**Contents**

**1. Definitions of chess pieces and symbols**

**4. Chess playing results**

**5. Statistics of basic layputs in the standard geometric chess**

**6. Pr**

**elimittary introdution of 12 beginning moves in Standard geometric chess**

*********

**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*

*b.***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.**

*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.

**: 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.**

*i. Piece “G”*

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.***like simple geometric chess**

*Repeating layout:***like simple geometric chess**

*b. Not score by moving can piece:***like simple geometric chess**

*c. Avoid*confusion in polymorphic layout*:*
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