# Two Squares

**Two Squares (Life in small groups)**

# Preface

KjeldPetersen: With this theory I would like to start a discussion if it is possible to view life and death in go in a different way. My attempt here is a semi-graph-theory approach, since I do not know better. I hope that this approach will be received positively and that someone with higher math skills could take over and describe it more accurately. Maybe this will help you; maybe you will find it rubbish. At least, please keep an open mind, and participate with constructive criticism. I will continue to update the article now and then, so I encourage you to come back in 6 month to read it again, and see if I have added new stuff.

What I want to do is try to see if it is possible to identify properties of different situations using the two square view. It may turn out to be just as difficult as any other method. I will start with examples that are almost filled, missing just one stone to complete the two squares. But I hope to extend the examples to including how to select moves in sparse positions so as to better ensure the formation of two squares, and hence life for your groups.

For a theory to be useful it must be give correct results when applied. So I challenge you to find counter examples for which this theory incorrectly determines the status of a group.

Note: Please put discussion in the /Discussion subpage or the forum.

I hope that I, with my Two Square semi-graph-theory, can contribute with some knowledge that let people understanding some of the fundamental principles in the game, that otherwise could stay hidden for long time. And maybe you will start to think in squares too after reading this article.

I played Go for very long time, before I started to understand the concept of good shape. Two Squares is an idea that I have, on how to describe principles of good shape.

# Introduction

Learning the game of go is hard. You have to spend hundreds of hours studying and playing games to slowly bit by bit to understand more and more about the game. Learn how patterns form, which patterns are strong and which to avoid completely.

Today there are many sources you can go to to gain knowledge. There are several online game servers, and a ton of books you can buy. There are go clubs in nearly all major city in the world. And maybe also on every habitable planet in the universe where intelligent life has evolved. And of course now there is also AI.

Go is a game of patterns. The alternating nature of the game and the constant battle for territory supremacy creates patterns that keep reappearing. Some patterns occur more frequently than others. Some patterns are more effective than others. Some patterns create life. Some patterns prevent life. Some patterns are counterproductive. In this article I will show one possible view on patterns for stones to survive on the board. The patterns that I will concentrate on are the square frame and rectangle frame as shown below.

Definition: A Square consist of 4 side positions and 4 elbow positions that are placed around an empty center position.

Definition: Rectangles are like squares except they have two more side positions and one additional empty center position. Another possible definition of a rectangle is that it actually is two overlapping squares, but since they have 3 positions in common, both squares link together and act only as one large one figure.

I will start explaining things with squares only. But later I will extend it to include rectangles also. The focus in the examples are on how squares and rectangles are formed and prevented. So in the examples below you should mostly focus on the square and rectangle parts of the boards. Not on the whole board. Not all examples are set up as actual game battle situations. At the start of the article very few are, but later I will show some actual game situation where the Two Squares graph theory can be applied. This article focuses on a theoretical approach about squares and rectangle shapes on the Go board.

# References

Here is a list of references related to this article:

- /Discussion
- the forum
- Bamboo joint
- Center of 3 Stones
- Crane's Nest Tesuji
- CarpentersSquare
- Double Keima
- Elephant Jump
- Empty Triangle
- Eye-Stealing Tesuji
- GeneralGraphGo
- Knights Move
- Ko
- Mouth Shape
- Ponnuki
- Sake Bottle Shape
- Seki
- StrangeThingsHappenAtTheOneTwoPoint
- Suicide
- Square Good Shape
- TwoHeadedDragon
- Table Shape
- Tiger Mouth
- One Point Jump
- Peep
- Peep at a Tiger's Mouth
- Oiotoshi
- Squeeze tesuji
- Net
- RabbittySix
- Snapback
- Horse Head
- Two Space Extension
- Large Knights Move
- Tippy Table
- Big Bulge
- Parallelogram
- TortoiseShell
- TrapeziumShape
- JGroup
- LGroup
- Damezumari
- Two Square Game analysis
- My other work about implementing rules with BinMatrix's
- Benson's Definition Of Unconditional Life
- Chris Bandy's Definition of Immortality

External links:

- The Go polynomials of a graph
- Learning on Graphs in the Game of Go
- Thomas Wolf's Seki with 2 Liberties per Chain in the game of Go paper and The Mathematics of Seki presentation

# Rules

By studying the rule it is possible to argue that certain constellations will emerge naturally. I will try to argue here that TwoSquares are the forms that will emerge.

## The board as a graph

The Go board can be represented as a graph of nodes and edges. Where each intersection of lines is a node on the graph, and the lines are edges on the graph. Placing a stone on an intersection means colouring that particular node. The graph is by default not coloured on any nodes (empty board).

## Stones and liberties

By colouring a node (placing a stone on the board) the graph is changes, and some properties emerges. The adjacent nodes to a coloured nodes are called liberties, as long as they are not coloured them self. Colouring adjacent nodes with the same colour (placing two black or two white stones next to each other) means that a group is formed. And the adjacent non-coloured nodes to a group is called the groups liberties. Coloured nodes that are encapsulated by other coloured nodes of the same colour is still part of the group. And as long as the group in total has non-coloured adjacent nodes (liberties) the group may exists on the graph.

## Capture

A a single coloured node or a group of single-coloured nodes without any adjacent non-coloured nodes, are not allow to exists on the graph (stones without any liberties are captured and removed).

## Illegal move

Colouring a node, either as a single node, or as a group, such that it has no adjacent non-coloured nodes is illegal. (Suicide is not allowed. NB: Some rules does allow it, but that is a minor detail.)

## Capture or illegal

Now said that coloured nodes can be captured, but also some non-coloured nodes can be illegal to colour, there are situations where both things happens simultaneous, and it has to be defined which has precedence. And the rules of Go does describe these situations. Capture always takes priority over checking if colouring a node is illegal.

## Repetitions

The capture over illegal-colouring creates a situation that has to be resolved. In the rules of Go this is described as a Ko. Ko is a situation where an endless repetition of recapturing could occur, but rules clear describe that this is now allowed, by saying that an immediately recapturing is not allowed.

## Simple protection (Elbows)

Nodes that are illegal to colour can be utilized for protection, but some measures must be taken for it to work correctly. Just setting up a single illegal position, like the Ponnuki below, is not sufficient for protection, due to the Ko property.

Some measures can be installed to prevent single coloured nodes from being captured in Ko. By connecting the nodes into a group over the angling connections (elbows). Now only the whole group can be captured as one. By having a group surrounding a non-coloured nodes, a momentary illegal position is part of the constellation.

However simple protection by connecting nodes with elbow nodes is not enough, as the group can be captured, as long as the momentary illegal node is taken last.

## Mutual protection

So a more complex measure is needed to fully protect a group. We need a mutual protection. And this is where the TwoSquares emerge.

A different type of mutual protection that can occur is Seki. This also relies on two points where neither player is interested in playing. But that is not because the points are illegal, but because who ever plays first loses his own stones.

# Life (point giving)

Life (point-giving life – we will have a look at non-point-giving life, i.e. seki, later) exists when 2 squares are played out so they have some kind of connection between them. The 2 connected squares below mutually ensure existence (life) by preventing the opponent from occupying any of the internal liberties. If the opponent were to play on any of the free position in the center of any square then it would be illegal (in most rule sets) or captured (in a few rule sets). In both cases it can’t remain on the board to occupy a liberty.

This mutually assurance of life among two squares is the reason why I have named this article Two Squares

## Closer distance

But squares doesn’t need to be so far apart that they need a string of stones to connect them. Placing the squares adjacent or to some degree overlapping is more efficient, as it requires fewer stones. Placing the squares on the side of the board even requires even fewer stones. But I will come back to that. Below are adjacent and overlapping placed squares.

Of course it goes without saying that if Two Squares overlap by too many positions (6) then the Two Squares form a rectangle.

## Omitting Elbow Connections

Squares don’t need to be fully formed with all eight stones at all times to be valid. Some of the corner stones can be omitted, and still ensure integrity for life. The side stones are essential, and cannot be omitted. And another thing that cannot be omitted, or rather should be omitted, is the center liberty, “the eye in the square”. This must remain unfilled at all times. The elbow connections however can be omitted to some extent, but the conditions for omitting them depends on the number of side stones that have separate connections to other squares. For a square with one connection over to some other square, one elbow connection can be omitted. Any one of the elbow connections can be selected, but only one.

A square with two connections to two other squares or two connections to just one other square can omit two of the elbows. But it must follow a certain structure. Only one single elbow between each pair of connections either way around the square can be omitted. So both clockwise and counter-clockwise between the two connections one single elbow can be omitted.

With three connections from one square to some other squares (one, two or three—it doesn’t matter), three elbows can be omitted. But still only one single elbow between each pair of connections can be omitted. In the diagram below, the elbow positions at the top of the square cannot both be omitted, since both elbows are between the same pair of connections.

And, of course, with four connections from a square to some other squares (one, two, three or four—it doesn’t matter), all four elbows can be omitted.

As you can see, the number of elbow that can be omitted is equal to the number of connections to other squares. It doesn’t matter if two, three or four connections run over to the same other square. The question is just how many distinct connections there are to some other square or squares.

## Double-edged sword

The thing about omitting corner stones is, however, a double-edged sword. A player may omit some corner stones when establishing squares and connections between them to save stones for other purposes, but the opponent can also occupy the corner positions of nearly formed squares, thereby forcing one to unwillingly establish more connections to other squares. More connections mean more stones spent on non-point-giving patterns.

One of the most obvious places where this happens is with the Eye-Stealing Tesuji. Black may still be able to create a square with the two corner stones occupied by the opponent, but the square must have two connections to one or two other squares in order to become a valid point-giving square.

## Overlapping with three positions

If you have two squares overlapping with three positions, then you may omit one shared corner stone or two unshared corner stones since there is only one connection () between the two squares. The shared corner-stone omission is acting as a single corner-stone omission for both squares.

## The center of a square

You may save some stones when forming two squares by overlapping them to some extent. (Overlapping two squares completely makes no sense if the group is to live.) In one case you can overlap the two squares by four positions and thereby make one square's center position (eye) a corner position of the other square and vice versa.

NB: That the center position of a square must remain empty takes priority over occupying the corner position of the other square, since it is the empty center positions of the squares that in the end ensure life for all the stones.

But this means that one corner of each square must be unoccupied. The question is now how many connections there are between the two overlapping squares, and hence, how many additional corner stones can be omitted in each square. The answer in this case is that there are two connections between the two overlapping squares (square markings), since two side stones are shared among the two squares. So two corner stones in each square can be omitted. Since one corner stone in each square is already omitted by the center position of the other square, only one additional stone can be omitted.

## The edge has minus-one elbow position

The edge of the board (the first line) has some special properties in regard to counting connections and omitting elbow positions. Squares with an unfilled centre position (eye) placed on the first line do not need the missing side and elbow positions of the square to form a valid square. But, as regards to the number of elbow positions outside the board, one elbow position should be considered already omitted. That means that two connections are needed in order to omit another elbow position on the board, and three connections are needed to omit another two elbow positions on the board.

The minus-one property of the edge can be determined by the fact, that 3 positions are removed by the edge, one side position and two elbow positions. The difference between the number of side-positions and elbow-positions is hence minus one. Another way to visualize this is by counting the remaining positions on the board. There are 3 side positions () but only two elbow positions (). Again the difference is minus one. The same applies for the corner, where there is a difference of minus one between the count of side-positions and the elbow-position.

All the (a) squares (in the diagram below) have only one single connection and can therefore only omit one elbow position. But the (a) square at the edge of the board has already one elbow position omitted by the property of the edge, and the square therefore cannot omit any of the elbow positions on the board.

The (b) squares all have two connections to other squares, so two elbow positions can be omitted. But since they are located on the edge of the board, one elbow position is already counted as being omitted, and only one additional elbow position can be omitted.

With the (c) square there are three connections, but, again, as it is located on the edge of the board, a maximum of two additional elbow positions can be omitted from this square.

## Two-Headed Dragon

A special position arises when you have two squares both with two connections running over to each other. Note: two or more squares of your opponent could be settled in between your two connections. Since both squares have two connections to some other squares, two corner stones can be omitted. This forms what is believed to be a Two-Headed Dragon, or a living shape with two false eyes.

I would like to argue here that talking about false eyes is not entirely correct. The state of the eye depends on the number of connections and how corner stones are and can be omitted. If the eye still follows the principle as described above, then it is still a true eye, even if it has two, three or all four corner stones omitted. As long as it is in balance with connections, and only one corner stone between each pair of connections is omitted, then all is fine.

Two squares that overlap at four points, as described in the section on the center of a square, also have two squares with two connections to each other. So it could be argued that this also is a Two-Headed Dragon even though there are no opposing stones between the two connections. Or should we call it Two-Headed Dwarf Dragon?

On a very small 2-by-2 board, two stones that are placed diagonally also form two squares with two connections to each other. One corner stone is omitted from each square by the center of the other square, and then another corner stone is omitted by the property of the edge of the board. This could then also be argued to be a very small Two-Headed Dragon (Two-Headed Baby Dwarf Dragon?).

## 1-2 Points

It is also very common that the 1-2 points are used for forming an eye in the corner. An eye is almost secured with just two stones, and a second eye is also well underway at the 2-2 point.

The 1-2 points are crucial in a lot of life-and-dead problems. And there is a saying that "Strange things happen at the one two point".

So why are the 1-2 points so important? A possible explanation could be found in the next diagram. If you split the dead position into 3-by-3 squares (and add some extra stones), then you can see that the white stone is places as a side stone in both squares. If you now compare this with the 1-2 points, then you will recognize that it is directly on the 1-2 point.

## Partially Formed Squares

The square structure can be recognized in many formations that occur in actual play. Below are some constellations of stones that are partially formed squares and occur frequently in games.

From left to right: Ponnuki, Table Shape, Bamboo joint, Tiger Mouth, One Point Jump, Knights Move, Center of 3 Stones, Sake Bottle Shape, Mouth Shape, Elephant Jump, Double Keima, Square Good Shape, and Empty Triangle.

Of all the different partially formed squares, the Empty Triangle stands out, since it is heavy towards a single corner. All the other partially formed squares do not have all of their stones fully connected as the Empty Triangle does. This makes the Empty Triangle less flexible when reusing the stones for other partially formed squares, and makes the position vulnerable for attack.

# Preventing Life (squares)

Preventing the opponent from forming life in certain parts of the board is easy: just occupy the space there yourself. That being said, you could also utilize the fact that missing corner stones in squares forces those squares to have more connections in order to live. And playing this way is often the case in the game of Go.

A single stone on the side of a square simply prevents it from becoming a life-giving square. It may happen that a large form is built around it, but for now that is outside the scope of this section.

If the opponent cannot occupy side positions, then maybe two or more corner positions can be occupied, thereby forcing more connections from the square. Each section of a square that is placed between two corners occupied by the opponent must be joined, and these sections need to connect to an external square in order to ensure integrity for this square.

The prevention methods and forcing-more-connections method can be recognized in various situations. In particular the Center of 3 Stones is very nice, because it affects two squares with one move and is essentially a double Eye stealing tesuji move.

Center of 3 Stones is a double Eye stealing tesuji, forcing two connections to other squares

Related concepts of preventing squares:

## Carpenter's Square

The vital point in the basic Carpenters Square is the corner of the square that is about to be formed with the five stones.

## Three overlapping positions

The opponent prevents two squares from being formed in this situation by occupying the overlapping side positions of both squares. Since the squares have no possibility of connecting to other remotely positioned squares, they cannot ensure integrity for their own life; hence, they are dead. Playing on the position yourself would of course ensure integrity for life with two fully formed, singly connected squares. We could actually omit some corner stones also, but for clarity I have left that out.

NB: I have added the detailed effect on each square in the lower section of the diagram below. A single white stone is enough to prevent Black from completing his two squares.

A well known life and death problem with three overlapping positions in the corner has the same solution as above.

This can also be recognized in a familiar life and death problem at the side of the board. Whoever plays first on the marked position ensures the corner.

## Four overlapping positions

The next example has four overlapping positions. A single opposing stone on one of the intersecting side stones (the second is marked ), and your two squares cannot be formed. The preventing method used for the situation is placed to the left and below the construction with the two overlapping squares. The stones marked with are the ones that are omitted by the empty center of the other square take priority over being filled.

This problem also arises in the corner, when the opponent plays on the 1-2 point (left) or at the 1-1 point (right), and prevents the two squares from being formed.

## Just one overlapping position

In the situation with just one overlapping position there is not much to say about how to prevent one of the squares. Either occupy a side position or occupy two corner stones. (But the one in the middle)

But there is something else to notice about this position. Both squares have two connections to the other square. From two side stones (closest to the middle) it is possible to run over to the other square. So it should be possible to omit two corner stones for each square. But only one corner stone between each pair of connections. So one of the stones must be the center stone, and the other can freely be selected among the other three corner stones in each square.

When the center stone is removed it also becomes a point, but if you look at it, it has no corner stones. So it must have four connections (by the definition above). And indeed it has. It has two connections to the upper square and two connections to the lower square, since all four side stones of the new square in the middle have shared side stones with other squares.

## Two overlapping positions

Corner stones that also are center positions of squares are obliged to be left out in order to ensure integrity. The next example has the opposite property. Here corner stones that also are side stones may not be omitted.

With two overlapping positions there are two connections between the two squares, since one square’s side stone is one of the other square’s corner stones. But placing side stones takes priority over omitting corner stones. So for each square only one corner stone can be omitted.

If a side stone is omitted anyway then we are moving over to rectangles instead, and that will come later.

## Recapitulate Two squares

For two connected squares some corners can be omitted depending on the number of connections, and if a corner stone is a center position at the same time, then they are forced to be omitted, or if a corner stone is a side stone at the same time, then they are prohibited from being omitted.

## Three and four squares

Three and four squares can also share one side position, and if the opponent get to play there first then all squares are prevented from being completed.

## Rabbitty Six

The Rabbitty Six is also a know position in Go. If White goes first then he playes on the center point with 1.

## Force more connections - kill

Playing in a way to force more connections will also kill a group if there is no way to make the necessary connections. The forcing method is shown on the right in the diagram

This way of killing can also be found in a life and death problem on the side of the board. Remember that the (a) square needs two connections after White plays the marked positions, because the edge counts as one corner stone has unwillingly been omitted, and it now require two connections to other squares to become a valid square (by the definition above)

If Black could make the connection to another square then he would survive. (N.B. Now you would probably argue that it would be better for White to play at 1. And, yes, that would be better in this diagram, but I am leaving it as it is for the clarity of the two square theory)

## The diagonal wand

Many life and dead problems includes, what I would call it, the diagonal wand. The first example that springs into mind is Snapback. There are two types of Snapback. One with diagonal corners taken, and the other type with sideways taken corner stones. Here I will concentrate on the diagonal taken corner stones.

In this situation the diagonal wand is still completely open, and the question is where to play first in order to make life for the black group? You properly quickly recognize, that if you play anything else than the middle point, then White will take the middle point and win.

So the solution is to throw a stone in at . White tries to capture it with and , but has to abandon the three stones, or more stones will be captured with Black on the marked spot

This situation came up in a game, and black made the mistake of playing hane and connected and then played elsewhere.

Black have to connect, and White played hane. Black created an eye on the side. White decented. Black had to protect the eye, and now White has the opportunity to play the second magic wand as a Ko.

One of the situations that springs into mind here is the Crane's Nest Tesuji, the sideways positioned corners is a vital part of the solution.

However first you must occupy the central point (1) (The eye if you will), and subsequently place stones on the corner positions (3) and (5).

## Partially Created Rectangles

Horse Head, Two Space Extension, Large Knights Move, Tippy Table, Big Bulge, Parallelogram, TortoiseShell, TrapeziumShape, JGroup, Two space jump in front of tiger mouth, 1 & 2 space extension

## Preventing rectangles

Preventing rectangles requires 2 or more stones. Looking closely, you will see that there are some similarities to how squares are prevented (Shown below the rectangles). If you simply cover one or the other 3 stones end of the rectangle shape you may recognise how squares are prevented also.

## Four spaces in the corner

Black’s four-space territory is killed by White with two stones, as in the diagram below. Both ends of the Black formation form a rectangle, and the White daggert simply kills the Black formation by using the same daggert for both rectangles.

## End Game

# Example of use in Life & Dead

## The L-group

One of the solutions Black would like to play in the L-group is to make a square and a rectangle with the two 1-2 & 2-1 marked positions . So if Black plays one of them then White should prevent Black’s square in the corner by playing the other marked position.

If Black plays 1, then White prevent the square in the corner with 2. I have added the preventive method. The marked black stone are the ones outside the board. White 2 also prepares to make a dagger into the rectangle with the marked white stone.

If Black continues like this with 3, then White drives a dagger into the heart of the rectangle with 4. I have added the preventive method of forming the rectangle to the right on the board.

Another solution that Black can aim for is to make the rectangle in corner and the square in the bend of the L. The solution for white is again the play the other circle-point (), that black attempts to use for making his group alive.

If Black plays first in the J-group he can quickly create multiple possibilities for two squares with a stone on any one of the circles. So the question is if Black can survive when White plays first.

White bends at , and Black tries to make two squares with his stone at , but White prevent this with .

Black then tries to make two rectangles with @(3,1) and @(2,2), White and prevent him from completing, because Black has to put his stones in self atari.

Another possibility is to create a large rectangle and a square in this way with and , but and will again prevent this.

## Other example 1

In this problem, White's stones are encircled by Black's rim of stones. And whether White live or dies depends on who plays first.

White can create an rectangle shape with 1. The second eye can be ensured by playing right or left of the lower single white stone. So that is miai.

However if Black plays first, then he can first play atari by throwing in at 1, and start forming the shown zig-zag preventing method. When White takes the stone with 2, then Black play atari from outside with 3. The square-preventing shape is shown in the second diagram below.

## Other example 2

In this next example we can see how black has managed to prevent white from making a living group in the corner, by placing a dagger into the rectangle in the corner. So white must try to find another way of creating Two Squares.

White need to take something away from black, so he can connect all his stones. The square marked here is still missing a side stone, and two corner stones. So maybe there are some possibilities here. Taking the side position will make black connect at (a), and then there is no way to capture anything.

So this corner position is the correct one. But what comes next? a, b or c? A simple atari at (a) will only make black connect. With (b) black connects again and survives.

So that leave us with the only option of playing a throw-in atari here. Black cant extend down to the edge, because that will not give him more liberties, and white can capture. So black can only capture, but then...

# Try it your self

Most problems use these ways (rotated and possibly reaching beyond the board) to prevent squares and rectangles from being formed.

Prevent this square by using one of the preventive methods. Play somewhere one the rim of the square :-)

## Problem 8

Make sure that Black can form a square and a rectangle in the corner. So where is the crusial common spot between the two diagrams?

## Problem 9

White could make a square and rectangle. Reduce White so he only can make Two Squares, then kill the group. Or reverse the moves

## Problem 10

Create your first square wisely, then White can't connect. And you can sit over for 2 moves to create your second square.

## Problem 11

# Non-point-giving Life (Seki)

Sekis can be constructed in different ways

- A Black string and a White string of stones that have as their only liberties a shared rectangle
- A Black string and a White string of stones that have as their only liberties 2 shared squares
- A Black string and a White string of stones that have as their only liberties 1 shared square, but also 1 independent square each
- Multiple Black and/or White strings that share 2 squares, but with different opponent strings
- Multiple Black and/or White strings that share 2 squares, but with different opponent strings, that terminate with 2 independent squares.
- Seki with stones in Atari
- (At least 2 more ways, that I will find examples for later)

# Work in progress

I'm working on more sections in TwoSquares/Sandbox

# See also:

- Two Square Game analysis
- My other work about implementing rules with BinMatrix's
- Benson's Definition Of Unconditional Life
- Chris Bandy's Definition of Immortality