In the Go game, you can see more shades of gray (continued No. 2 - Mirror, No. 3 - Holes)

    Mirror


    In a previous article, I talked about the interaction of stones with a wall. Now I will show how it looks in theory.
    I will cite four examples in turn. As a wall (1st line), I use the middle of the 13x13 board to draw a reflection behind it.



    1) In the first figure, there is one stone. Its corridors of influence see a mirror (1st line). This is a discrete mirror. How does it work? Look, on the wall I marked the points of the mirror with the icons of the cross, square and circle, and on the other side of the mirror I noted the points that the black stone in front of the discrete mirror sees. The peculiarity is that only those points are visible that lie exactly in the direction of the vector from the stone to the point on the mirror. Thus, the stone through point "A" sees its reflection.When this condition is met, the influence of the stone interacts with the reflected stone at their intersections. For example, in paragraph “A” there are four directions of influence at once, and in the remaining paragraphs (as you can check), the interaction of influences gives a total of two.


    2) In the second figure, two stones are located on the 4th line. Two stones are interconnected without breaks; we will call such formations a monad (something whole, indivisible). The effect of each monad stone is considered separately. A stone marked with a cross sees through its influence points on a mirror marked with crosses. We build vectors and mark those points on the other side of the mirror that this stone sees as crosses (not all points are marked, but only necessary for example). We will do the same with the second stone using the circle. It is seen that the reflected two stones do not fall into the visible zone. Consequently, two stones standing on the 4th line do not interact with reflection and their influence does not increase.


    3) In the third figure, three stones representing the monad are located on the 4th line. Repeating the analysis of the visibility of points, we find that the stones marked with a cross and a circle see in the reflection (through point “B” on the mirror) the corresponding reflected stones marked with a circle and a cross. Since these stones belong to one monad, the monad of three stones in 4 lines interacts with its reflection and enhances the influence.


    4) In the fourth figure, two stones that do not make up the monad are located on the 4th line. Despite the fact that they see pairwise reflections of each other through point “C”, the influence does not increase, since they do not belong to the same monad.

    A couple of examples of how a mirror strengthens a group of stones. A group I call stones that have holes in pairs (see the definition of a hole below).



    Hole


    We define such a concept as a “hole”. Usually, “players in Go” (goists, goshniks, ... go ... - how affectionately for the Russian language to call a player in Go?) By a hole is understood as an opportunity to divide stones into two groups, speaking the language of theory, to prevent stones from forming a monad. Once again, we formalize the concept, and then examine the holes for methods of attacking them by the enemy (in the next article).
    First, imagine a gobana in the form of a graph, where points will play the role of vertices, and the lines between them will be edges. When setting a stone, the peak acquires the corresponding state (for now, it will just be a stone at the top). The length of the edges between the vertices is zero. The length of the path is determined by the number of intermediate peaks not occupied by stones.
    Definition:A hole will be referred to as points between stones of the same color, located on the shortest path between them (the smallest possible number of vertices), if the following condition is satisfied for these points. On all possible paths between stones that go through the minimum number of vertices, on a vertex located in the middle of the path if the path contains an odd number of vertices, or on two middle vertices if the path contains an even number of vertices, you can put a stone of the same color and this the stone will see both initial stones as its corridors of influence.
    The figures below show all possible combinations of two stones where there is a hole.

    Legend:
    • A square is a peak at the shortest distance between the stones from which the set stone sees both stones as its influence.
    • Red color - a vertex located in the middle of the path if the path contains an odd number of vertices, or at two middle vertices if the path contains an even number of vertices.
    • The cross is the peak at the shortest distance between the stones from which the set stone does NOT see both stones as its influence.

    At stones:
    • 1 and 2 - at the shortest distance there is only one peak, it is the middle and both stones are visible from it. This is a hole.
    • 3 and 4 - at the shortest distance there are two peaks, they are the middle and both stones are visible from them. This is a hole.
    • 5 and 6 - at the shortest distance there are two peaks that can be selected in three ways, they are the middle and both stones are visible from them. This is a hole.
    • 7 and 8 - at the shortest distance there are three peaks that can be selected in four ways, the middle is marked in red and both stones are visible from it. This is a hole.
    • 9 and 10 - at the shortest distance there are three peaks, the middle is marked in red and both stones are visible from it. This is a hole.
    • 11 and 12 - at the shortest distance there are four peaks that can be selected in five ways, the middle is marked in red and both stones from the peaks marked with crosses are NOT visible from it. This is NOT a hole. (Later, consider this position on the edge of the board and find out what changes due to interaction with the edge.)
    • 13 and 14 - at the shortest distance there are four peaks, the middle is marked in red and both stones are NOT visible from it. This is NOT a hole. For example, triangles indicate the peaks of which both stones are visible, but they do not lie on the shortest path.
    • 15 and 16 - at the shortest distance there are five peaks that can be selected in six ways, the middle is marked in red and both stones are NOT visible from it. This is NOT a hole. Note that the peaks from which both stones are visible are not the middle.


    In the second figure I show what did not fit in the first.

    At stones:
    • 1 and 2 - at the shortest distance there is one peak that can be selected in two ways, the middle is marked in red and both stones are visible from it. This is a hole.
    • 3 and 4 - at the shortest distance there are three peaks that can be selected in five ways, the middle is marked in red and both stones are visible from it. This is a hole.
    • 5 and 6 - at the shortest distance there are four peaks that can be selected in seven ways, the middle is marked in red and both stones from the peaks marked with crosses are NOT visible from it. This is NOT a hole.
    • 7 and 8 - at the shortest distance there are five peaks that can be selected in many ways (count how many there are), the middle is marked in red and both stones from the peaks marked with crosses are NOT visible from it. This is NOT a hole.
    • 9 and 10 - at the shortest distance there are six peaks that can be selected in many (I haven’t counted myself yet) methods, the middle is marked in red and both stones from the peaks marked with crosses are NOT visible from it. This is NOT a hole.

    Further, obviously, as the distance between two stones increases, the hole will not appear.

    Later there will be an article on how defects of the “hole” type reduce the effect. Soon there will be an article on defects such as "short life."

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