Kō fights are a most interesting phenomenon in go. Go is already special in that the simple stone-capturing rules lead to the emergent property of two-eyed groups being uncapturable, and this property in turn greatly affects go gameplay. The rule prohibiting the repetition of an earlier board position, i.e., the kō rule, carries a similar weight, as it allows for a myriad of different strategies and tactics relying on kō fights.
What is perhaps even more interesting is that there are several variants of the kō fight with their own characteristics – approach kō fights, two-step kō fights, and the ten-thousand-year kō, to name a few. This post concentrates on the last of the mentioned variants.
The ten-thousand-year kō is called ‘mannen kō’ in Japanese, of which the English term is a direct translation. In Japanese, the word ‘man’ means ‘ten thousand’, and it is commonly used in the language when one wants to emphasise ‘a lot’. For example, ‘I’ve told you a thousand times’ might become ‘ten thousand times’ in Japanese, as the words for thousand (sen) and ten thousand are similarly easy to say. In fact, the Japanese counting system also works in powers of four, not three, which can confuse westerners – for example 100,000 becomes 10,0000, or ‘ten man’.
In other words, it makes sense to translate the 10,000 in ‘ten-thousand-year kō’ simply to ‘thousand-year-kō’ in English, as it rolls off the tongue more easily and carries the same nuance of a really really long time.
What does a mannen kō, or a ‘thousand-year extremely-long-time’ shape look like? See below.
Dia. 1 shows a tsumego shape that most dan players know by heart. In fact, the reading involved is so complex but the bottom line useful enough that probably all go players who know this shape have simply memorised it.
Dia. 2 shows the solution. Black’s attachment of 1 is the correct first move, and white 2 in response works the hardest at building eye shape for White.
Instead of 6, White could also consider playing at ‘a’, but this would lead to a direct kō fight for White’s life – 6 instead leads into an iconic thousand-year-kō shape, which is better for the defender.
After white 8, we have a thousand-year-kō – but what is really going on?
If Black plays elsewhere and White wants to play something locally, he can only start a direct kō fight with 1 Dia. 3. Black captures at 2 and, if Black wins the kō fight by getting to connect at ‘a’, the white group is dead.
Dia. 4. If it is Black’s turn, she captures at 1. Now there is a kind of kō fight going on, but if Black ends up connecting at ‘a’, the shape locally becomes seki – meaning that the White group is then unconditionally alive.
Instead of Dia. 4, Black might consider to first play 1 in Dia. 5. White still cannot do anything but start a direct kō, so he rather plays elsewhere; then, Black captures with 3. White again cannot do anything locally, so he plays elsewhere. Finally, then, Black can play at 5, after which White captures at ‘a’ with white 6 and we have a direct kō fight.
The important thing to realise is that this kō fight is a lot better for White than the one in Dia. 3: Black has had to play two more stones, and if White wins the kō fight, he also ends up with two more points of territory.
Mathematically, too, the kō fight in Dia. 5 is not a big deal. After white 6 at ‘a’, White is one stone away from having nine points of territory while Black is two stones away from having 18 points of territory. This means that the expected score is exactly zero points. If, instead of 5, Black simply accepted that the corner will end up in a seki, Black would eventually get to capture the white stone at ‘a’ and therefore the score would be one point for him – which is one (expected) point more than if Black starts the kō fight.
In an actual game, however, Black will often want to aim at the prospect of starting the kō fight shown in Dia. 5. If initiated in the endgame, and White doesn’t have a surplus of big kō threats, White can find it hard to play two nine-point moves while Black wins the kō. However, if we take into account that Black also had to prepare the thousand-year kō with black 1 and 3, that means that White just needs four 4.5-point moves to balance out the whole exchange.
Dia. 6 is where things start to get really interesting. This, too, is a problem that most strong players have learned by heart. What difference do you think the open liberty at ‘a’ can make?
Now, when Black initiates her attack in Dia. 7 with 1, White can resist with 2–4; 4 is a move that is almost impossible to conceive if one hasn’t seen it before.
Dia. 8. If Black tries to capture the stone White just played, she will try 1–3.
However, when White then recaptures with 4 in Dia. 9, Black has nothing more to do; the escape of black 5 is stopped by the atari of white 6, and now we see what value the liberty at ‘a’ has.
Therefore, going back to the situation after Dia. 7, it is correct for Black to create a seki with 1–6 like in Dia. 10. This is classically considered the correct solution.
The ai, however, complicates matters for us. KataGo, for example, is firmly of the opinion that in Dia. 6, it is best for White to go for a thousand-year-kō position instead of a seki – even though White is not strictly speaking alive in the former. Dia. 11 shows KataGo’s recommended sequence, and it judges this roughly two points better for White than the seki variation. Apparently, the forcing moves of black 3 and 5 in Dia. 10 are something that KataGo hates to give away for free.
I am particularly intrigued by KataGo’s preference in that, so far I haven’t met a single other professional player who agrees with it. My research, on the other hand, suggests that KataGo is not making a reading or counting mistake, but that it really does judge the position more accurately than humans.
Lastly, Dia. 12 shows another classic go problem that shows up in real games from time to time.
Dia. 13 shows the classic solution: up to 6, White is alive in seki. Again, KataGo thinks this is wrong.
According to KataGo, White should exchange 2–3 in Dia. 14 and then switch elsewhere. We now have a strange thousand-year-kō brewing up.
Dia. 15. Like in the normal thousand-year-kō, White has no incentive to continue playing in this position. Black, therefore, will later play 1 and 3, starting a kō fight – only, this is not a standard kō but a two-step kō (i.e., after white 4, either player needs two more moves to finish the kō in their favour).
Thanks to KataGo’s insight, it seems we can coin a new type of a kō: ‘thousand-year two-step kō’.
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