Two Squares (Life in small groups)
KjeldPetersen: With this theory I would like to start a discussion if it is possible to view the life and dead in the game go in a different way. My attempt here is a semi-set-theory-math approach, since I do not know better. I hope that this theory will be received positively and that someone with higher math skills could take over and describe it more accurately. Maybe this theory will help you; maybe you will find it rubbish. At least, please keep an open mind, and participate with constructive criticism.
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.
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 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 plannet in the universe where intelligent life has evolved.
Go is a game of patterns. The alternating nature of the game and the constant battle for territory surpremecy creates patterns that keep reappearing. Some patterns occures more frequent than others. Some patterns are more effective than others. Some patterns creates life. Some patterns prevents 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: Squares consists of 4 side stones and 4 corner stones, that are placed around an empty center position.
Defenition: Rectangles are like squares except they have two more side stones and one additional empty center position.
I will start explaining things with squares only. But later I will extend it to include rectangles also.
Life (Point giving life. We will have a look at none-point giving life Seki later.) exists when 2 squares are played out so they have some kind of connection between them. The 2 connected squares below mutually ensures exsistence (life) by preventing the opponent from occupying all liberties. If the opponent would 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 some few rule sets). In both cases it cann’t remain on the board to occupy a liberty.
But squares doesn’t need to be so distant positioned so 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 adjecent and overlapping placed squares.
Squares doesn’t need to be fully formed with all eight stones at all time to be valid. Some of the corner stones can be omitted, and still ensure integrity for life. The side stones are essential, and can not 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 time. The corner stones however can be omitted to some extend, but the condition 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 corner stones can be omitted. Any one of the corner stones can be selected, but only one.
A square, that has two connections to two other squares or two connections to just one other square, can omit two of the corner stones. But it must follow a certain structure. Only one single corner stone between each pair of connections either way around the square can be omitted. So both clockwise and counterclockwise between the two connections one single stone can be omitted.
With three connections from one square to some other squares (one, two or three, it doesn’t matter) three corner stones can be omitted. But still only one single corner stone between each pair of connections can be omitted.
An of cause with four connections from a square to some other squares (one, two, three or four, it doesn’t matter) all four corner stones can be omitted.
As you can see the number of corner stones that can be omitted is equal to the number of connections to other squares. It doesn’t matter if two, three or four connections runs over to the same other square. The question is just how many distinct connections to some other squares are running out of a square.
The thing about omitting corner stones is however a double-edged sword. A player may omit some corner stones, when he establish squares and connections between them, to save stones for other purposes, but an opponent can also occupy the corner positions of nearly formed square patterns and thereby forcing a player to unvillingly have to establish more connections between squares. More connections means more stones spend on none-point giving patterns.
You may save some stones when forming two squares by overlapping them to some extend. (Overlapping two squares completely makes no sense.) In one case you can overlap the two squares by four positions, and hereby make one squares center position (eye) a corner position of the other square. And vica 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 ensures life for all the stones.
But this means that one corner stones of each squares can not be occupied. 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 is this case is that there is two connections between the two overlapping squares, so two corner stones in each square can be omitted. Since one corner stone in each square is allready omitted by the center positions of the other square, only one addition stone can be omitted.
The edge of the board (first line) has some special properties in regards to counting connections and omitting corner stones. Squares with the unfilled centered position (eye) placed on the first line, does not need the missing side and corner stones of the square to form a valid square. But in regards to the number of corner stones outside the board one should be perceived as already omitted. That means that two connections are needed in order to omit yet another corner stone, and three connections are needed to omit two corner stones.
All the (a) squares have only one single connection, and therefor can only omit one corner stone. But the (a) squares placed on the edge of the board has already one corner stone omitted by the property of the edge, and therefor cannot omit any of the corner stones on the board.
The (b) squares all have two connections to other squares, so two corner stones can be omitted. But since they are located on the edge of the board, one corner stone is already counted as being omitted, and only one additional stone 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 corner stones can be omitted from this square.
A particular position arises when you have two squares both with two connections over to each other. Note: Two or more opponent squares 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.
Two squares that overlaps with four position as described in the section with the center of a square, also had two squares with two connections to each other. So it could be argued that this also is a Two-Headed Dragon. Except that there is no opponent stones in between the two connections.
On a very small 2-by-2 board two stones that are placed diagonally also forms two squares with two connections to each other. For both squares one corner stones is omitted by the center of the other square, and the 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.
The square structure can be recognized in many formations that occur in actual play.
Preventing that the opponent forms life in cirtain parts of the board is easy. Just occupy the space there your self. That being said, you could also utilize the fact that missing corner stones in squares forces the necessity of squares to have more connections over to other squares in order to have integrity for forming life. 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 becomming a life giving square. It may happen that a large form is build around it, but that is for now outside the scope of this section.
If the opponent cannot occupy side positions, then maybe two or more corner stones can be occupied, and thereby forcing that more connections from this squares must be constructed. Each section of a square that is placed between two corner that is occupied by the opponent, must be joint as one, and they 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
The opponent is preventing two squares from being formed in this situation, by occupying the overlapping side positions of both squares. Since the squares has no possibilities to connect to other remotely positioned squares, they cannot ensure integrity for life them selfs. Hence dead. Playing on the position your self whould of course ensure integrity for life with two fully formed single connected squares. We could actually omit some corner stones also, but for clearity I have left that out here.
NB: I have added the method for preventing the squares in the lower section of the diagram below. A single white stone is sufficient for preventing black from completing his two squares.
A well known life and dead problem with three overlapping positions in the corner has the same solution as above.
This can also be recognized in a know life and dead problem at the side of the board. Who ever plays first on the market position ensures the corner.
The next example is with four overlapping positions. A single opponent stone on one of the intersecting side stones (The second is marked with a circle) and the 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 X are the ones that are omitted by the empty center of the other square take priority over being filled.
This problem is also arise in the corner, where the opponent place his stone on the 1-2 point, and prevents the two squares from being formed.
In the situation with just one overlapping position there is not must to say about how to prevent one of teh 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 has two connections to the other square. From two side stones (closest to the middle) it is posible to run over to the other square. So it should be able 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.
Corner stones that also are center positions of squares are obligated to be left out in order to ensure integrity. The next example has the opposit property. Here corner stones that also are side stone can not be omitted.
With two overlapping positions there are two connections between the two squares, since one squares side stone is one of the other squares corner stones. But that side stones are placed 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.
For two connected squares some corners can be omitted depending on the number of connections, and if a corner stone is a center positions at the same time, then they are forced to be omitted, or if a corner stone is a side stones at the same time, then they are prohibted from being omitted.
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.
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 dead problem on the side of the board. Remember that the (a) square needs two connections after white playes 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. (NB: Now you would properbly 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)
White's stones, are in this problem No. 1, cased in by Black's rim of stones. And the outcome of if 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 the when White takes the stone play atari from outside with 3. The square preventing shape is shown in the second diagram below.
Black to play in every diagram
Most problems uses these ways (rotated and possibly reaching beyond the board) to prevent squares from being formed.