Endgame Strategy (02)

[Robert Jasiek]

CLASS OF POSITIONS 2

White to move // White (in contrast to the class of positions 1)
D = 0
K > 0
Kb >= Kw >= 0
Tb >= 0
Tw >= 0

STRATEGY FOR EACH PLAYER

1. answer ko threat
2. connect ko
3. capture ko if legal
4. ko threat if there is some ko
5. pass

ANALYSIS

Case 1: Kd MOD 3 = 0
----------------------

Algorithm:
1 Repeat Kw times:
1.1 White connects ko
1.2 Black connects ko
2 Repeat Kd/3 times:
2.1 White captures ko
2.2 Black connects ko
2.3 White connects ko
2.4 Black connects ko
3 Passes

Proposition 2.1: The score due to the algorithm is: 2/3 * Kd

Proof: Like for proposition 1.1.1.

Case 2: Kd MOD 3 = 1 && Td > 0
----------------------------------

Algorithm:
1 Repeat Kw times:
1.1 White connects ko
1.2 Black connects ko
2 Repeat (Kd - 1) / 3 times:
2.1 White captures ko
2.2 Black connects ko
2.3 White connects ko
2.4 Black connects ko
3 Repeat 2 * Tw + 1 times:
3.1 capture ko
3.2 ko threat
3.3 answer ko threat
4 Black captures ko
5 White passes
6 Black connects ko
7 Passes

Proposition 2.2: The score due to the algorithm is: 2/3 * (Kd - 1) +
2

Proof: During (1), the score is 0. During each of the (Kd - 1) / 3
repetitions of (2), Black gets 2 points more than White. During (3),
the final ko is not yet fixed. During (6), the final ko is fixed and
Black gets 2 points more than White.

Case 3: Kd MOD 3 = 1 && Td <= 0
-----------------------------------

Algorithm:
1 Repeat Kw times:
1.1 White connects ko
1.2 Black connects ko
2 Repeat (Kd - 1) / 3 times:
2.1 White captures ko
2.2 Black connects ko
2.3 White connects ko
2.4 Black connects ko
3 Repeat 2 * Tb times:
3.1 capture ko
3.2 ko threat
3.3 answer ko threat
4 White captures ko
5 Black passes
6 White connects ko
7 Passes

Proposition 2.3: The score due to the algorithm is: 2/3 * (Kd - 1) -
2

Proof: During (1), the score is 0. During each of the (Kd - 1) / 3
repetitions of (2), Black gets 2 points more than White. During (3),
the final ko is not yet fixed. During (6), the final ko is fixed and
White gets 2 points more than Black.

Case 4: Kd MOD 3 = 2
----------------------

Algorithm:
1 Repeat Kw times:
1.1 White connects ko
1.2 Black connects ko
2 Repeat (Kd - 2) / 3 times:
2.1 White captures ko
2.2 Black connects ko
2.3 White connects ko
2.4 Black connects ko
3 White captures ko
4 Black connects ko
5 White connects ko
6 Passes

Proposition 2.4: The score due to the algorithm is: 2/3 * (Kd - 2)

Proof: During (1), the score is 0. During each of the (Kd - 2) / 3
repetitions of (2), Black gets 2 points more than White. During (3)
to (5) together, the score remains constant.

--
robert jasiek