Calculation and Conceptualization

by admin on April 13, 2014

This week I recorded a ChessLecture that should come out in a month or so, called “How to Tell When the Moment is Right.” The question I was looking at in the lecture is, how do you tell when it’s time to calculate detailed variations, and how do you tell when you shouldn’t calculate variations and you should just go on general principles?

Of course, I don’t want to give away the content of the lecture. But I will say this: I’m a calculator. On almost every move of every tournament game, I’m trying to calculate variations, even in positions where there is nothing to calculate. I think that this is one of my flaws as a chess player, and it’s the reason that I get into time trouble so much.

As I thought about it some more after the lecture, I realized that this dichotomy goes way beyond chess. There are two complementary approaches to almost any problem that requires conscious reasoning. We could call them the “top-down” approach and the “bottom-up” approach or, as in the title to this post, conceptualizing and calculating.

The top-down approach is to start out with the big picture and then modify it with specific information. This is the viewpoint of, say, a CEO, who has to set the direction of his company but cannot be bothered (except in exceptional circumstances) with the details of what individual people do.

The bottom-up approach is to start out with the nitty-gritty details and assemble them into a big picture. This is the viewpoint of a farmer. Sow lots of plants, water them day after day, and eventually you feed the world.

The dichotomy is very strong in mathematics, which is something I was also thinking about a great deal this week. In mathematics, too, my style was always to calculate, calculate, calculate. On a few rare occasions — about three or four in my career — the calculations came together into a glorious big picture. One of those times was just a couple years ago, when I was working on the problem that recently turned into this paper (my first mathematical research paper in 18 years). Roughly a third of the way through the paper there is something called the Symmetry Lemma, which started out as a mass of calculations but then got abstracted into a result that barely involves any calculation at all.

But calculation doesn’t always lead to a big picture. Usually, it doesn’t. One virtue of calculation is that at least you get a lot of small pictures out of it. You aren’t left empty-handed. Notice that the paper I just referred to is 56 pages long. Of that, maybe six pages are devoted to the big-picture result that I just mentioned, the Symmetry Lemma. The rest is just massive calculations. These have value, too; they solve the problem I was trying to solve. On the other hand, they don’t point to any general, overarching principles beyond this particular problem. To invoke the chess analogy, the paper is like a single chess game that I won. Imbedded within it, from moves 20 to 26, is this general principle called the Symmetry Lemma that I could use to reason about other, similar positions (and maybe win them, too).

There is another approach to mathematics, the grand systematizing approach, which in my view is represented best by the French school and especially Alexandre Grothendieck. If you haven’t heard of him, you should definitely read this amazing two-part series by Allyn Jackson that appeared in the AMS Notices in 2004. There’s an anecdote in the second part that illustrates what I’m talking about. Grothendieck always reasoned in terms of grand structures and never talked about specific examples, which were beneath him. Once, Jackson writes,

In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”

Of course, the punch line is that 57 isn’t prime. (It’s 3 x 19.) People started calling it a “Grothendieck prime.”

If you’re a calculator, the joke is on Grothendieck. How can this world-famous mathematician not know his basic arithmetic? But if you’re a conceptualizer, the joke is on the person who was talking with Grothendieck. Of course, the CEO of a corporation can’t be bothered with how many light bulbs they’re using or who is installing them. Similarly, Grothendieck can’t be bothered with which numbers are actually prime.

As an aside, I think that number theory (a field very close to Grothendieck’s work) is a prime example of a part of mathematics where the conceptualizers have won. The great theorems of number theory are proved by creating vast abstract concepts and then specializing. The subject is replete with terms like sheaves and schemes and idèles and adèles and étale cohomology that mean almost nothing to me. Look at how Andrew Wiles proved Fermat’s Last Theorem, the greatest number theory result of our lifetime. He proved that “all semisimple elliptic curves are modular,” and deduced the numerical result known as Fermat’s Last Theorem as a very incidental example. You don’t need to understand what “semisimple,” “elliptic curve,” and “modular” mean to get my point. These are very broad, conceptual terms.

I wrote at the outset that calculating and conceptualizing are complementary. The beauty of mathematics is that there is plenty of room for both of them. And chess, too, is a game of sufficient richness that both approaches are feasible and reasonable. We all learn some general chess concepts: develop your pieces, control the center, move your king to safety, look for your worst-placed piece and try to make it better, bishops are better than knights in an open position, etc.

I think that there are “conceptual” players who have an even broader repertoire of concepts, although they might not be able to formulate them precisely and they may be mistaken at times (a little bit like Grothendieck thinking 57 was prime). For example, I think of Jesse Kraai as a great conceptualizer. That was one of the things that made his ChessLectures so wonderful and so special: He was always looking for simple rules, expressed in common English (like the “angry bishop”), that governed the game. But I also think that he did a lot more calculating than he realized. He would use his general principles to get to a critical position, and then there would be this little tactic here and that little tactic there, which he would almost brush off in his lectures as inconsequential. Perhaps they were inconsequential compared to the general themes he was elaborating. But if you don’t spot those tactics and calculate them accurately, all of your “general principles” and “simple chess” won’t do you any good.

Anyway, as said before, I’m in the opposite situation. I need to work more on trusting and using general principles. That will be my goal for my next tournament, the Larry Evans Memorial in Reno from April 18 to 20.

For the rest of you, it might be good to pause and ask yourself for a moment: What kind of player am I? Which facet of the game do I need to work more on? If you’re frustrated by the fact that you seem to know more chess than your opponents but you keep making tactical errors or missing opportunities, then y0u’re probably a top-down person and you need more work on calculating. If you’re quick with the combinations and can solve tactics puzzles like an expert, yet somehow you never get the positions where you can use those abilities, then you’re probably a bottom-up person and need more work on conceptualizing.

Print Friendly


“How a Master Eats an Expert”

by admin on April 5, 2014

At Mike Splane’s most recent chess party, Craig Mar showed us a really nice game he played years ago. Even though he doesn’t play tournament chess any more, he is a really good teacher and the games he shows us are usually as relevant as they were when he played them.

This time he told us he was going to show us “How a Master Eats an Expert.” All of the moves that Black plays in this game are superficially reasonable, and yet he ends up in huge trouble and loses a miniature. How did this happen? Well, let’s see.

Craig Mar – NN, Queen’s Gambit Declined

1. Nf3 Nf6 2. c4 e6 3. Nc3 d5 4. d4 Be7

There are many options here for Black. I always used to play 4. … c5, called the Semi-Tarrasch, taking advantage of the move order, which allows Black to not get an isolated queen pawn.

5. Bg5 h6 6. Bh4 Nbd7 7. e3 O-O 8. Qc2 b6?!

Position after 8. ... b6. White to move. Position after 8. … b6. White to move.

FEN: r1bq1rk1/p1pnbpp1/1p2pn1p/3p4/2PP3B/2N1PN2/PPQ2PPP/R3KB1R w KQ – 0 9

Although you can find grandmaster games where Black has played this move, I think that it is slightly misguided. Both sides are playing a waiting game, trying to get more information. Craig avoided 8. Bd3 because it gives Black information and allows him to play 8. … dc, winning a tempo. Similarly, Black could play a noncommittal move like 8. … c6 and throw the ball back into White’s court. Or he could play 8. … c5, which again puts the onus on White to figure out how he is going to open the center. With 8. … b6, Black announces that he is developing his bishop on b7. That allows White to take on d5, a move that now has strategic value because it will lock in the bishop on b7. In this sort of position, the bishop on b7 has nothing to do.

Not surprisingly, in ChessBase White wins 58 percent of the games after 7. … O-O but 66 percent after 8. … b6.

9. cd ed 10. Bd3 c5 11. O-O Bb7

Position after 11. ... Bb7. White to move. Position after 11. … Bb7. White to move.

FEN: r2q1rk1/pb1nbpp1/1p3n1p/2pp4/3P3B/2NBPN2/PPQ2PPP/R4RK1 w – - 0 12

Talking about this position at the chess party, Craig said that Black’s position feels loose. There’s a lot of space behind those d5 and c5 pawns, which might turn into “hanging pawns.” There is also some incipient weakness on Black’s kingside, due to the weakening move … h6. This is a type of formation that is especially vulnerable to a knight coming to f5.

12. Rad1 Rc8?!

Wow, Black’s winning percentage on ChessBase just plummeted from 33 percent to 17 percent! How can such a natural move be bad? Well, the trouble is that White has a bishop that can get on the h3-c8 diagonal. Black’s rook doesn’t really have a comfortable square on the c-file. All of this goes back to the unfortunate decision to fianchetto the queen bishop too early.

13. Bf5 cd 14. Nxd4 Ne4 15. Bxe7 Qxe7

Position after 15. ... Qe7. White to move. Position after 15. … Qe7. White to move.

FEN: 2r2rk1/pb1nqpp1/1p5p/3p1B2/3Nn3/2N1P3/PPQ2PPP/3R1RK1 w – - 0 16

Now Craig plays a move I really like. To motivate it, he asked us, “What piece would you really like to have on f5?” And of course there is also a huge hole on d6. Remember Craig’s comments about all the empty space in Black’s position. Now that is coming into play.

16. Bxe4! …

A lot of people don’t like to give up a bishop for a knight. But take a look at the pieces that are left! White’s two knights turn into monsters.

16. … de 17. Nf5 Qe6 18. Nd6 Ba6

In the only game in ChessBase to reach this position, Vakhidov-Kalogiannis 2000, Black waved the white flag and sacrificed the exchange with 18. … Rxc3. It didn’t work any better.

19. Nxc8 Bxf1 20. Rd6! …

This move speaks to the form and confidence of a master. White could win a pawn with 20. Qxe4 and would most likely grind down Black in the endgame. But Craig smells blood in the water and doesn’t settle for just winning a pawn.

20. … Bd3 21. Qd1! …

Another kind of move I like: retreating to attack! The queen is heading over to the kingside, where it will exploit those weak squares I talked about earlier. It’s just wonderful to see how all the ingredients work together in this game.

21. … Qe8 22. Nd5 Kh7 23. N8e7 Ne5 24. Qh5 Ng6

If 24. … f6 it looks as if 25. Qf5+ followed by Re6 is decisive.

Position after 24. ... Ng6. White to move. Position after 24. … Ng6. White to move.

FEN: 4qr2/p3Nppk/1p1R2np/3N3Q/4p3/3bP3/PP3PPP/6K1 w – - 0 25

It seems as if Black has gotten himself out of trouble, because the rook sacrifice Rxh6+ is no longer possible. At the same time, he is finally threatening to trade off one of White’s rampaging knights.  How can White break through?

(Space inserted if you want to think about it.)

25. Nf6+! Black resigns

Everyone can see that White would like to play Nf6+ to fork the king and queen. Everyone can see the Black’s g-pawn defends that threat. What not everyone can see is that White can do it anyway! One of the secrets of mastery is that when your opponent “prevents” a move, you find a way to play it anyway. Here the point is that after 25. … gf 26. Nf5, Black simply can’t get him out of the threat of Qxh6 followed by Qg7 mate.

A really stylish performance by Craig!

Note: Typo on move 17 corrected 4/8/2014.

Print Friendly



March 30, 2014

Back when I wrote my Chess Life article about the Bryntse Gambit (White’s queen sac on move six in the Grand Prix Sicilian, 1. e4 c5 2. f4 d5 3. Nf3 de 4. Ng5 Nf6 5. Bc4 Bg4?! 6. Qxg4!) I described it as an “anti-computer” opening, because very few human players have the gumption […]

Print Friendly
Read the full article →

Anand Earns a Rematch

March 29, 2014

As most of my readers probably know already, Viswanathan Anand, the former world champion who was dethroned last year by Magnus Carlsen, has earned a rematch with Carlsen by winning the Candidates in dominating fashion. The tournament isn’t even over yet, but with one round to go Anand has already clinched first place! Even more […]

Print Friendly
Read the full article →

Newspaper article + hyperbolic chessboard

March 24, 2014

Today the Santa Cruz Sentinel had an article about me! Amazingly, there wasn’t a single thing about the article that was embarrassing or cringe-worthy. That is a credit to the writer, Bonnie Horgos. She had a lot of things to cover — chess, hula, science writing, animal care, bomb scares — all in 600 words […]

Print Friendly
Read the full article →

Mobility Recovery After Pelvic Fracture (Dog Video!)

March 17, 2014

Going off-topic today… For four years, Kay and I have been foster caregivers for kittens for the local animal shelter. We call ourselves the Mackenzie Finishing School for Felines. However, we usually have a hiatus between December and April, because cats just seem not to have kittens then. We’re dog people too, so this year […]

Print Friendly
Read the full article →

Two Milestones

March 12, 2014

I’d like to send congratulations to two Bay Area players who scored impressive recent accomplishments. First, Daniel Naroditsky was recently named the winner of the 2014 Samford Fellowship, which is more than just a scholarship — it’s a living wage that lets talented young players focus on nothing but chess (if they want) for a […]

Print Friendly
Read the full article →

The Perils of Opening Preparation II

March 9, 2014

Continuing my series of posts of favorite games from the Santa Cruz Cup…   The third Santa Cruz Cup, in 2005, was a really weird one for me. I finished first, but only because of unbelievable blunders by my two main rivals, Ilan Benjamin and Juande Perea. First, a word about the format. After the […]

Print Friendly
Read the full article →

The Perils of Opening Preparation I

March 7, 2014

At Mike Splane’s last chess party, about two weeks ago, I was excited to see two people who have been almost absent from the Santa Cruz chess scene for six years: Eric Fingal and Juan Diego (Juande) Perea. In Eric’s case the reason for his absence was personal events that I probably shouldn’t write about. […]

Print Friendly
Read the full article →

Tabiyah Time!

March 3, 2014

A “tabiyah” is an opening position that arises when both sides play their “most natural” moves, or a position that can arise from a multitude of different move orders. In many cases it is arrived by mutual consent, although this doesn’t have to be the case. And it’s usually a position where one side or […]

Print Friendly
Read the full article →