Name of the game: Symmetry.

Objective of the game: To improve the ability of analytical thinking.

Definition of the game: As inferred from its name this game is played according to basic symmetrical rules. It is played with nine stones (17x17) in two-dimensional world.

 At the very beginning of the game; the stones should be placed in symmetrical forms wherever the players want.

When they have formed symmetry, the game is over on condition that every stone has been moved from their initial points-without considering the stones of the rival. Whichever party has first realized this form wins the game.

Sample Symmetrical Moves:

1.     AB = BC   The symmetry of A to B is C or that of C to B is A.

2.  AB = BC   The symmetry of A to B is C or that of C to B is A.

3. AB = BC   The symmetry of A to B is C or that of C to B is A.

4. AB = BC   The symmetry of A to B is C or that of C to B is A.

Sample Symmetrical Positions:

1. (x4 , y12)                 (x12 , y12)               (x8 , y12)

 

    (x4 , y8)                    (x12 , y8)                 (x8 , y8)

 

    (x4, y4)                      (x12, y4)                 (x8 , y4) 

2.                                              (x₁₀ , y₁₄)

                                (x₉ , y₁₁)
                                                                      (x₁₄ , y₁₀)
               (x₇ , y₉)      (x₉ , y₉)      (x₁₁ , y₉)
(x₄ , y₈)
                                (x₉ , y₇)

                (x₈ , y₄)

3.                (x₁₃ , y₅)               (x₁₇ , y₅)   

                     (x₁₇ , y₁)

       (x₁₃ , y₃)(x₁₅ , y₃)(x₁₇ , y₃)

            (x₁₅ , y₁)

      (x₁₃ , y₁)               (x₁₅ , y₅)    

4.  (x₄ , y₁₄)                                           (x₁₂ , y₁₄)

                               (x₇ , y₁₁)

     (x₄ , y₈)                (x₈ , y₈)               (x₁₂ , y₈)

                                (x₉ , y₅)

    (x₄ , y₂)                                          (x₁₂ , y₂)

5.                        (x₈ , y₁₂) 

               (x₆, y₁₀)                                      (x_14,y₁₀)
                      (x₇ , y₉)(x₉ , y₉)(x₁₁ , y₉)
(x₄ , y₈)                                      (x₁₂ , y₈)

                                     (x₁₀ , y₆)

6a.                                        (x₃ , y₁₇)

                                 (x₂, y₁₆)             (x_4, y₁₆)

                      (x₁ , y₁₅)          (x₃ , y₁₅)           (x₅ , y₁₅)

                                 (x₂ , y₁₄)            (x₄ , y₁₄)

                                             (x₃ , y₁₃)

 6b.            (x₆ , y₆)    (x₈ , y₆)    (x₁₀ , y₆)

                             (x₇ , y₅)    (x₉ , y₅)    (x₁₁ , y₅)  

                                       (x₈ , y₄)    (x₁₀ , y₄)    (x₁₂ , y₄) 

7a.          (x₁₃ , y₅)                    (x₁₅ , y₅)                     (x₁₇ , y₅)   

   (x₁₃ , y₃)                    (x₁₅ , y₃)                     (x₁₇ , y₃)

   (x₁₃ , y₁)                    (x₁₅ , y₁)                     (x₁₇ , y₁) 

7b.                                   (x₁₃ , y₁₇)

                                        (x₁₃ , y₁₅)

     (x₉ , y₁₃)     (x₁₁ , y₁₃)     (x₁₃ , y₁₃)      (x₁₅ , y₁₃)      (x₁₇ , y₁₃)

                                        (x₁₃ , y₁₁)

                                         (x₁₃ , y₉)

7c.    (x₁ , y₁₇)       (x₃ , y₁₇) 

        (x₁ , y₁₅)       (x₃ , y₁₅)       (x₅ , y₁₅)       (x₇ , y₁₅)       (x₉ , y₁₅)  

                                                                 (x₇ , y₁₃)       (x₉ , y₁₃)

8a.                                    (x₅ , y₅)

                (x₂ , y₄)              (x₈ , y₄) 

             (x₁ , y₃)     (x₅ , y₃)     (x₉ , y₃)             

                            (x₂ , y₂)              (x₈ , y₂)

                                         (x₅ , y₁)

8b.                                         (x₇, y₅)    (x₉, y₅)

                                         (x₆, y₄)    (x₈, y₄)

                                              (x₇, y₃)

                                        (x₆, y₂)    (x₈, y₂)

                                 (x₅, y₁)   (x₇, y₁)

8c.                                (x₉ , y₇)    (x₁₁ , y₇)

            (x₆ , y₆)     (x₈ , y₆)    (x₁₀ , y₆)   (x₁₂ , y₆)    (x₁₄ , y₆)

                                    (x₉ , y₅)    (x₁₁ , y₅)   

8d.                                    (x₃ , y₉)

                       (x₂ , y₈)              (x₄ , y₈)

                                   (x₃ , y₇)

                                   (x₃ , y₅)

                                   (x₃ , y₃)

                      (x₂ , y₂)              (x₄ , y₂)

                                   (x₃ , y₁)

9.                       (x₁₁ , y₅)    (x₁₃ , y₅)     (x₁₅ , y₅) 

               (x₉ , y₃)                (x₁₃ , y₃)                 (x₁₇ , y₃)

                          (x₁₁ , y₁)    (x₁₃ , y₁)     (x₁₅ , y₁)

10.    

(x₂ , y₁₀)             (x₆ , y₁₀)             (x₁₀ , y₁₀)

                     (x₅ , y₇)               (x₉ , y₇)               (x₁₃ , y₇)  

                                (x₈ , y₄)               (x₁₂ , y₄)               (x₁₆ , y₄)

11.                                     (x₉ , y₁₅)

                                          (x₉ , y₁₁)

(x₃ , y₉)           (x₇ , y₉)      (x₉ , y₉)      (x₁₁ , y₉)            (x₁₅ , y₉)

                                           (x₉ , y₇)
 

                                           (x₉ , y₃)

Verification of the game: There are two types of verifications that can check whether the stones are symmetrical in their first and final positions:

1.     When the obtained symmetrical form is divided into two parts in the center, the stones which are located on one side should be symmetrically moved to the other side one by one.

2.     The total of the values of x and y have to be equal to multiples of nine.