I wonder if any one has solved the fewest stones to break all ladders problem. This is probably pretty vague, maybe just any ladder from the third or forth line going inwards. I am thinking sort of analogously to the queens problem, placing 8 queens such that they weren't mutually attacking.

Useful factoid: san-ren-sei (three star points) breaks all ladders.

Yes. The handicap points are exactly at the maximum distance from each other so that they break all ladders. If the space between them were one wider, a ladder could pass though the gap.

Therefore, 8 stones on the corner and side star points break all ladders going up from anywhere.

Note that the stones need to be on the fourth line for this to work. On the third line:

Click Here To Show Diagram Code

[go]$$W White can bend the ladder at the last moment.
$$ ----------------------+
$$ . . . . . . . . . . . |
$$ . . . . O X 1 . . . . |
$$ . # . O X X O # . . . |
$$ . , O X X O . , . . . |
$$ . . . O O . . . . . . |
$$ . . . . . . . . . . . |
$$ . . . . . . . . . . . |
$$ . . . . . . . . . . . |[/go]

breaks all ladders going one direction.

So an 8 stone handicap breaks all ladders? If W plays tengen first does that enable them all?

_________________
Still officially AGA 5d but I play so irregularly these days that I am probably only 3d or 4d over the board (but hopefully still 5d in terms of knowledge, theory and the ability to contribute).

Click Here To Show Diagram Code

[go]$$c
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . X . . . . . X . . . . . X . O . |
$$ | . . . . . . . . . . . . . . . . O X . |
$$ | . . . . . . . . . . . . . . . O X X O |
$$ | . . . . . . . . . . . . . . O X X O . |
$$ | . . . . . . . . . . . . . O X X O . . |
$$ | . . . . . . . . . . . . . . O O . . . |
$$ | . . . X . . . . . X . . . . . X . . . |
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$$ | . . . X . . . . . X . . . . . X . . . |
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$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Probably we can all agree the first diagram is a ladder. If so, we can't use stones on the third line....

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[go]$$Wc A Ladder
$$+ - - - - - -
$$| . 2 3 . . .
$$| 1 X O . . .
$$| . O X . X .
$$| . . . , . .
$$| . . X . . .
$$| . . . . . .[/go]

If the next diagram is also a 'ladder' we can't use the 3-2 point.

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[go]$$Wc A Ladder?
$$+ - - - - - -
$$| 2 3 . . . .
$$| X 1 X . X .
$$| O . . . . .
$$| . X . , . .
$$| . . . . . .
$$| . X . . . .[/go]

However, it seems to my eye that filling the second row with alternate stones as below will break all ladders. This requires four rows of eight stones or 32 stones in total.

Click Here To Show Diagram Code

[go]$$Bc All Ladders Broken?
$$+ - - - - - -
$$| . . . . . .
$$| . X . X . X
$$| . . . . . .
$$| . X . , . .
$$| . . . . . .
$$| . X . . . .[/go]

_________________
Dave Sigaty
"Short-lived are both the praiser and the praised, and rememberer and the remembered..."
- Marcus Aurelius; Meditations, VIII 21

Click Here To Show Diagram Code

[go]$$Wc All Ladders Broken?
$$+ - - - - - -
$$| . . . . . .
$$| . X O X 1 X
$$| . . . O . .
$$| . X . , . .
$$| . . . . . .
$$| . X . . . .[/go]

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

So where do we go from here?

Click Here To Show Diagram Code

[go]$$Wc All Ladders Broken?
$$+ - - - - - -
$$| . O X 1 X .
$$| . X O . . .
$$| X . . . . .
$$| . . . , . .
$$| X . . . . .
$$| . . . . . .[/go]

I'm assuming a rather different idea about the role of ladder breakers.

Edit: OK, I think I see your point, mine isn't a ladder. So we can't really get away from the first line?

_________________
Dave Sigaty
"Short-lived are both the praiser and the praised, and rememberer and the remembered..."
- Marcus Aurelius; Meditations, VIII 21

I was thinking about posing a subtly different question. What placement of stones gives maximal ladder breakage for X stones. The easy case I think would be for one stone, at tengen. Two stones? Three stones etc.

We could consider some kind of minimal ladders, for instance a bend, block, bend pattern.

Click Here To Show Diagram Code

[go]$$Wc A Minimal Ladder?
$$+ - - - - - -
$$| . . O . . .
$$| . O B . . .
$$| O X B W . .
$$| O X O . . .
$$| . O . . . .
$$| . . . . . .[/go]

I'm not saying that's the "true" meaning of ladder or anything like that. But it might be an interesting question, and perhaps we can avoid having to put as many stones on the board as Bill had to

_________________
Occupy Babel!

Click Here To Show Diagram Code

[go]$$c
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
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$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . X . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------
$$ {LN D19 T4}
$$ {LN A16 Q1}
$$ {LN Q19 A4}
$$ {LN T16 D1}[/go]

Click Here To Show Diagram Code

[go]$$c
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . X . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . X . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------
$$ {LN A1 T19}
$$ {LN A7 N19}
$$ {LN T13 G1}
$$ {LN A19 T1}
$$ {LN A13 N1}
$$ {LN G19 T7}[/go]

For more stones I think directionality and edge cases might come into play, as well as overlap.

Here is a configuration for three with minimum overlap, I would have to count to figure out the optimum centering.

Click Here To Show Diagram Code

[go]$$c
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . X . . . . . X . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . X . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------
$$ {LN A2 S19}
$$ {LN A8 M19}
$$ {LN A16 Q1}
$$ {LN D19 T4}
$$ {LN A10 K1}
$$ {LN K19 T10}
$$ {LN A14 F19}
$$ {LN F1 T14}[/go]

Click Here To Show Diagram Code

[go]$$c
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . X . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . X . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------
$$ {LN A1 T19}
$$ {LN A7 N19}
$$ {LN A13 N1}
$$ {LN A19 T1}
$$ {LN A7 G1}
$$ {LN G19 T7}
$$ {LN A13 G19}
$$ {LN G1 T13}
$$ {LN N19 T13}
$$ {LN N1 T7}[/go]

There that is better. Every position is covered for ladders in some direction, a majority of the board is covered in two directions. I suspect tengen would work its way back into the picture at five stones and triple directional coverage.

@SmoothOper: Please define 'ladder'.

_________________
Dave Sigaty
"Short-lived are both the praiser and the praised, and rememberer and the remembered..."
- Marcus Aurelius; Meditations, VIII 21

I guess 7-4's would do pretty well, but not any more efficiently than star points.

Click Here To Show Diagram Code

[go]$$c
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . X . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . X . . . . . . . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . X . . . . . . . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . X . . . . . X . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------
$$ {LN A1 T19}
$$ {LN A7 N19}
$$ {LN A13 N1}
$$ {LN A19 T1}
$$ {LN A7 G1}
$$ {LN G19 T7}
$$ {LN A13 G19}
$$ {LN G1 T13}
$$ {LN N19 T13}
$$ {LN N1 T7}[/go]