3-move tsumego rule

zermelo · #1

I noticed a simple rule that has helped with some tsumego. It may be close to trivial, and surely others have thought of it, but nevertheless I’ve found it useful for eliminating starting moves that cannot work.

The 3-move tsumego rule: Assume that the problem is black to kill. Now let A, B, C be such intersections that if white would start at A, the only way for black to kill with 2 moves would be playing both B and C. Now the solution to the original problem has to start with black play at A, B, or C.

An example:

Click Here To Show Diagram Code: [go]$$W Black to kill. $$ ------------------ $$ | . O . . . . X . . $$ | a . b c O O X . . $$ | . O O O X X X . . $$ | X X X X . . . . . $$ | . . . . . . . . . $$ | . . . . . . . . .[/go]

If we would let white play at a, black would need to play both b and c to kill. So we know that black has to start either at a, b, or c. Otherwise white would get a, and black would not get both b and c.

I don't think this is quite a theorem as such, because there could be cases where move order matters, if e.g. B would make A illegal, or because of captures.

You can also generalise the rule in different ways. E.g. if after white A, black would need to play either B and C, or B and D, then the correct starting move is one from A-D. It should work in ‘white to connect’ or other problems in the same way.

John Fairbairn · #2

I do look forward to discussion of this excellent post.

Matthew Macfadyen (ex-European Champion) used to mention this principle a lot and he used it in his seminars (?around 1975~1980). As I recall, he said he got it from someone else, and some others claimed to know about it, but I never understood why it was not more widely known. My brain baulks at thing like that, so I've never tried to use it - frankly, didn't even understand it. Maybe others had the same disinclination. Or perhaps people then were not doing tsumego much. That may sound far-fetched but the only collection available was the small Maeda series.

I think Matthew referred to it as a 2-move rule but I can't remember his exact wording.

Bill Spight · #3

zermelo wrote:

You can also generalise the rule in different ways. E.g. if after white A, black would need to play either B and C, or B and D, then the correct starting move is one from A-D. It should work in ‘white to connect’ or other problems in the same way.

Edited in toto for clarity.

The original rule is that if there is a White stone on A, Black to kill must play two stones in a row on B and C. We assume that Black to play can kill, without the White stone on A. Let us indicate this symbolically this way.

WA -> ((BB, BC) <-> kill) ; where the comma indicates "and"

But we are assuming a kill if Black plays first, which would allow her to play two stones while White plays one stone. So that reduces to

WA -> (BB, BC)

which in turn reduces to

¬WA v (BB, BC) ; where ¬ indicates "not" and v indicates "or"

To prevent White from playing on A, Black can play on that point or perhaps make the White play on A illegal, and if he does not do so on his first play he must play on B or C in order to play on both.

The second case is this.

WA -> (BB, (BC v BD))

Unless White to play can prevent Black from playing either C or D, or a play at one of them by White prevents Black from playing the other, then we may regard them as miai. That means that Black does not need to play at C or D first.

Tryss · #4

The problem with this rule is that it can be more complicated to find the moves A, B and C that verify the hypothesis than to solve the problem.

"if white would start at A, the only way for black to kill with 2 moves would be playing both B and C. " is quite specific.

For exemple, these moves A, B, C are not fitting the rule :

Click Here To Show Diagram Code: [go]$$ $$ ------------------------ $$ | . . . a O c . . . . . . $$ | . X X X O X . . . . . . $$ | . O O O b . . . . . . . $$ | . . . . . . . . . . . . $$ | . . . . . . . . . . . . $$ | . . . . . . . . . . . .[/go]

Indeed, if white play A, there is two possibilities for black to live :

Click Here To Show Diagram Code: [go]$$W $$ ------------------------ $$ | . . 4 1 O . . . . . . . $$ | . X X X O X . . . . . . $$ | . O O O 2 . . . . . . . $$ | . . . . . . . . . . . . $$ | . . . . . . . . . . . . $$ | . . . . . . . . . . . . $$ ------------------------ $$ | . . . 1 O 4 . . . . . . $$ | . X X X O X . . . . . . $$ | . O O O 2 . . . . . . . $$ | . . . . . . . . . . . . $$ | . . . . . . . . . . . . $$ | . . . . . . . . . . . .[/go]

So, what would be the A, B, C points for this problem? :study:

Bill Spight · #5

John Fairbairn wrote:

I think Matthew referred to it as a 2-move rule but I can't remember his exact wording.

That's how I heard it. Let White play first, and then give Black 2 moves in a row to kill.

zermelo · #6

Tryss wrote:

The problem with this rule is that it can be more complicated to find the moves A, B and C that verify the hypothesis than to solve the problem.

Yes, it does not work everywhere. But I noticed I used it already twice while looking through less than 10 Xuanxuan Qijing problems, so I was reminded of it. The above problem is from that collection.

Another was this:

Click Here To Show Diagram Code: [go]$$B Black to move $$ --------------------------------------- $$ | . . . . . . . . . . a . b . O O X . . | $$ | . . . . . . . X . O X O O c O X . X . | $$ | . . . . . . X . X X O O . . O X . . . | $$ | . . . , . . . . . , X X X X X , . . . | $$ | . . . . . . . . . . . . . . . X . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |[/go]

It seems to be often useful, when there is some obviously vital-looking point, like point a in this and the previous problem. THen quite often you can find a pair of black moves to go with it.

Bill Spight · #7

Tryss wrote:

The problem with this rule is that it can be more complicated to find the moves A, B and C that verify the hypothesis than to solve the problem.

"if white would start at A, the only way for black to kill with 2 moves would be playing both B and C. " is quite specific.

For exemple, these moves A, B, C are not fitting the rule :

Click Here To Show Diagram Code: [go]$$ $$ ------------------------ $$ | . . . a O c . . . . . . $$ | . X X X O X . . . . . . $$ | . O O O b . . . . . . . $$ | . . . . . . . . . . . . $$ | . . . . . . . . . . . . $$ | . . . . . . . . . . . .[/go]

Indeed, if white play A, there is two possibilities for black to live :

Click Here To Show Diagram Code: [go]$$W $$ ------------------------ $$ | . . 4 1 O . . . . . . . $$ | . X X X O X . . . . . . $$ | . O O O 2 . . . . . . . $$ | . . . . . . . . . . . . $$ | . . . . . . . . . . . . $$ | . . . . . . . . . . . . $$ ------------------------ $$ | . . . 1 O 4 . . . . . . $$ | . X X X O X . . . . . . $$ | . O O O 2 . . . . . . . $$ | . . . . . . . . . . . . $$ | . . . . . . . . . . . . $$ | . . . . . . . . . . . .[/go]

So, what would be the A, B, C points for this problem? :study:

What you have shown is that Black should start at 1 or 2.

We can regard the different 4 points as miai for capturing the two White stones. OC, that does not mean that Black will actually be able to capture them.

Bill Spight · #8

zermelo wrote:

I noticed I used it already twice while looking through less than 10 Xuanxuan Qijing problems, so I was reminded of it. The above problem is from that collection.

Another was this:

Click Here To Show Diagram Code: [go]$$B Black to move $$ --------------------------------------- $$ | . . . . . . . . . . a . b . O O X . . | $$ | . . . . . . . X . O X O O c O X . X . | $$ | . . . . . . X . X X O O . . O X . . . | $$ | . . . , . . . . . , X X X X X , . . . | $$ | . . . . . . . . . . . . . . . X . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |[/go]

It seems to be often useful, when there is some obviously vital-looking point, like point a in this and the previous problem. THen quite often you can find a pair of black moves to go with it.

Consider also these three points.

Click Here To Show Diagram Code: [go]$$B Black to move $$ --------------------------------------- $$ | . . . . . . . . . . . . b . O O X . . | $$ | . . . . . . . X . O X O O . O X . X . | $$ | . . . . . . X . X X O O c a O X . . . | $$ | . . . , . . . . . , X X X X X , . . . | $$ | . . . . . . . . . . . . . . . X . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |[/go]

If White "a", Black "b" and "c" are necessary to kill.

Problem solved. (Assuming that Black to play can kill, OC.

)

John Fairbairn · #9

Quote:

I noticed a simple rule that has helped with some tsumego. It may be close to trivial, and surely others have thought of it, but nevertheless I’ve found it useful for eliminating starting moves that cannot work.

@zermelo: One thing that interests me a lot is how other people think. Would you be willing to explain how you came up with this insight? It may be a re-discovery but it's still a discovery! And also, your view of it as a 3-move principle rather than the 2-move principle I first heard makes it easier for me to comprehend.

Kirby · **#10**

So let me see if I understand the rule...

1.) You find a move 'A' that white might play if white were going first.

2.) Usually this leaves the state such that black can still kill in 2 moves. These two moves are 'B' and 'C'.

3.) Solve the problem from the beginning in the normal way, but consider 'A', 'B', and 'C' as potentially good starting moves.

Is that it? If so, how is 'A' selected? Just what appears to be the good starting move for white?

dfan · **#11**

I'll rephrase:

1) You find a move A that would make White live if she played first (e.g., the 1-2 point in zermelo's original diagram; it makes a corner eye and one at the 2-2 point).

2) Often you are now in a state where a) Black cannot kill White if he just plays one move; b) if Black is given two moves in a row, they must be at B and C to kill (not anywhere else).

3) You now know that A, B, and C are the only starting moves that could kill.

(Say you play a different move D. White plays A, and you only have one move to play both B and C, which is the only way to kill. If you could play E to kill now, condition 2b would not have been satisfied.)

Knotwilg · **#12**

Me too I have a hard time understanding either the rule or its novelty.

My understanding of your rule
0) It's your turn to play
1) Somehow you find A, a move for the opponent, which then requires you to play both B & C to kill
2) You infer that A, B, C are good starting moves (the only starting moves?) to consider for killing

First, a question on symmetries
- If the property of this triplet is fully symmetric, then all of A, B, C are good first moves

Then a question on completeness
- How does finding such a triplet narrow the search tree to something particularly promising?

I have a more structured approach, I think, which is also more fundamental

1) Try solving the L&D problem by narrowing the eyespace (that's for killing, likewise, expand for living)
2) this leads to 1 sequence of consecutively narrowing/expanding; it usually reduces to a well known eyeshape
3) one of the players can play the vital point of that eye shape - this is a candidate vital point for the problem
4) In a second stage, promote this vital point above any of the narrowing/expanding moves
5) cuts are stronger vital points than no-cuts
6) iterate

Killing by narrowing is straightforward and it's also good practice in games because you strengthen the outside position and don't throw away stones if a group is revived (by a ko, or so). Finding the vital point is harder and can be found by narrowing the space.

I think that your A, B, C moves will often have 2 edge points and 1 vital point.
]

dfan · **#13**

What I took from the presentation was:

1) Sometimes White has a very clear move (A) to immediately live if it is her turn. If this is the case, it is worth checking whether you could then kill at all given two moves in a row (B and C).

1a) If this is so, you have just narrowed your list of candidate moves down to three (A, B, C).

1b) If you can't even kill given two moves in a row, then you have narrowed the list of candidate moves down to one (A).

2) If there wasn't such a White A, then go back to your regularly scheduled method of solving tsumego (e.g., starting by reducing the eyespace).

This principle isn't a general system of solving tsumego, it's just a nice thing that can narrow your search tree when it comes up.

Bill Spight · **#14**

John Fairbairn wrote:

Quote:

I noticed a simple rule that has helped with some tsumego. It may be close to trivial, and surely others have thought of it, but nevertheless I’ve found it useful for eliminating starting moves that cannot work.

@zermelo: One thing that interests me a lot is how other people think. Would you be willing to explain how you came up with this insight? It may be a re-discovery but it's still a discovery! And also, your view of it as a 3-move principle rather than the 2-move principle I first heard makes it easier for me to comprehend.

My apologies to zermelo for not congratulating him on his wonderful insight. :clap:

I heard it from a Japanese 6 dan back when Japanese amateur 7 dan overlapped with pro strength.

As for the correctness of the rule, it only works if we assume, as with a problem, that Black to play can kill (or connect, or whatever is the goal). If White plays A, Black must play both B and C to kill. B and C are the two necessary moves in that case. B and C have to be such that Black cannot guarantee both with a single play, and White to play cannot with one move prevent Black from playing either.

Given those conditions, Black's first play must be either B or C, or a play that prevents White from playing A. Otherwise White can reply at A, making miai of B and C, and Black cannot then kill.

Bill Spight · **#15**

Kirby wrote:

So let me see if I understand the rule...

1.) You find a move 'A' that white might play if white were going first.

2.) Usually this leaves the state such that black can still kill in 2 moves. These two moves are 'B' and 'C'.

3.) Solve the problem from the beginning in the normal way, but consider 'A', 'B', and 'C' as potentially good starting moves.

2) is incorrect. Moves B and C must be both necessary and sufficient to kill. Otherwise White A might just be a bad move.

Bill Spight · **#16**

dfan wrote:

This principle isn't a general system of solving tsumego, it's just a nice thing that can narrow your search tree when it comes up.

It narrows the conscious search tree for humans. The requirement that the two moves are necessary means that we have to know that all other combinations of two moves do not kill. Humans either have that knowledge already, deduce it, or perform the necessary search unconsciously.

Knotwilg · **#17**

OK, so the heuristic is "I spot a move A for White, so that Black's only way to kill with 2 moves is B & C"
That sounds like a heavy calculation in itself.
I thought it was about finding one such triplet but then from the answers understood it must be unique (B & C)

I applied both this heuristic and the systematic method on the mentioned problem. The systematic approach gave the first move faster than the heuristic. The real trouble in the problem is finding move 3, which neither the systematic approach nor the heuristic find easily, if at all.

Bill Spight · **#18**

Knotwilg wrote:

OK, so the heuristic is "I spot a move A for White, so that Black's only way to kill with 2 moves is B & C"
That sounds like a heavy calculation in itself.

If done consciously, yes. This is probably not a good heuristic for computers.

Quote:

I applied both this heuristic and the systematic method on the mentioned problem. The systematic approach gave the first move faster than the heuristic. The real trouble in the problem is finding move 3, which neither the systematic approach nor the heuristic find easily, if at all.

Let me illustrate the heuristic by the use of logic.

Click Here To Show Diagram Code: [go]$$W Black to reply and kill with two moves $$ --------------------------------------- $$ | . . . . . . . . . . 1 a 2 . O O X . . | $$ | . . . . . . . X . O B O O 4 O X . X . | $$ | . . . . . . X . X X O O . . O X . . . | $$ | . . . , . . . . . , X X X X X , . . . | $$ | . . . . . . . . . . . . . . . X . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |[/go]

We find

by the heuristic in tsumego of making one point eyes. That is the most efficient eye shape in a cramped space.

Once that is done, we deduce that :b2:

is necessary to prevent White from making a second one point eye at "a". Then we deduce that, given :w1:

and

,

is necessary to prevent White from making a second eye enclosing :b2:

. To make these deductions we only have to read one move deep for each player. (Edit: At each step, that is. I guess we have read to a depth of 3.

)

The best bots today do not use deduction, but humans do. Deduction can save us a lot of conscious effort.

Note that we have not shown that :b2:

and

kill White, only that they are necessary to do so, given :w1:

. That's all we have to do to find our candidate moves.

Bill Spight · **#19**

Let me try starting with the heuristic of reducing or expanding eye space.

Click Here To Show Diagram Code: [go]$$B Black to kill $$ --------------------------------------- $$ | . . . . . . . . 5 3 4 . 6 . O O X . . | $$ | . . . . . . . X . O X O O 2 O X . X . | $$ | . . . . . . X . X X O O . 1 O X . . . | $$ | . . . , . . . . . , X X X X X , . . . | $$ | . . . . . . . . . . . . . . . X . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |[/go]

Failure 1.

Let's back up to :b3:

and try 4 or 6.

Click Here To Show Diagram Code: [go]$$B Black to kill, II $$ --------------------------------------- $$ | . . . . . . . . . 4 3 . . . O O X . . | $$ | . . . . . . . X . O X O O 2 O X . X . | $$ | . . . . . . X . X X O O . 1 O X . . . | $$ | . . . , . . . . . , X X X X X , . . . | $$ | . . . . . . . . . . . . . . . X . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |[/go]

We know already or can see without conscious reading that White is alive. Failure 2.

Click Here To Show Diagram Code: [go]$$B Black to kill, III $$ --------------------------------------- $$ | . . . . . . . . . . 4 . 3 . O O X . . | $$ | . . . . . . . X . O X O O 2 O X . X . | $$ | . . . . . . X . X X O O . 1 O X . . . | $$ | . . . , . . . . . , X X X X X , . . . | $$ | . . . . . . . . . . . . . . . X . . . | $$ | . . . . . . . . . . . . . . . . . . . | $$ | . . . . . . . . . . . . . . . . . . . |[/go]

Obvious two eyes. Failure 3.

So we back up to :b1:

and try 3 or 4 in the last diagram.

Our search has led us to two of the three candidate moves that the two move rule did. And, being humans, we have learned some things along the way which may help us in further search.

Knotwilg · **#20**

Thanks Bill, that's really helpful.

I actually tried zermelo's example with the systematic approach. The first move was easy enough to find that way, but the next was not.

This example is quite good in that the first move is less easy to spot but once you found it, the next move is easy.

So, with some delay, kudos to the OP!

3-move tsumego rule

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