to not touch anything major, and just push my h-pawn.

26...Ra2 An unusual way to defend the a-pawn.

27.h4 Kc6

I have no idea why I did this.

I have no idea why he did this.

29.Rg7 Qd8
30.h5 a5
31.h6 Kb5

What is Black doing with his king?

I have no idea what came over me. It is foolhardy to open the second rank, and I soon get into trouble because of it. I should just push the h-pawn, of course. I think I was motivated by the fear that his queen might move off the d-file and I’d lose the pin, and also that 32...Bd5 can currently be met by 33.c4+

32...f5 In an over-the-board game, I might have missed the threat of Qd8-h4. This is why he ran with his king – so that Qxe6 isn’t check.

33.Qh5 Bxb3

Because I was so upset over the course of the game, I decided afterwards that this was a mistake, and I should have played 34.Rb1. This seems correct, although the position has still gotten even more nerve-wracking than it needs to be.

This was all part of my plan when I played 32.c3, and I was expecting Black to resign now. After all, I’ll be a full rook up in a moment.
Huh! Sort of like a spite check! A spite mate threat. I just defend, and then... he... takes my rook on b3. Son of a bitch! Emotional pandemonium!

Luckily, in correspondence chess one can go downstairs and watch back-to-back episodes of Wings and Bewitched to calm ones nerves.

You might want to take a bit of time now and figure out how White gets out of this situation.

36.Qf3+ e4

37.Rb8! exf3
After 37...Qxb8 38.Qxf5, saving the bishop loses the rook to Qe6+, and 38...Qg8/g3 is met by Qxe4+ and cxd4, when Black is in a world of hurt, since White’s queen covers the perpetual (after 38...Qg3) by Qf2-h4 and back. This last is also quite hard to notice over the board, since my main focus for Qxe4 is guarding g2.
37...Qg5 may be the best swindling try (38.Qf1?? Qh4#) but 38.Qh3 holds things down surprisingly well.

38.Rxg8 And the rook covers g2

And here I thought that 38...f2 39.cxd4 Re2 was a better chance, though in correspondence it’s not difficult to find 40.g4 with Kg2 to follow.

39.Re8 fxg2+
40.Kg1 Bg3
41.h7 a4
Around here, I decided that a computer would not have shed material so fast, and gave up the idea that he was using one.

42.Ra1 42.Rd3 may be more accurate, chasing the bishop instead of the rook. His rook can (and should) just move back and forth on the 7th rank.

43.Kxg2 Bd6
44.h8Q Ra2+
45.Kf3 1-0

After the game, I mentioned the quick play and the computer idea, and he said “Oh, no! It’s just that I play at work, so I always have to move very fast, before someone sees me.”

Thursday, June 11, 2009

Pretty Drawing Line Busted

I've added something at the end here, since Blogspot isn't allowing me to post comments to my blog. This seems to be a known problem that they haven't fixed -- there are a few threads in their help forums complaining about it.

Alas, I busted my favorite line in this whole endgame on my way home from work yesterday. You will recall one critical position, which in my home notes is labeled “The Last Crossroads” (and I have now added that title in the original post).
The Last Crossroads

Here I gave three different tries for Black:

61...Nd3, toying with queening the queenside pawns
61...Nd4+ 62.Kb1 Ndb3, forcing the queening of a queenside pawn, and
61...Nd4+ 62.Kb1 Nce2, threatening mate and forcing a draw. This last line is the one that I thought the most correct, and also the most attractive. However, it seems that White has a superior option: after (61...Nd4+ 62.Kb1 Nce2) 63.Ka2 Nc3+ 64.Kxa3 b1/Q

The line that I gave in my original post is 65.Rxb1 Nxb1+ 66.Kb2 Ne6 =. However, this particular position always bothered me, in that sort of nagging way that all too often turns out to be significant. White sort of has a little breather here, because his king isn’t being checked. However, what can he do other than take the queen, right? – because he’s threatened with mate on the move. Wrong! He can under-promote:
65.f8/N+! Kh8
66.Rxb1 Nxb1+
67.Kb2 Nd2
68.Kc3 N2f3

and for the moment Black has preserved his extra knight, but he’s not happy about it. For one thing, White can win the knight back at will by pushing his e-pawn. At the moment, that would also lose the h-pawn, so probably White should start by protecting that by pushing his g-pawn (perhaps preceded by Kd3-e3/e4. None of Black’s pieces can really move (especially once White plays Kd3-e3, tying down both knights). Black may be able to draw if he gets two pawns for one of his knights, but in all these lines it's White who's playing to win.

It’s no good trying to avoid this with

65.f8/N+! Kg8
66.h7+ Kf7
67.Rxb1 Nxb1+

and now Black has to come back with Kg7 to keep the h-pawn from queening, so White gets to take the Nb1. 66...Qxh7 67.Nxh7 is also awful. So it looks like Black would have to go for complicated swindling chances in one of the other tries.