Rapid calculation

Some time ago, on Sensei’s Library I wondered about the possibility of developing savant like abilities to calculate variations in go. Recently I brought up the question again here and hyperpape referred me to a speech by a rapid numerical calculator, Professor A. C. Aitken, in 1954. (URL: http://stepanov.lk.net/mnemo/aitkene.html ) I also found a talk by celebrated rapid calculator, George Bidder, in 1856. (URL: http://www.devonperspectives.co.uk/geor ... lation.pdf ) Neither Aitken nor Bidder were savants. Aitken became a college professor, Bidder became a civil engineer. Aitken’s experience may be more pertinent to us, as adults, since he did not develop his skill at rapid calculation until age 13. Bidder was adept at age 7.

From these two talks I have drawn four lessons that I believe apply to the calculation of variations at go and other games, concerning

1) relaxation
2) meaning
3) memory, and
4) visualization

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Relaxation

Aitken makes a point of the importance of relaxation. Once you have developed discipline and concentration, relaxation is the next step. Paraphrased from the Q and A after Aitken’s speech: “It was like Yoga. It was practised in the physical plane but in the mental plane as well. One had to concentrate at first-to put chains on oneself. But there came a time when one in no way felt the chains at all. Then one had moved away from concentration to relaxation.” Aitken seems to be talking about both memorization and calculation.

Bidder did not make a point of relaxation, but did refer to the strain of calculation, and his attempts to avoid or lessen that strain.

IMO, relaxation is important during games and when doing problems or other training. IMX, it is not so easy to cultivate, however.

Aitken pointed out that schools do not teach relaxation when doing arithmetic and other mental tasks, which is a shame.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Meaning

Bidder brings up the importance of meaning. Numbers meant something to him. They were not just strings of digits. He makes a comparison to words and letters, pointing out that many of us would gladly memorize a page of poetry, but not a page of letters. As a child he was not taught any notation for numbers, and he considered that a boon for developing his skill. By contrast, school children have been taught to manipulate digits in a mindless fashion.

Aitken does not talk about meaning per se, but his interest in calculation derived from his interest in algebra, which he began to learn at age 13. His techniques are thereby infused with meaning. His whole approach was mindful.

There does seem to be a kind of yoga here, doesn’t there? Concentration, relaxation, and mindfulness.

The breadth and depth of go tells against brute force search. (More on brute force later.) Meaning helps us to assess the results of our search, and it also guides it. That is why I, as well as others, stress the importance of understanding problems, not just solving them. What does the problem mean? What do the moves in the variations mean?

IMX, amateurs, particularly SDKs, often fail to read out a situation correctly, because they never even consider the right play or plays. Why? Because those plays do not mean anything to them. The answer is not brute force search, but acquiring meaningful knowledge. Which leads us to the next topic, memory.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Memory

Experienced players are often able to stop their search when they reach a position that they recognize. In choosing plays they recognize tesuji. In this way long term memory assists rapid calculation.

Short term memory is also important, to keep track of intermediate results. Bidder multiplied from left to right — although, since he did not understand number in terms of digits, that was not how he would have put it — since that way put less strain on his short term memory. And the natural tendency when calculating variations is to do a depth first search, since that also puts less strain on short term memory. Aitken believed that short term memory was very important for mental calculation. In the Q and A he said that “He thought this ability to put an {intermediate} answer in storage was what distinguished the calculator from what might be called the man in the street. The man in the street forgot the stages between.” Aitken went on to contrast himself with Bidder in that regard.

I find this quite interesting and hopeful because Aitken started at such a late age. Research suggests that go professionals have sculpted their brains at an early age to have a large workspace (short term memory) devoted to go reading. The fact that Aitken seemed to accomplish a similar feat, starting at a relatively late age, suggests that we can do so, too, if to a lesser degree.

How to develop this ability?

I do not have my copy of Kotov’s “Think like a grandmaster” handy, but, IIRC, one of the things that he did in his training was to find highly tactical positions in games that had been analyzed and spend up to 30 minutes reading out as much as he could, and then writing down the variations that he had considered and then checking them against the analysis. At first he was way off, but he got better. In terms of building a workspace in short term memory, it is not so important that he made a correct analysis as that he calculated and remembered a large number of variations.

Since breadth first search requires more short term memory than depth first search, using it to develop short term memory may be a good idea. Also, brute force search is not bad, either, since the point is not finding the right answer efficiently.

I gave brute force, breadth first search a try, to see how it was. I confined my search to eight empty points. To a depth of three ply, there are 336 sequences. I found that no strain at all, but I doubt if I held all of them in memory. Instead, I think that I recreated most of them on the fly. That hardly matters, in terms of speed and correctness, however. Also, the number of positions to keep in mind is much less than the number of sequences, and that fact can be exploited. Readily seeing the sequences is what is important.

And that leads us to our last lesson, on visualization.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Visualization

Visualization of go positions and sequences is a topic that has come to the fore recently, and I am not going to disparage it. I wrote recently of the time when I lay in bed at night visualizing go variations on the ceiling. However, as I said, despite having a good sense of spatial relationships, for me it is mainly a question of feeling rather than visualizing. I found it interesting that neither Bidder nor Aitken relied much on visualizing numbers. Bidder did not know any visual representation of them, and Aitken said that his memory was in the main auditory and rhythmical. He added, “The fastest mental calculators seem to have been of auditory, not of visual, type,” and gave some examples of both auditory and visual types. I was reminded of my girlfriend’s father when I was in Japan, who was a 2 dan (around Japanese 4 dan today). Often when he calculated variations he would nod his head in time and go “Kou, kou, kou. . . .” at a rate of about 90 beats (moves) per minute. A relaxed method of reading, I suppose.

The takeaway, I guess, is that rapid reading does not necessarily depend upon visualization. OC, you have to visualize the resulting position well enough to count eyes and liberties, perhaps even territory. But having a clear picture of each intermediate position is probably not necessary.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Bill

Great topic to discuss. I'd like to get off on the right foot so wish to clear up one point: "Research suggests that go professionals have sculpted their brains at an early age to have a large workspace (short term memory) devoted to go reading."

It may be that you are using terms in a different way from me, but short-term memory to me is what some call 'working memory' and is that area of the brain which acts as a filter between physical perception and long-term memory. Unless I've misunderstood, the classic fact about STM is that all of us are limited by what its discoverer Miller called "the magical number seven" - that is, we can only hold a maximum of about seven things in working memory at any one time. The reasons the number is magical are (1) that it applies to all of us with just rather rare exceptions who range between 5 and 9, and (2) the number is fixed - all the training in the world won't budge it significantly.

What follows (for me) from that is that I have to question what you mean by "large workspace (short-term memory)".

It is of course possible to fool the STM (the magical version) by turning a long string of small things into a short (<8) string of larger things. Is that what you mean? (I call it 'chunking' but I don't know if that is a generally used term.) But if that is the case we need to acquire a special ability to create memorable chunks, and again it seems, within the limits of my understanding, that STM cannot easily be used to create these chunks on the fly and that, even where effort is put into that, there is still an upper bound of about 7 for the number of chunks. Am I right so far?

Assuming I haven't fallen off the track yet, it would seem to me that the most efficient go players are not those who have a larger working memory but rather those who are able to supplement their inherently small WM by being able readily to access ready-made chunks in long-term memory.

That observation would fit what Kotov was doing, for example. His "effortful" practice was creating chunks that were parked in long-term memory and so they became an available extra tool for him.

It would also fit what you discuss under visualisation. I agree that imagery is not necessarily involved, and that's why I personally prefer 'chunks' to Yoda's 'images'. But presumably it doesn't really matter whether images, sounds, rhythm or other stimuli are used as specific memory joggers - any combination that works for each individual is what counts?

The most interesting point for me in what you said was the notion of the task having to be meaningful for the problem solver. It's one of those things that sound totally obvious once you hear it, but never think of it for yourself (there must be a word for such things and I'd love to know what it is). I'm intrigued by the notion that meaningfulness here can't really be as broad as winning or losing yet must be more than just solving a problem - so what is it for each of you? For me, what drives reading about go (too desultory to be called study) is making myself more equipped to 'appreciate' go as a spectator.

Finally an observation on your "eight empty points" illustration. In no way contradicting what you said, I'd like to suggest that it skipped a very important first step (which in practice I think almost all weaker players tend to miss out, incidentally). In very many problems (both L&D and tesuji) it is possible to identify a portion of the enemy's structure (i.e one of the strings of stones) that can be captured and so make the whole enemy group unviable. This can restrict the number of empty points to play in dramatically, and of course brings the focus much closer to the origin, thus taking an enormous load off short-term memory. It may be that there are other such tricks.

I have heard that the human brain is said to have a very short term memory (a few seconds), a short term memory (some minutes) and a long term memory. I cannot know in which of these a rapid reader would temporarily store intermediate results:)

Meaningful stored information / aims of reading or partial reading steps: OC, I agree. What the meanings / aims can be can fill many books explaining meaning and other explicit information. Since different intermediate steps / ways of problem solving can often lead to the same conclusion, it is possible that different players use somewhat different domains of meanings or aims for intermediate steps. I would be surprised if rapid calculation did not have such a variety.

This is very intriguing to me. In my last US Open game, I completely missed that one of my cutting groups was in grave danger. It had two directions to run, so I just didn't see it. When my teacher looked at the position, she inquired, "Wasn't something about that group telling you it was in danger, not even a tingle?"

My first reaction was, "OMG, she's saying pros have a Spidey-sense for danger!" like everybody's favorite web-crawling superhero. In other words, one should develop a proper go instinct. Then follow that instinct with proper reading. The instinct tells one where to look.

I guess this is part of what you are referring to as "meaningful knowledge."

Regarding visualization, I still find this incredibly difficult and frustrating. I'd like to think you are correct to minimize its importance, but I'm not sure I buy it yet. I will keep doing loads of problems to better develop this skill.

_________________
- Brady
Want to see videos of low-dan mistakes and what to learn from them? Brady's Blunders

Sure, pros have a Spidey sense. The good news is, so do we. The bad news is, ours is not as good as theirs.

Going back to my undergraduate cognitive psych class, so this may well have been superseded by now. I think that intuition, of which Spidey sense is a part, goes beyond conscious knowledge, but grows together with it. So enlarging your knowledge enlarges your intuition.



I remember reading out a fight once, but miscounting the resultant liberties, so that I lost a group instead of killing an invasion. Luckily, I got thickness and sente and was able to launch an attack to win the game. But if my visualization of the final position in my reading had been better I would have not had to do so. So I am not about to downplay it. But as I said, I have a good spatial sense even though I have below average visualization.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

I seek meaning in go on several levels, and I imagine that is true for most who frequent this forum. I care about the cultural impact of go, both from a historical and contemporary perspective. I appreciate go as a possible metaphor for life; exploring go can be a way to learn about myself. And I'm thankful that go can be a way to form relationships with people whom I would not otherwise cross paths. But while these things serve an important function in keeping me connected to the game, I don't think it's this level of meaning that Bidder (or Bill) is discussing.

Bidder seems to be emphasizing understanding as opposed to rote memorization. As an example, he talks about the "number of facts" that need to be memorized to calculate large multiplication tables versus memorizing the multiplication tables up to 12x12, and discusses that fact that he taught himself multiplication by a process of exploration rather than being forced to learn the concept in school. I think is similar to the way people on this board emphasize understanding the "fundamentals--" rather than trying to memorize more and more positions, we look for principles we can apply in many situations. While this applies to all moves in go, the point is perhaps most clear when discussing amateur use of joseki: rote memorization of moves may appear to get a good result, but if you don't understanding the meaning of the moves they can't really be applied properly in a game.

Since I studied neuropsychology I have heard that the brain has a number of memory systems, more than just two or three.



I agree.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

I now have software that I use to download and store material from the internet that I think I may cite later on. But I read about that research before I started downloading possible references. I think that it was in the '90s in Japan. OC, it is only one study, vaguely recalled, so it is only suggestive. The part about sculpting the brain is my own interpretation. The part about having a specialized area of working memory is not.

Sculpting of the brain begins in the womb. Our genes may dictate that certain neuronal connections be made, but do not specify the details of how those connection are made. What happens is that a number of attempts at connection are made, and the successful ones get stimulated. The unsuccessful, unstimulated neurons die off. This kind of brain sculpture by stimulation and cell death continues at a great rate during childhood. And, from what I have heard in recent years, it continues at a slower rate through the teenage years into adulthood. (Since neuronal death continues throughout life, you could say that it never ends.) Anyway, it is quite possible that children, through calculating go variations, could sculpt a specialized area in their brains for doing just that. Aitken's ability to recall intermediate results suggests that he was able to sculpt such an area for numerical calculation, even though he started at the late age of 13.



I have not kept up with neuropsychology, but I believe that we now know that things are rather more complicated than that. In problem solving, as I recall, we can say that there are two systems at work, if not more. One is conscious and more or less linear; the other is unconscious and more or less parallel. When I talk about what Victor Mollo calls seeing, it is the unconscious, parallel system that has done most of the work. (Old hat to a raja yogi, I suppose.) Now, when we talk about the calculation of variations, the conscious, linear system is what we have in mind. But both systems are always at work, and both require short term memory to do their job.

What I imagine go pros are doing when they read is to use a specialized workspace in their brains for both conscious and unconscious processing, including the requisite short term memory.



That holds up pretty well, I believe. However, I know that hypnosis can be used to stretch those limits. Also, it would not necessarily apply to a specialized workspace.



Well, I am willing to let pros fend for themselves. As for the rest of us, I think that learning chunks is a great idea.

I also think that the combination of concentration and relaxation can enable us to do more than we think we can.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Hard to relax when your opponent is trying to kill you

Seriously, though, I find it much easier to relax and concentrate during private study. When playing a game, especially an important one, it is very difficult to relax.

_________________
be immersed

Relaxation is a peculiar concept, and big business at the moment, but maybe a misnomer. The true goal, if we take our cue from martial arts, is actually what I call "relaxed alertness" (relaxing and concentrating sounds like the same thing). This was well illustrated in a Japanese samurai serial I was watching the other day. An old samurai was passing the baton (swords actually) onto his grandson, who had never been in battle before. Then he gave him some advice. He said, "When you face an opponent, laugh at him." He then explained that apart from the psychological effect that might have, the most important point was that laughter relaxes your body and so sharpens your reflexes.

I've heard of a similar trick used by bomber pilots in World War II who were sent on near-suicidal missions night after night. The trick was to assume they were already dead. Again there's a psychological effect, but also a physical one: relaxation (no need to worry about staying alert in this case!) would give a pilot an edge in a dog fight.

So maybe, for some people, the trick for alertly relaxing in go is to assume you've already lost.

I do recall some miracles from some of my games, where I had lost a large dragon, and then came back (the last game I played in the US Open this year was exactly this situation). From my own experience, I can feel a difference in mindset between the beginning of the game, when the situation is even, and after the point that I've basically lost. When the game is even, I am constantly worried about the balance of territory, who is ahead, who has more on the board: How much is that influence worth? Did I get enough from that corner?

At the point when I fall behind by a large amount, my mindset switches. I no longer worry about who is ahead or how much this particular piece of territory was worth. Instead, I focus on the opponent's weaknesses - does he have a group I can cut off? Is there a way I can break up his moyo? Can I make one of his groups weak and attack? I'm not sure if my state of mind after falling behind is more alert - maybe it is. But I certainly feel that I focus more on my opponent's weaknesses than about the balance of territory. And sometimes that gives me a chance to come back.

Maybe this is a little off topic, or different than what you were getting at when you brought up a strategy for being more alert.

Nonetheless, I can certainly relate to a change in mindset in my own play when I'm ahead, even with my opponent, or behind.

_________________
be immersed

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

When I do this in the games I post for review here, you all get on my case for my bad attitude.

Well, you still have to play to win despite the assumption.

But Fedya, you never relax when you think you are losing!

_________________
Dave Sigaty
"Short-lived are both the praiser and the praised, and rememberer and the remembered..."
- Marcus Aurelius; Meditations, VIII 21

I wanted to say something about meaning.

What is the meaning of this corner position, assuming all the Black stones to be alive?

Click Here To Show Diagram Code

[go]$$W
$$ --------------
$$ | . . . . O X .
$$ | . . . . O X .
$$ | O O O O O X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

It looks like an 8 point corner for White, but I have learned that if Black plays first she can get a seki or 10,000 year ko. White to play can secure 7 points.

Click Here To Show Diagram Code

[go]$$W Independent life
$$ --------------
$$ | . 1 . . O X .
$$ | . . . . O X .
$$ | O O O O O X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

This White play looks obvious, and I gave the position no further thought.

Until I started playing around with brute force search. Then I considered this play.

Click Here To Show Diagram Code

[go]$$W Senseless (?)
$$ --------------
$$ | 1 . . . O X .
$$ | . . . . O X .
$$ | O O O O O X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

I had ignored this play as meaningless, but now I was forced to look at it.

Click Here To Show Diagram Code

[go]$$W Two eyes
$$ --------------
$$ | 1 . . . O X .
$$ | a 3 . . O X .
$$ | O O O O O X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

threatens to make an eye at “a” with .

BTW, this is something that I just saw. To understand positions and plays you sometimes need to consider tenuki.

Click Here To Show Diagram Code

[go]$$W Two eyes
$$ --------------
$$ | 1 . 4 5 O X .
$$ | a 2 3 . O X .
$$ | O O O O O X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

This is another sequence that I saw. prevents the eye at “a”, but ends up getting killed as part of a larger eye.

Click Here To Show Diagram Code

[go]$$W Seki
$$ --------------
$$ | 1 2 3 . O X .
$$ | 4 . 5 6 O X .
$$ | O O O O O X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

This is another sequence I saw. It ends in a familiar seki.

Click Here To Show Diagram Code

[go]$$W No seki
$$ --------------
$$ | 1 2 3 . O X .
$$ | 4 5 . . O X .
$$ | O O O O O X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

To prevent that, White has to play at to kill the Black stones.

Click Here To Show Diagram Code

[go]$$W Ko
$$ --------------
$$ | 1 4 2 . O X .
$$ | a 3 . . O X .
$$ | O O O O O X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

Another sequence I saw. After , if White plays to try to make the eye at “a”, makes an approach ko or 10,000 year ko.

Click Here To Show Diagram Code

[go]$$W Life
$$ --------------
$$ | 1 3 2 . O X .
$$ | . 4 5 . O X .
$$ | O O O O O X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

But White has to make two eyes for life. Unless you have learned the corner eye shape, it may not be obvious. It is curious in that it is an eye but not a liberty.

Click Here To Show Diagram Code

[go]$$W Life (continued)
$$ --------------
$$ | 1 3 2 . O X .
$$ | 6 4 5 . O X .
$$ | O O O O O X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

@ 3

can capture two White stones, but then catches two White stone in a snapback, to retain the eye. But after the sente sequence, - , the White chain has no liberty in the eye. That could matter in some circumstances, as White may not have time to play the snapback.

Click Here To Show Diagram Code

[go]$$W Strange life
$$ --------------
$$ | 1 . . . O X .
$$ | . . . . O X .
$$ | O O O O O X .
$$ | X X X X X X .
$$ | . . . . . . .[/go]

So the apparently meaningless actually makes two eyes in a strange way. The meaning of is revealed in the sequences that follow.

Frankly, I was delighted to find the sequence with the strange eye in the corner. It has two forms in which two stones are in atari. The snapback is one of those chunks that make calculation easier. That eye is something that every kyu player should learn about, as well as the seki.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.