## Thursday, 7 September 2017

### STANDARD GEOMETRIC CHESS

Contents

1. Definitions of chess pieces and symbols
2. Cases of scoring
3. Some notes
4. Chess playing results
5. Statistics of basic layputs in the standard geometric chess
6. Prelimittary introdution of 12 beginning moves in Standard geometric chess
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1. Definitions of chess pieces and symbols

Each side has nine pieces and one Bien piece, originally arranged on the chessboard as shown in Figure 3.1.
* In the geometric chess, when placing a piece in a small square, we understand that the center of the chess piece always coincides with the center of the square.
The center of each square always exists but it is only determined when there is a piece in that square.
* In the standard geometric chess, most chess pieces have no fixed "geometric value" that they have a highly customizable character.
* After performing a certain move, the chess player has the right to choose the appropriate "geometric values" to assign to the chess pieces (both its pieces and the competitor’s pieces) in order to reach a target as given. Chess players can choose different values ​​to assign to each chess piece. These temporary "values" only exist momentarily, when the competitor moves to the new move, they have the right to change the appropriate values ​​according to the specific conditions of layout.

Fig 3.1: Layout of chess pieces on the standard geometric chess board

a. Point Piece: Point piece has a circular symbol in the middle of the chess piece, representing a point on the chess board, and the point also coincides with the center point of Point piece. Each side has 2 Point pieces.

b. Axis piece : The axis represents a custom segment (value> 0) located on the vertical axis or horizontal axis of the chessboard (the concept of vertical axes, horizontal axes, diagonal axes is shown in the general introduction). The center of the Axis piece is a point on that straight line. This means that the position of the Axis piece may be the beginning, the end, or the middle of the straight line it produces. Which axis is the straight line represented by the Axis Piece? Its length is selected by the person making the move. Know at least two points that the beginning and the end of a straight line can determine that straight line. The Axis Piece is denoted as a straight line. Each side has 1 Axis Piece.

c. Cross Axis piece (short name – Cross piece): The Cross piece represents a custom segment (valued at> 0) located on a particular diagonal of the chessboard (at the center of each small square we identify two diagonal of the chessboard - only 4 squares in the four corners of the chessboard can only identify a diagonal axis). The position of Cross piece is a point on that segment. The characteristics of the straight line represented by the Cross piece is the same as the straight line represented by the Axis Piece. The Axis Piece represents the straight line on the longitudinal axis and the horizontal axis, while the Cross piece represents the straight line on the cross axes of the chess board. The Cross Piece is denoted as a straight line. Each side has 1 Cross piece.

d. Square Corner Piece (short name - Square Piece): The Square piece represents a custom right angle, the central position of Square piece is the top of that right angle, two sides of the right angle are straight lines of varying length (price value> 0), this angle can rotate around its peak to the proper position. The person making the move has the right to determine the square corner represented by the Square piece. The Square piece only represents one square corner, not two square corners. The Square piece is denoted by two squares at opposite ends with continuous lines and dashed lines. This symbol indicates that the Square piece can create a right-angled corner on either the horizontal axis and the longitudinal axis or a right-angled edge on the two diagonal axes of the chess board. Each side has a Square piece.

e. Acute corner piece (short name – Acute piece): The Acute piece represents a custom 45 degree angle, the central position of Acute piece is the apex of the 45 degree angle, two sides of the 45 degree angle are straight lines of varying length (value> 0), this acute corner can rotate around its top to the proper position. The person making the move has the right to determine a 45-degree angle created by the acute piece. The Acute piece is marked by two 45-degree angles to the top with straight lines and dashed lines. This symbol indicates that the acute piece can create a 45 degree angle with edges on the horizontal axis, the longitudinal axis or the diagonal axis. Each side has 1 Acute piece.

f. Triangular piece: Triangular piece shows a custom triangle (the length of the edges> 0), this triangle has a peak at the center of the Triangular piece and two other tops identified by the players. The Triangular piece is denoted by a triangle. Each side has a Triangular piece.

g. Quadrilateral piece: Quadrilateral piece performs a custom quadrilateral (the length of the edges is 0), this quartet has a peak at the center of the Quadrilateral piece and the other three summands are identified by the players. Quadrilateral piece is denoted by a quadrilateral. Each side has a Quadrilateral piece.

h. Round piece: Round piece represents a custom circle (with radius> 0),  Round piece position is the center of the circle and the radius of the circle is identified by the players. The Round piece is denoted by a circle with a definite center. Each side has a

Round piece.

i. Piece “G” Piece G is an extra piece, not used for the competition, Piece G is placed outside the main chess board (on the edge of the chess board). Piece G is used to mark (Receipt) scores gained by each side during the competition. The upper part of piece G is denoted by G (the first letter of the phrase Geometric chess), its underside is denoted by the star.

Initially, piece G displays a G-shape, when one side wins a match, its  piece G is flipped over to show the star. Therefore, piece G just recognizes the score for the side which wins in the matches. Each side has an edge piece.

2. Cases of scoring

In the standard geometric chess, the scoring cases include: Straight, straight, triangular, square and circular.

a. General regulations

* Only consider the points for the side which has just done the move.
* A geometric form that is considered valid and scored for the side that has just ended the move must have the following characteristics:

+ The chess pieces for layout must consist of pieces of both sides and the number of pieces of the side making the late move must be greater than or equal to that of the competitor.
+ All angles, edges, vertices (if any) of each participating piece must be fully incorporated into the formation of that shape.
Geometric shapes formed by the chess pieces must follow logical logic.
+ Angles, edges, lines should not overlap

* To make it easy to see, the illustrations in the next section refer only to the pieces involved in the puzzle and ignore the remaining chess pieces on the chess board.

b. Straight line (1 – 2 points )

A straight line segment is created when there are at least 3 pieces on either a horizontal line or a vertical line or a diagonal line of the board and they form a geometrical line in which the distance between two chess pieces consecutive is always equal.

* Figure 3.2 illustrates a number of straight-line sequences consisting of 3 pieces.

Fig 3.2: Straight line with equal distance

+ Straight section as mentioned in Figure 3.2 is made up of two Point pieces and one Axis piece, including the chess pieces on both sides and the distance between two adjacent chess pieces equals one square. Of the three pieces involved in straight line alignment, two Point pieces are only two discrete points while the Axis piece creates a straight line on the horizontal axis and connect three pieces together. This equilateral straight line is located on the horizontal axis of the chessboard, therefore, it is impossible to replace the Axis piece by the Cross piece. When the positions of the Point pieces and the Axis pieces are changed, a straight line is obtained.

+ Straight section as mentioned below 3.2 is made up of one Cross piece and two Point pieces. The chess piece in the middle of two pieces remains two times the diagonal of the square. This is a straight line on the diagonal axis, therefore, it is impossible to replace the Cross piece with the Axis piece.
- It is supposed the red party has made the last move to make a straight line as shown in Figure 3.2. And also, the number of red pieces is 2 and the number of black pieces is 1 (that is, red number is bigger than black number). Then the red side will be added 1 point.
- If the black side finishes the last move and creates a straight line as in Figure 3.2, the black side is not scored (because the black number is less than the red number) and the red side has no points.

* Figure 3.3 illustrates some cases where a straight line segmentation is valid.

Fig 3.3: Straight line with equal distance

+ As mentioned in Figure 3.3, there is the straight line that is created from two Axis piece and one Point piece. Although here use two Axis pieces for the formation but still not overlap is because the player has the right to customize the length of the line generated by the Axis piece.
Figure 3.3 is a straight line segment that is created from two Cross pieces and one Point piece.
* Figure 3.4 illustrates the case of "Stop piece" when straight line spacing. If you leave the red square as mentioned in Figure 3.4, you will get a straight line segment. Likewise, if we remove the black triangles in the straight line as mentioned in Figure 3.4, we also get a straight line segment.  Square piece and Triangular piece in this case are known as Stop pieces.

Fig 3.4: "Stop piece" in the case of straight line with equal distance

* Figure 3.5 illustrates two straight lines of four chess pieces. When a straight line is drawn with 4 or more pieces, 2 points are counted.

Fig 3.5: Straight line with equal distance with 4 chess pieces

+ Straight section as mentioned Figure 3.5 with the number of chess pieces on both sides is equal to 2, therefore, any side making the last move created is also considered valid and added two points.
+ The straight line is mentioned in below the figure 3.5 with red number is 3 and the number of black pieces is 1: if this line is made by the red side, it is calculated 2 points, if the black side making the last move created, neither side has extra points.

c. Triangular layout (1 point)

Triangle is a shape with 3 vertices and 3 edges. In the geometric chess, a triangle is considered valid when at each vertex there must be a positioning chess piece and the pieces must create three sides of the triangle, the sides of the triangle must lie on the horizontal axis, the vertical axis or diagonal of the chessboard.
* There are many ways to arrange a valid triangle. Figure 3.6 illustrates how two triangles are valid.

Fig 3.6: Triangular layout with chess pieces

+ The triangle as mentioned in Figure 3.6 is made up of Triangular piece and 2 Point pieces. Triangular piece can create a custom triangle with a peak at position of the triangular piece and two other vertices selected by the player, therefore, if you choose two Point pieces to make the other two peaks, then we have a triangular shape pear. If the red side has the last move to create this triangle, one more point is added (because red pieces are more than black pieces).

+ The triangle as mentioned in Figure 3.6 is made up of one Square, one Axis, and one Point piece. This is a square triangle with two sides created by the Square piece and the bottom is located on the horizontal axis formed by the Axis piece, Point piece only acts as a top of the triangle. If the black side has the last move to create this triangle, it will add up to 1 point (because the number of black pieces is more than red pieces).

* Figure 3.7 illustrates invalid triangular layout.

Fig 3.7: Invalid triangular layout

+ The triangle as mentioned in Figure 3.7 has three sides created by the Triangular pieces, therefore, the straight line generated by the Axis piece will have to be folded onto one side of the triangle causing the overlapping phenomenon to be considered invalid.
+ The triangle as mentioned in Figure 3.7 has two square edges created by Square piece, Acute piece which creates one bottom and one square edge, thus causing overlapping and invalidation.

* Figure 3.8 illustrates the existence of “Stop piece" when triangulating.

Fig 3.8: "Stop piece" and “non-Stop piece” in the case of triangular layout

+ The triangle as mentioned in Figure 3.8 is considered invalid because in addition to the three pieces at the top of the triangle, there is another piece (quartet piece) lying on one side of the triangle. The Quartet piece is called Stop piece.
The triangle as mentioned in Figure 3.8 are still valid because the Round piece is in the interior of the triangle and should not be considered Stop piece.

+ Thus, if there are 3 chess pieces at the top of the triangle, there are other chess pieces positioned on the sides of the triangle, then the triangle is considered invalid. Chess pieces inside the triangle are not Stop pieces.

d. Square layout (2 points)

The square is the right shape with 4 equal sides and 4 right angles at its vertices. In the geometric chess, a valid square must have four chess pieces positioned at four vertices, and these chess pieces must qualify for the edges of the square, the right squares on the chess pieces. With the chess pieces present on the chessboard, we have many combinations to arrange a valid square.

* Figure 3.9 illustrates the arrangement of two valid squares.

Fig 3.9: Valid square layout

+ The square as mentioned above Figure 3.9 is created by a Quartile piece and three Point pieces. Three Point pieces determine the three vertices of the square while the Quartet piece identifies the remaining one and customises the edges of the quadrilateral into the four sides of that square. Since the number of pieces involved in this square is 2, each party making the last move created is also given 2 points.

+ The square as mentioned below Figure 3.9 is made up of one Square, two Cross and one Point piece. This square is made up of three black pieces and one red pieces: if the black side makes the last move creating this shape, it will be added two points, and if the red side makes the last move created, neither side will be added points.

* Figure 3.10 illustrates the existence of “Stop piece" in the case of square layout. When there is 1 piece, in addition to four pieces that join the square, lying on a certain edge of the square, this square is considered invalid.

Fig 3.10: "Stop piece" and “non-Stop piece” in the case of  square layout

+ The square as mentioned in Figure 3.10 is considered invalid because the Triangular piece is located on one side of the square.
+ The square as mentioned in Figure 3.10 is still valid because there are no “pieces" on the sides of the square, only the Axis piece is on the inside of this square.

e. Round layout (2 points)

In the geometric chess, a circle is considered valid when the center is identified and at least three points lie on that circle, all of which must have a positioning geometric chess, the Circle must be at the center of the circle, The radius determined by the Round piece and the remainder must lie on the vertical, horizontal or diagonal axis of the chessboard. With the chess pieces present on the chess board, there are only a few combinations to arrange a circle. No other army can replace the Round piece position at the center of the circle.

Fig 3.11: Valid round layout

* Figure 3.11 illustrates how to make valid round layout for Round and Three Point pieces. This circle is centered on the Round piece, the radius created by the Round piece and three Point pieces are on the diagonal of the chessboard. The black side or the red side making the last move to create this circle is also added two points (because the number of pieces involved in the formation of two sides are equal)

Fig 3.12: Invalid round layout

* It is supposed the red side creates the circle as shown in Figure 3.11 and is added two points. If the black side moves then the point involved in this circle from position B4 to position F4 as shown in Figure 3.12 does not create a valid new circle for the point. But if the black side moves the second point from another place to the F4 position as shown in Figure 3.13 to create a fourth point of centeredness, then this new circle is considered valid and scoring.

Fig 3.13: Valid round layout with 5 chess pieces

* Figure 3.14 illustrates how to make invalid round layout. Although this is also a round centered at the Round piece and defining four points in the center, the radius created by the Round piece and the Point pieces are not located on the axes of the chess board.

Fig 3.14: Invalid round layout

Figure 3.15 illustrates the existence of “Stop piece" in the case of round layout.

Fig 3.15: "Stop piece” in the case of round layout

+ In Figure 3.15, in addition to three Point pieces, there are other chess pieces - in this case the Cross piece - which are also located in the center of the circle. The Cross piece (or any other piece - aside from Point pieces) is called “Stop piece”.

e. Polymorphic layout

In many cases, when we move a chess piece to a new location, we can create two or more basic geometric shapes - then we have a polymorphic sort. Thus, the polymorphic layout is a composite of individual shapes.

* When making the polymorphic layout, the maker must follow the following principles:
+ All edges, angles, vertices (if any) of the pieces participating in the layout must be fully involved in the arrangement.
+ Each individual shape, when separated, must be a valid shape following the basic layout principles described in the previous sections.
+ Individual pairs can share one or two pieces together.
+ Score is calculated as the total score of individual shapes.

* An example of a polymorphic layout that shares a chess piece.

+ You pay attention to the Point piece on the red side in Figure 3.16. When there is no Point piece at position E6, there is no valid geometry, if the red side moves the Point piece to position E6, it will create 3 valid triangles that share the same vertex. With this move on the red side is added 3 points (each triangle is calculated 1 point).

Fig 3.16: polymorphic layout with the same chess piece

+ Figure 3.17 illustrates two cases where polymorphism is used to share a chess piece. As mentioned in Figure 3.17, when the red side moves Triangular piece to C9, it creates two valid triangles using Triangular piece (here, Triangular piece is assigned two different geometric values to serve the purpose of layout). With this move, the red side gains 2 points.
As mentioned in Figure 3.17, when the red sidemoves Square piece to the G4 position, it creates two valid triangles using the same Square piece (where the Square piece is assigned two different geometric values for the purpose of layout). With this move, the red side gains 2 points.

Fig 3.17: polymorphic layout with the same chess piece

* Illustrative example of a polymorphic layout sharing two pieces

+ You pay attention to the Point piece in the red side on Figure 3.18. There is no valid geometry in the E4 position if the red side moves to position E4 to produce three geometrically valid shapes: triangle and square. These single shapes use the same 2 pieces (Point pieces at E1, E4). With this move, the red side is added 4 points (1 point for straight line spacing, 1 point for triangle and 2 points for square).

Fig 3.18: polymorphic layout sharing two pieces

+ Figure 3.19 illustrates two more polymorphic cases using two pieces together. As mentioned in Figure 3.19, when the red side moves the Acute piece to C7, it creates two valid triangles using the Acute piece & Axis piece (where the Acute piece and Axis piece are assigned two different geometric values each other for lining purposes). With this move, the red side gains 2 points.

As mentioned Figure 3.19, when the red side moves the Triangular piece to the F4 position, it creates two valid triangles using the Triangular & Point piece (here the Triangular piece is assigned two different geometric values for the purpose of layout). With this move, the red side gains 2 points.

Fig 3.19: polymorphic layout sharing two pieces
3. Some notes

a. Repeating layout like simple geometric chess
b. Not score by moving can piece: like simple geometric chess
c. Avoid confusion in polymorphic layout:  like simple geometric chess