Mathematics on Opening Rules - 五子棋

1. (a,b,c,d,e) is defined by black 1 in a*a, white 2 in b*b, black 3 in c*c, white 4 in d*d, and black 5 in e*e. In which a,b,c,d,e are all odd integers and if any of them is zero, it means the opening stage is ended before that move.
2. <A,B,C,D> is defined by
after black 1, the corresponding player has a right to choose white and stop the opening my put A white stones, the other player choose any of them.
after white 2, the corresponding player has a right to choose black and stop the opening my put B black stones, the other player choose any of them.
after black 3, the corresponding player has a right to choose white and stop the opening my put C white stones, the other player choose any of them.
after white 4, the corresponding player has a right to choose black and stop the opening my put D black stones, the other player choose any of them.
If any of them is zero, it means no alternative is provided at that move.

3. 5-swap opening rules and N-5th-moves opening rules are defined as usual.

Ex1:
a. Rule 3 is (1,3,5,7,9) 5-swap opening rule
b. Rule 4 is (1,3,5,7,9)-<0,0,0,5> 5-swap opening rule
c. Rule 8 is (1,3,5,5,15)-<0,0,0,3> N-5th-moves opening rule
d. Rule 10 belong to (15,15,15,0,0)
e. Rule 13 is (1,3,5,15,15) N-5th-moves opening rule
f. Rule 14 is (1,3,5,15,15)-<0,0,0,5> N-5th-moves opening rule
g. Rule 21 is (1,5,9,5,9) 5-swap opening rule
h. Rule 23 is (7,7,7,7,7) 5-swap opening rule
i. Rule 24 is (1,3,7,15,15)-<0,0,0,12> N-5th-moves opening rule
j. Rule 25 is (15,15,15,15) 5-swap opening rule
k. Rule 28 is (1,3,5,7,15) N-5th-moves opening rule
l. Rule 29 is (1,3,5,15,15)-<0,0,0,4> N-5th-moves opening rule

4. The (a,b,c,d,e)-<0,0,0,D> N-5th-moves opening rule is

- The first player puts three stones where black 1 in a*a, white 2 in b*b, and black 3 in c*c
- The other player has a right to swap.
- The white player puts the 4th move in d*d on board and declares whether there will be 1, 2, 3 ...., D fifth moves offered in the game.
- The other player has a right to swap.
- The black player puts as many 5th moves in e*e on the board as it was declared before.
- The white player chooses one 5th from these offerings and plays the 6th move.
After that the opening stage is over and the game continues normally by renju rules.

5. The (a,b,c,d,e) 5-swap opening rule is
- The black player puts the first move within a*a central square.
- The other player may swap.
- The white player will put the 2nd move within b*b central square.
- The other player may swap.
- The black player will put the 3rd move within c*c central square.
- The other player may swap.
- The white player will put the 4th move within d*d central square.
- The other player may swap.
- The black player will put the 5th move within e*e central square.
The other player may choose color.
After that the opening stage is over and the game continues normally by renju rules.

6. The (a,b,c,d,e)-<A,B,C,D> 5-swap opening rule is
- The black player puts the first move within a*a central square.
- The other player may swap or stop the opening stage by putting A white stones.
- The white player will put the 2nd move within b*b central square.
- The other player may swap or stop the opening stage by putting B black stones.
- The black player will put the 3rd move within c*c central square.
- The other player may swap or stop the opening stage by putting C white stones.
- The white player will put the 4th move within d*d central square.
- The other player may swap or stop the opening stage by putting D black stones.
- The black player will put the 5th move within e*e central square.
- The other player may choose color.
After that the opening stage is over and the game continues normally by renju rules.

7. The (a,b,c,d,e)-<0,0,0,D> N-5th-moves opening rule and the (a,b,c,d,e)-<0,0,0,D> 5-swap opening rule have exactly the same number of variants.

8. The jousekial ability of a person in (a,b,c,d,e)-<A,B,C,D> opening rule is the ratio of the variants he/she knows or can derive easily to all variants in (a,b,c,d,e)-<A,B,C,D>.

9. An expert is a person of nearly 1.0 jousekial ability.

10. A person of X jousekial ability can be considered to be a beginner to a person of Y jousekial ability, if Y is large enough to X.

Ex. An expert of Old RIF rule is a beginner of (1,3,5,7,9) 5-swap opening rule.

Discussions:

1. Theoretically, an expert in the (a,b,c,d,e)-<0,0,0,D> N-5th-moves opening rule is also in the (a,b,c,d,e)-<0,0,0,D> 5-swap opening rule and vice versa.

2. Only experts can play well to N-5th-moves opening rules
(see http://renju.compute.com.tw/~renjuclass/cgi-bin/forum/topic.cgi?forum=25&topic=46)

3. Beginners also can play well to 5-swap opening rules
(see http://renju.compute.com.tw/~renjuclass/cgi-bin/forum/topic.cgi?forum=25&topic=46)

Suggestions:

1. Free Renju Rule (multi-groups) is (15,15,15,15,15) 5-swap opening rule
2. Free Renju Rule (single-group) is (15,15,15,15,15)-<A,B,A,B> 5-swap opening rule ( Selection of A and B is still an issue. <1,4,1,4> may be tried)
3. It is also possible to enlarge the numbers in () step by step

Comments:
If the direction to Free Renju is not observed, it is very possible to found the International Federation of Free Renju

C.C. Tsai 2006,2,19





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