Recently on the ngd Discord server there has been some discussion about how to precisely read out capturing races, so I thought to write a post on them. As the topic is broad, it is of course impossible for a single post to cover even nearly all of it, but hopefully this will at least get the reader started.
Dia. 1 shows a basic capturing race. Four white and black stones have gotten cut off from their allies, and, in order to survive, they need to capture the opposing stones first. Both the white and the black stones involved in the race have four liberties, as shown in Dia. 2, and therefore whoever starts filling the opponent’s liberties first will win this race.
Capturing races such as in Dia. 1 are easy to read out; however, most capturing races in actual games will have shared liberties, which complicate matters somewhat.
Dia. 3 features a capturing race with three shared liberties. Unless you know the trick to how to evaluate this position quickly, you may need a moment to read everything out. On the other hand, because this position is still relatively clean, the reading involved is also straightforward.
After a quick initial reading, you can probably see that there is no way for Black to capture White first: this is also backed up by the fact that Black only has five liberties to start with, while white has seven. However, we still need to read whether White can capture Black; and, if he can, does he need to add a move for it or is Black in fact already captured.
Dia. 4 shows what happens if White starts filling Black’s liberties first: finally, White successfully captures Black because he has one liberty more.
Dia. 5 shows what happens if instead Black gets to play first. After black 7, both players have two shared liberties with no outer liberties, and the final result is therefore a seki.
Therefore, in Dia. 3, the black stones are half-dead, and the white stones are alive no matter what happens.
From Dias. 4 and 5 we can already derive our first general rule for solving capturing races: fill the opponent’s liberties starting from the outside, i.e., fill non-shared liberties first. Like with all general rules, of course, there are cases where a more specific rule can take priority, but generally this rule is a very useful one.
Dia. 6 shows how White fails if he starts filling shared liberties first. After white 5, both players have two liberties left, but now it is Black’s turn: White will get captured first.
Dia. 7 instead shows Black’s failure. This time, up to white 6, Black is unable to form a seki and simply gets captured.
Although arguably this capturing race is simple enough that we can just read it out, if we wanted to solve it by counting liberties, we would do the following.
First, let’s say we consider what happens if White tries to kill Black. Dia. 8 shows White’s ‘attacking liberties’: White gets all of his outer liberties, as well as one shared liberty, marked with 5. When you look back to Dia. 4, you notice that when Black is about to get captured, White still has a liberty at that intersection. In the White failure diagram of Dia. 6, White has filled this liberty prematurely, and this is what causes him to get captured first.
As is also evident in Dia. 9, Black has five liberties that White must fill if he wants to capture Black. This means that, when White attacks Black, the liberties are 5-5, and therefore if White plays first he wins the race.
If Black were to dream of attacking White, she would have the three liberties shown in Dia. 10. On the other hand, then White’s defending liberties would total a whopping 7, meaning that White could ignore Black three times and still not die.
From this liberty-counting analysis, we could fairly quickly see that White can capture Black, but Black cannot capture White. By extension, because we know that White wins his capturing race barely (5-5), therefore, if Black plays first, White becomes unable to capture Black. This means that then a seki is inevitable.
There are cases where the above counting method does not work: namely, when one of the players has an eye and the other does not. Dia. 12 shows an example of this. Such capturing races are monstrously unfair to the player who does not have an eye; although we can only see four liberties for White and seven liberties for Black, this race is actually a really close shave.
In capturing races where one group has an eye and the other does not, the group with an eye gets the full benefit of shared liberties while the group without an eye gets none. If Black can play first in this race as in Dia. 13, we get 1–5, and can then see that Black barely wins by one liberty.
Dia. 14 shows Black’s actual liberties in this position; she only gets the four outer liberties. Therefore, currently liberties in this position are 4-4, and whoever plays first captures the opponent.
In real games, capturing races tend to be much more complicated, with groups ‘maybe’ getting an eye or not, and with some liberties not being easily fillable. For your enjoyment, I have prepared a rather difficult capturing race problem, based on a shape that appeared in one of Oscar’s tournament games. What’s going on in Dia. 15?
To start getting an idea on what should happen, we can pick the player who looks easier to play, and see what happens if they try to capture the opponent. For Black, 1–6 is a straightforward starting sequence that efficiently limits White’s liberties.
For Black’s attack, we can therefore probably imagine the position in Dia. 17; this is still complicated, but less so than the one in Dia. 15.
Since this capturing race has a shared liberty, it can make sense for White to form an eye to get the one-sided benefit of the shared liberty. In this case, we can imagine 1–2 in Dia. 18, and, furthermore, we may as well let White force the triangle-marked exchange to get a clearer idea of Black’s liberties.
In Dia. 18, then, ‘a’ is a liberty for White but not for Black: we can therefore count that White has four liberties and Black has four, and it’s Black’s turn to play. Black therefore wins this capturing race by one move.
If White hopes to survive, then, he cannot afford to form the eye with 2 in Dia. 18.
Back in the situation in Dia. 17, we can then count the players’ liberties as follows. For White, we see three unconditional liberties marked 1, 2, and 3 in Dia. 19. Then, either Black will play ‘a’, ‘b’, and then ‘c’ to capture, in which case his group to the right gets the benefit of the ‘c’ liberty; or else, Black can play ‘c’ and then ‘b’, but his group will not benefit from the ‘c’ liberty.
For Black’s liberties, in Dia. 20, to simplify, we may want to imagine that 1–4 are played. This leaves Black with three unconditional liberties (unmarked), and then White will either need to play ‘a’, ‘b’, ‘c’, and then ‘d’ to get the benefit of the ‘d’ liberty; or else White will play ‘d’ and ‘c’ directly, but not benefit from ‘d’ himself.
Contrasting Dias. 19 and 20 against each other, we see that Black has one more ‘conditional liberty’, and to offset that White is basically forced to try to play ‘d’ in Dia. 20; but after this, the liberties are 4-4 and it’s Black’s turn, so Black will win the race.
This all seems to suggest that, in Dia. 15, Black has already won the capturing race; but this is not quite the case. Enter a white capturing race tesuji.
If White gets to continue from Dia. 17, he can play 1–10 as normal in Dia. 21, and then stake everything on a kō fight with 11 and 13. Although White takes a slightly bigger risk like this, 11 and 13 allow him to omit one move in the race.
Having noticed White’s tesuji, it is therefore correct for Black to fill White’s liberties as in Dia. 22. Up to 9, Black wins the race with at least one liberty because White is unable to play at ‘a’.
Going back to Dia. 15, what if it was White’s turn? Can White then capture Black? White might for example consider 1–3 in Dia. 23, although this is arguably not White’s optimal starting sequence (more on this later).
For Black, it anyway looks like there is no loss to be made with the 4–9 forcing sequence in Dia. 24, so we can probably imagine that it gets played immediately.
This time, when Black eventually turns at 10 in Dia. 25, we can see that White’s forming an eye with 11 is enough. After 11, White has four liberties, and Black’s group to the right only has three liberties because it does not benefit from the mutual liberty.
If White does not form an eye as in Dia. 26, and fills liberties normally with 11 on, we notice that eventually he again needs to go for the kō with 17 and 19. White will not get captured outright, but this is of course much worse than the clean capture in Dia. 25.
Optimally, White should actually start with the connection of 1 in Dia. 27. This did not make a difference earlier, as White anyway won the race in Dia. 25; but, if for example Black had an extra liberty, like at ‘a’ here, 1 in Dia. 27 would still allow White to win the race. This is because Black’s forcing sequence of 4–9 in Dia. 24 actually removes a White liberty for free, and white 1 in Dia. 27 avoids this.
Dia. 28 shows the rest of the sequence, with White’s having an eye again making the key difference. Finally White wins the race by one move.
As we can see from this real-game problem, there are plenty more techniques involved in capturing races than is possible to cover in a single post. This time the ‘conditional liberties’ in Dias. 19 and 20 turned out to make a big difference, as did White’s kō setup tesuji in Dia. 21; and the connection of 1 in Dia. 27 is far from an obvious move, too, although maybe conceivable if you think in terms of reverse sente (as in the endgame concept, just extended to liberties in capturing races).
Besides the above, the topic can also be expanded for example with larger eyes, which get priority over smaller eyed-groups, and one-sided double kō fights. More of these may get featured in future posts on the topic.
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