## 'Sufficient but Not Necessary: Two Eyes and Seki in Go' printed from http://nrich.maths.org/

You may first like to read about the rules of go in behind the rules of Go or some of the strategies of games in Going first

This article examines a fundamental tactical principle in the game of Go, that of 'two eyes' for long-term survival of a group of stones. While 'life' - the capacity to fend off all assaults whether external or internal - is very closely connected to the capacity of a group to form what are called two eyes if challenged, these two ideas aren't logically equivalent. The loophole is a complication in the form of local stalemate, called seki in Japanese.

## Two eyes suffice for life.

The rules of the game of Go don't by any means tell you all you need to play the game successfully. That much one might expect from experience with any game with even a little hidden complexity. Worth special attention in Go is the existence of a basic tactical principle that on first sight could pass for a 'derived rule': a logical consequence of the primary rules.

Certainly without an understanding of the principle involved, called 'two eyes', it is essentially impossible to follow skilful play in Go. It is not however the case that the 'two eyes' requirement for the long-term viability of a group of stones can be deduced from the rules; and it would be wrong to present it in that way. A group having two eyes, or sure of being able to form two eyes, has an assured future. That is, two eyes are sufficient for life. But, as will be explained later, one cannot say that two eyes are necessary, because there is the chance of survival in seki (se-key), a kind of local stalemate. Therefore scrupulous statements about go are often complicated by the need to treat seki as an exception. This is the kind of price to be paid, quite generally, for the power and completeness of mathematical reasoning when it applies.

In the very simplest cases, it is easy to see what is implied by two eyes.

Here is a group of black stones that has two eyes: the two internal vacant points. White cannot capture it. The black stones form a single chain, with two liberties left even when White has played round the outside. White's plays on the remaning liberties would be suicidal, which makes them illegal moves as the rules are usually set up.

To make that claim watertight, I'll state more fomally than sometimes is done the procedure for updating the board after a play.

• Step 1: The newly-played stone is put on the board;
• Step 2: Check whether it performs any captures, i.e. reduces any enemy chains to zero liberties from one. If there are captures resulting from the play, remove all those stones from the board.
• Step 3: Only at this point check whether the play of the new stone leaves a legal position, that is, one in which each chain on the board has at least one liberty. If not, the play is rejected as illegal (suicidal).

This careful way of saying it lends itself to the writing of a computer program to update the board, and we can take that as meaning the process has been described with enough formality. A more relaxed and informative way of expressing the point here is 'a stone may only be played into a captured position if in so doing it completes the capture of some enemy stones'. Suicide is ruled out.

Placing a clack stone at Black 1 to capture five white stones is permitted.

With all that under our belt, it should be clear why the group with two eyes introduced above is safe. White cannot legally play inside it, and Black hasn't the slightest reason to (it is always legal to pass instead of playing in Go, by the way).

This example shows that safe formations can exist on the Go board. One point about them to make immediately is illustrated by this next case.

Here Black is again safe, with two chains. Black shouldn't connect them up to make one chain because a play inside would reduce the eyes and liberties to one as well; and White could capture immediately. Therefore the term 'group' is applied, to help distinguish a collection of chains that may be collectively secure, from a chain considered on its own.

Two eyes are at a premium, but the real question in practice is whether (and how) it is possible to make two eyes in a formation that is subject to attack. You can be sure that as the board fills up, each group will eventually find its scope for developing into vacant areas quite restricted. The question of whether it has two eyes will then come up in a logically deeper form: 'for each way my opponent may attack it, does that group have a way of forming two eyes?'

For example, here Black is already safe.

White has just these two significant attacks (in the other cases the invading stone may be captured at once). They both end up as dead-end positions for further progess, as may be checked using the update scheme: White has no more legal moves inside Black's group.

The situation as generally described may be identified as a typical challenge/response statement (life means every attack has an adequate reply that prevents ultimate capture). This makes survival logically similar to many definitions in mathematical analysis. If one compares go to another game of territory, Othello, which also depends importantly on safe formations, one can see greater logical depth in Go. The corners in Othello are immune from capture once occupied, as are certain convex blocks based in the corners. But all these cases are very easy to recognise directly from the capture rule.

Both from the point of view of playing go at all well, and to understand the exact scope of the two eyes concept, one has to investigate deeper into the idea of playing inside groups.

For example this play inside definitely denies Black the chance to form two eyes. It seems as if White has further work to do, to capture Black's stones. White will need two extra plays inside to threaten Black.

At this point Black must capture White's stones to avoid immediate capture.

Now we are back to the initial position, except that the marked black stone is there. White should play at the key point again.

With one more play by White, this position is reached. Now for the second time Black must capture, but it should have become clear that this leads nowhere. White can play back inside, Black takes once more but is down to a single liberty, White recaptures taking everything (what go players call a snapback).

Mathematically one can see what is going on as reminiscent of a quadratic progression. In fact there is an obvious arithmetic progression: N spaces inside means you play N-1 stones to threaten the group, those N-1 stones are captured leaving a space of size N-1, and you start again. That implies that the number of plays required actually to take off a vulnerable group is given by a quadratic polynomial in the number of internal spaces.

One wrinkle ought to be mentioned.

This is the right way to attack Black in this case with five spaces. White must take care, though.

This would be the wrong plan for filling in. When Black captures White's four stones, Black will be safe, as was seen before.

Therefore White should make some other choice. This way is good, and there is one more for you to look into.

So there is actually a systematic way of filling in, inside groups that are prevented from making two eyes, and leading to their final capture. That makes it tempting to say that two eyes are necessary for safety.

The joker in the pack is this type of position. What is going on? With this larger internal space neither player can now make progress.

White provoked the capture of the internal stones, Black captured with the marked stone. Now when White plays back inside at the point 1, the resulting space of seven points will still give Black two eyes. The attack with 1 and 3 is typical, but leads nowhere after Black 4. (If you are getting good at go you'll see Black could even ignore White 3 quite safely.)

So finally this is an example of the positions called in Japanese seki (impasse). Neither player will want to carry on playing inside. The two vacant points will not count as territory, according to the definition given in 'Behind the Rules'. The black and white groups are both safe, and presumably will remain so for the rest of the game.

## Conclusion.

The game of go has a simple mechanism, rooted in well-defined abstract ideas of connection (as was explained in 'Behind the Rules of Go'). It is tempting to believe that the play of the game will depend on equally clear-cut concepts. This discussion of the principle of 'two eyes' in go has shown that not to be the case, at least in too simple-minded a way. The principle isn't accurately described as a derived rule of go, as it would be if it were a possible theorem provable by treating the game's rules as axioms. If you play go, the two eyes factor in practice dominates tactics and strategy alike for survival of stones. There is however a gap in rigour between that statement, and the true facts about the game.