3-move tsumego rule

[Bill Spight]

I did not include the ko diagram originally, even though I suspected that the best Black can do is ko, because the logic of the rule relies upon the assumption that Black to play can kill. We have all seen positions where the play to kill is different from the play to make ko.

However, upon reflection I think that the logic should also work with the disjunctive assumption that Black to play can kill or make ko. You just need to check for both ko and the kill for each choice of White play.

$$W B to play
$$ ------------------
$$ | . . . 4 5 . . . .
$$ | . X X O . 1 . . X
$$ | . X O 2 O . . . X
$$ | . X O . O . X . X
$$ | . X O O . a . X .
$$ | . . X O O O X . .
$$ | . . X X X X X . .
$$ | . . . . . . . . .

Click Here To Show Diagram Code

[go]$$W B to play
$$ ------------------
$$ | . . . 4 5 . . . .
$$ | . X X O . 1 . . X
$$ | . X O 2 O . . . X
$$ | . X O . O . X . X
$$ | . X O O . a . X .
$$ | . . X O O O X . .
$$ | . . X X X X X . .
$$ | . . . . . . . . .[/go]

and appear to be necessary to make ko after , which means, in conjunction with the position starting with Wa, is the first move, with Black to kill or make ko.

$$B Variation 1
$$ ------------------
$$ | . . . . . . . . .
$$ | . X X O 2 . 4 . X
$$ | . X O 1 O . . . X
$$ | . X O . O . X . X
$$ | . X O O . 3 . X .
$$ | . . X O O O X . .
$$ | . . X X X X X . .
$$ | . . . . . . . . .

Click Here To Show Diagram Code

[go]$$B Variation 1
$$ ------------------
$$ | . . . . . . . . .
$$ | . X X O 2 . 4 . X
$$ | . X O 1 O . . . X
$$ | . X O . O . X . X
$$ | . X O O . 3 . X .
$$ | . . X O O O X . .
$$ | . . X X X X X . .
$$ | . . . . . . . . .[/go]

After we cannot assume that Black can only make ko, so we try .

$$Bm5 (continued)
$$ ------------------
$$ | . . . 7 . 6 3 5 8
$$ | . X X O O 2 O 4 X
$$ | . X O X O 1 . . X
$$ | . X O . O . X . X
$$ | . X O O . X . X .
$$ | . . X O O O X . .
$$ | . . X X X X X . .
$$ | . . . . . . . . .

Click Here To Show Diagram Code

[go]$$Bm5 (continued)
$$ ------------------
$$ | . . . 7 . 6 3 5 8
$$ | . X X O O 2 O 4 X
$$ | . X O X O 1 . . X
$$ | . X O . O . X . X
$$ | . X O O . X . X .
$$ | . . X O O O X . .
$$ | . . X X X X X . .
$$ | . . . . . . . . .[/go]

at 9

If at 11, at 9. (White still dies.)

By convention the rest of the board is empty and White dies.

$$B Variation 2
$$ ------------------
$$ | . . . 3 . . . 9 .
$$ | . X X O 4 2 8 6 X
$$ | . X O 1 O . 7 . X
$$ | . X O . O . X . X
$$ | . X O O . 5 . X .
$$ | . . X O O O X . .
$$ | . . X X X X X . .
$$ | . . . . . . . . .

Click Here To Show Diagram Code

[go]$$B Variation 2
$$ ------------------
$$ | . . . 3 . . . 9 .
$$ | . X X O 4 2 8 6 X
$$ | . X O 1 O . 7 . X
$$ | . X O . O . X . X
$$ | . X O O . 5 . X .
$$ | . . X O O O X . .
$$ | . . X X X X X . .
$$ | . . . . . . . . .[/go]

White dies this way, too.

$$B Ko
$$ ------------------
$$ | . . . 3 4 . . . .
$$ | . X X O 5 2 . . X
$$ | . X O 1 O . . . X
$$ | . X O . O . X . X
$$ | . X O O . . . X .
$$ | . . X O O O X . .
$$ | . . X X X X X . .
$$ | . . . . . . . . .

Click Here To Show Diagram Code

[go]$$B Ko
$$ ------------------
$$ | . . . 3 4 . . . .
$$ | . X X O 5 2 . . X
$$ | . X O 1 O . . . X
$$ | . X O . O . X . X
$$ | . X O O . . . X .
$$ | . . X O O O X . .
$$ | . . X X X X X . .
$$ | . . . . . . . . .[/go]

[Bill Spight]

I'm trying to convince that this always works for different selections of 'A' - or the boundaries of when it works.

By the logic of the rule, the necessity of B and C to achieve the goal, given A, is equivalent to the assumption that the goal is achievable. The derivation of A as a possible first play depends upon the observation that if the first play is none of A, B, or C, then the defender can play at A and make miai of B and C. A has to be a good play.

$$Wc Black to kill.
$$ ------------------
$$ | . O C . . . X . .
$$ | . C 1 C O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .

Click Here To Show Diagram Code

[go]$$Wc Black to kill.
$$ ------------------
$$ | . O C . . . X . .
$$ | . C 1 C O O X . .
$$ | . O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

In this position looks like a vital point, threatening to make 3 one point eyes (marked).

$$Wc Black to kill (1)
$$ ------------------
$$ | . O . 5 7 6 X . .
$$ | 2 8 1 . O O X . .
$$ | 4 O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .

Click Here To Show Diagram Code

[go]$$Wc Black to kill (1)
$$ ------------------
$$ | . O . 5 7 6 X . .
$$ | 2 8 1 . O O X . .
$$ | 4 O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

Then and suffice to kill. But are they necessary?

$$Wc Black to kill (2)
$$ ------------------
$$ | . O . 4 . 6 X . .
$$ | 2 . 1 . O O X . .
$$ | 5 O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .

Click Here To Show Diagram Code

[go]$$Wc Black to kill (2)
$$ ------------------
$$ | . O . 4 . 6 X . .
$$ | 2 . 1 . O O X . .
$$ | 5 O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

No, they are not. This and also kill.

$$Wc Black to kill (3)
$$ ------------------
$$ | . O . 5 . 4 X . .
$$ | 2 . 1 . O O X . .
$$ | 6 O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .

Click Here To Show Diagram Code

[go]$$Wc Black to kill (3)
$$ ------------------
$$ | . O . 5 . 4 X . .
$$ | 2 . 1 . O O X . .
$$ | 6 O O O X X X . .
$$ | X X X X . . . . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .[/go]

If at 6, at 5

We have seen this position before. White is dead.

But it looks like is necessary, as it occurs in all variations. Besides, after , a White play at 2 would make 2 one point eyes.

Now the fact that is necessary after suggests that it is a good first move. The relation to the All Moves As First heuristic is no accident.

Edit: Now, to see that only is necessary after in all the variations took either logic, which I did not use, or going through a number of variations. As knotwilg points out, proving necessity may entail just as much search and effort as simply proceeding directly to reading. However, if you can see or show necessity without reading, you may be able to save time and effort.