The following is a Ready Reckoner, or list showing the values of common moves and formations. Using and memorising the Ready Reckoner should help one to develop high-speed positional judgement. Please amend the details and add to it as you see fit. If possible, please add the formation and its value/s along with an explanation in the main section, and append an in-a-nutshell version in the "Concise Ready Reckoner" at the end.
Caveat utor! Don't go blaming me if you lose games because you relied on these evalutions and found they did not work in those situations :-)
1) Push on the second line = 3 points
In this common position, resulting from a 3-3 point invasion underneath a hoshi stone, Black will not usually want to push at a because it only increases his corner territory by one point, while helping White to build her thickness with b. I often tell novices to keep on pushing if their opponents insist on running along the second line, since the compensating thickness is almost always worth much more than the single points of profit that the pushes on the second line each generate.
2) Push on the 3rd line = 5 points
Pushing along the third line is twice as good as pushing along the second, but still not particularly great, as the opponent's thickness tends to outweigh the profit. For this reason, you need to seek to get ahead as quickly as possible.
3) Push on the 4th line = 7 points
Now we're talking. A much better deal, since the opponent's fifth-line influence is unlikely to be efficient enough to compensate for the profit made by your moves on the fourth line.
4) Push on the 5th line = 9 points
It can be excellent value, except that fifth-line territory tends to be more vulnerable to invasion.
- Bill: None of these values is right. See Counting Crawls.
- Tamsin: Thanks for pointing this out. I've made some amendments; please would you or somebody else fill in the ?s. Thanks.
- Bill: I have made corrections, thanks. In general, a push or crawl just underneath an opponent's wall gains around 2N - 1 points, where N is the line of the push. However, and particularly with the higher plays, there are usually better plays available.
- For instance, such a push on the 5th line would gain about 9 points. A keima to the 4th line is almost always better, however. So we can conclude that it gains more than 9 points. That means that it is worth more than 9 points by miai counting, or 18 points by deiri counting, which is what most people have learned. Not small. :-)
- Tamsin: Thanks for pointing this out. I've made some amendments; please would you or somebody else fill in the ?s. Thanks.
The following list shows two values, one being the actual profit that a formation provides ("ap") and the other being the potential value of the structure ("pv"). Obviously, these potential values are only useful as estimates, and should not be relied upon unthinkingly. Please note, too, that pv includes the ap: when you play the 3453 Enclosure below, you expect to make 12 points at least (ap), and maybe about 13 further points as a result of its influence (to give a pv of 25).
1) 3453 Enclosure = 12 points ap, 25 points pv
2) 3454 Enclosure = 12 points ap, 25 points pv
The actual profit of this corner is about the same as the 3453 Enclosure. However, it is more prone to invasion than the former set-up, while its potential value might be slightly higher due to its greater influence.
3) 4464 Enclosure = 15-30 points?
I find it difficult to decide how to evaluate this in terms of actual profit and potential value. Obviously it's a very big play, but how much of that size is actual profit and how much is due to its potential I am not qualified to guess at. (I'd appreciate some help here from a strong dan player please!)
4) 4463 Enclosure = 15-30 points?
Ditto previous comments, except that against this enclosure the invasion at x is now only a ko. This promise of a greater actual profit is balanced by the lesser influence exerted by the 6-3 stone. (Again, I'd like help please to understand and explain this better!)
5) 443563 Enclosure = 14 points ap, 30 points pv
This enclosure often crops up in handicap go, as in the following sequence:
Black plays 4 to try to prevent the thematic invasion at x. Unfortunately for him, this aji is still not completely erased, while three moves in the opening to secure just fourteen points of profit does not seem to be especially good value for money. White's structure up to 5 exerts considerable influence and has a potential territorial value also of about 10 points.
How much is thickness worth? This is a tentative attempt to provide some means of getting an evaluation in numerical terms. I'm well aware of the problems associated with this, but I think there's some value in it, even if it is only to stop one giving away too much thickness or to give one faith in it when one is exchanging territory in return for thickness.
"The value of a wall is approximately the number of points surrounded by the square that can be built from the thickness plus its appropriate extension."
Therefore, a wall of 5 plus a 6-space extension will be worth about 30 points:
In this position, you can imagine a square being completed by the points marked. The amount of territory this encloses is about 30 points (5 X 6). Now, obviously black will not crudely try to turn this thickness into territory by playing straightaway at the x points, as that would be to play too close to thickness. However, provided he uses the thickness sensibly, he can expect it to yield that 30 points either as actual territory in that part of the board or through its effects on fighting elsewhere.
Now, if you don't get to make an ideal extension, you can see that the thickness is likely to yield less than optimal profit. So, a 5-space wall with a decidedly cautious 4-space extension is only worth about 20 points -- not too good compared with 5-space wall plus 6-space extension.
What is the practical value of this? Well, suppose you're invited to make a wall in exchange for territory. First, work out how much territory you're giving up, then see whether you can make a good extension (or, better still, will already have in place a good extension) from the thickness and calculate its approximate value according to this method. If this value compares favourably with the territory, then the exchange should be a good one.
Thickness does not always mean a wall. Indeed, walls are not always thick. A thick position is simply a structure without weaknesses. How can one account for its value? Experienced players will know that strong formations are valuable because they make the game easy to play - the presence of a friendly strong group is very comforting when engaged in a fight. Even without extensions, such groups gain one points simply through the effect of their dead weight pushing down on the opponent. Consider the following joseki:
In this variation, White has invaded the corner at the 3-3 point, but Black has decided to hold on to the corner using a double-hane tesuji. Inexperienced players, who have a tendency to want to retain the corners no matter what, often play this line indiscriminately. However, although black keeps the corner for about a dozen points, White gets a ponnuki along the side -- a classic thick shape. Now, everybody knows the proverb that a ponnuki is worth thirty points, but here this structure is stuck on the edge, where its value is somewhat smaller. Nevertheless, White can often still be happy with this exchange. Why is that? For starters, the ponnuki will assist White with whatever fighting happens along the top half of the board; indeed, she might be able to get some territory here. Second, Black's corner has plenty of bad aji. So, how to weigh it all up? I would guess that a ponnuki on edge would be worth about half as much as one in the centre, and so I would give White 15 points. Then, I would like to add to this some figure to reflect White's good aji in the corner. Let's say between 5 and 10 points. This computes to about 20-25 points.
- Bill: Considering that this ponnuki is bounded on two sides, your 15 point estimate is quite reasonable. :-) However, White is attackable: for instance, at a. I would feel nervous about leaving the White group as is. I also think that you overestimate White's aji in the corner. The most that Black can get in the corner is 12 points. You are saying that White owns half the corner??? 3 points is a better guess for White's share. Moi, I would subtract it from my estimate for Black rather than add it to my estimate for White.
Now, the above makes for an interesting parallel with Rob van Zeijst's QARTS theory. He argues that a weak group without eyes will cost you about 20 points in the course of a game; therefore, using turning this principle around and using the above example as my evidence, I would like to argue that a strong group will earn you 20 points, by virtue of its simply being there. In other words, the typical deadweight value of a thick group may be reckoned as 20.
The above discussion takes a simple view of thickness. But what to do when there are complicating factors? After all, things are rarely that simple in go :-) For example, what allowances should one make for flaws in one's thickness? (As Ishida says, "Imperfect thickness can be erased completely"). Should one subtract a certain amount from the thickness value according to its flaws, in a similar way to applying QARTS when evaluating the losses incurred by weak groups of various kinds? Suggestions please!
CONCISE READY RECKONER
1) Push on second line = 3 points
2) Push on third line = 5 points
3) Push on fourth line = 7 points
4) Push on fifth line = 9 points - bad aji
(There are often better plays.)
1) 3453 Enclosure = 12 ap, 25 pv
2) 3454 Enclosure = 12 ap, 25 pv
3) 4464 Enclosure = 15-30?
4) 4463 Enclosure = 15-30?
5) 446353 Enclosure = 14 ap, 30 pv
Value = length of wall (or thick structure) x ideal extension
Deadweight value (i.e., the effect of its presence alone, regardless of territorial extensions) = 20 points (reversing the QARTS principle that a weak and eyeless group will cost 20 points).
Nb. All thickness values are susceptible to diminution due to flaws, bad aji, etc.
Charles Matthews This page raises a number of issues. There is some Japanese literature I have seen with titles like 'How Much Is That Move?', which does give numbers to moves early in the game. That is most easily done for genuine endgame plays, of course. To include opening plays you need to double the reach of your scale, extrapolating from plays worth around 15 points gote (7.5 miai counting) to plays worth nearly 30 points gote (15 miai counting). There is something a bit spurious about this, in that we all know the one about 'before the endgame, the gods have placed the middlegame'.
Tamsin Agreed, Charles. The idea is to provide a rule of thumb, nothing more. Nothing is certain in life, nor is it certain in go. However, it's surely helpful to know what common formations are typically worth in order to be able to make high-speed assessments of positions in gameplay. Nothing can replace a detailed analysis of the position, but shortcuts have their uses, too.
Charles Matthews The thing is that mostly the enclosing plays will have similar values - if they are made in a sensible overall context - because the tighter enclosures form a more secure base for future fighting. So that when you look at the value of a possible invasion in points terms, that doesn't tell you the whole story.
Tamsin A lot of this stuff is admittedly a little dubious, especially with regard to the quantitive assessment of thickness. But, then again, Sensei's Library is a place for putting forward ideas and testing them. I'm not a very strong player (but I hope to become one), but I am trying to understand the major issues of the game as best I can. I believe it is important to think for oneself about these crucial issues, and this is why I am constructing my own theories on evaluation. I hope very much to learn from people's responses to these ideas.
The enclosing move, second play in the 3453 enclosure, is called 20+ points. More detailed argument gives 22 points, based on getting a two-space extension both ways by right.
Then the big point in the middle of the side in the orthodox fuseki formation is also called 20+ points, with some reasoning that it is at least halfway to a 40 point side.
Black 1 here is called a 25 point play, that is, even larger. It takes territory while setting up an attack on the marked stones. White has apparently left this position before playing the slide to a. That must be why Black 1 is flagged with such a high value.
Tamsin-- Thanks for the above, Charles. The enclosure values you give above seem roughly equivalent to the "pv" ratings I assigned them. Ishida made honorary Honinbo, so there must be something to be said for taking a quantitive view...
One thing that having a Ready Reckoner produces is a greater awareness of big points in general. At my level, people often start fights in non-urgent areas, while missing easy-to-play big points. Again, people often underestimate the value of thickness, because for most of them it has only a vague, uncertain kind of value -- but if one could give various types of thickness (and, as in QARTS, weak groups) some sort of numerical evaluation, it could help a player achieve some sort of judgement, rather than "playing blind", as it were. Sorry to labour the point, but if, for instance, by appying my theory, I could see that my move "x" worth 15 points' profit would provide the opponent with a thick, influential group (20-points "deadweight" value) then I would think twice before making the exchange.
Bill: I want to echo Charles's sentiments in general. That being said, a ready reckoner has some heuristic value. ;-)
However, I think that there are some systematic biases here. They are the reverse of beginner's bias, which overvalues territory and undervalues thickness.
(Note: Later corrected. :-)) You say:
<< 1) Push on the second line = 1 point
After the exchange, B a - W b, Black has indeed increased his territory count by 1 point plus. (Now W c is worth less than W a would have been.) And, arguably, B a is sente. But it is a losing sente, because W b gains much more for White.
In the opening or middle game, if crawling along the second line is not important for the security of a group, nor does it carry a big threat, it is not worth making. In the endgame you can calculate its value, which can be surprisingly large.
Crawling along the 3d or 4th line is much less likely to be sente, more likely to be correct. (Although crawling may be inferior to sliding or jumping.) If such a move is worth making in the opening or middle game, it should be worth at least 10 points (miai). Your way of reckoning assumes that the play is sente and ignores what it loses in exchange.
As for thickness, indeed there is an interaction between thick stones that cooperate together. However, that effect is not multiplicative, so that the value of a wall is not proportional to the square of its length. The effect is most pronounced with only a few stones which support each other. As the wall lengthens, the effect approaches zero.
Still, for practical purposes, your reckoning of thickness is not bad. For instance, I think that this wall is worth around 20 points, not so much because of an extension, but because of the possibility to wrap around (partially) with the hane at a.