User:Selinger
Proposed article: Pivoting template
The problem with this article draft, as currently written, is that the stated condition, "Red can either connect A to the edge, or else occupy and connect B to the edge", is weaker than what many of the templates satisfy. Most templates satisfy the stronger condition "Red can continuously threaten to connect A until Red occupies and connects B to the edge." There are situations where templates satisfying the weaker condition would be losing, but templates satisfying the stronger condition are winning. So one either needs several different notions, or specify the strength of each template. or state the stronger condition and select only templates that satisfy it. All choices seem awkward.
A pivoting template is a kind of edge template that guarantees that the template's owner can either connect the template's stone(s) to the edge, or else can occupy a specified empty hex and connect it to the edge.
Example
This template guarantees that, with Blue to move, Red can either connect A to the edge, or else occupy and connect B to the edge. Its carrier is minimal with this property.
Proof: Red's main threat is to bridge to c and connect to the edge by ziggurat or edge template III1b. Therefore, to prevent Red from connecting to the edge outright, Blue must play in one of the cells a,...,g.
If Blue plays at a, Red responds at b and connects outright by edge template IV1a.
If Blue plays at b, Red responds with a 3rd row ladder escape fork:
If Blue plays at c, d, or f, Red responds as follows and is connected by edge template V2f. If Blue plays on the right instead of 3, Red responds as if defending template V2f.
If Blue plays at e or g, Red responds at c and gets a 2nd or 3rd row ladder, which can reach B by ladder escape fork.
Usage
Pivoting templates can be useful in many situations, but are especially useful in connection with flanks.
[Todo: Add an example.]
