Pivoting template

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.

More precisely, a pivoting template is a pattern that has a stone A and an empty hex B, such that the template's owner can continuously threaten to connect A to the edge until the point where the template's owner either connects A to the edge or occupies B and connects B to the edge. To be considered a "template", its carrier should moreover be minimal with this property.

Example
The following is a pivoting template.



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.

2nd row


3rd row




4th row






5th row




Pivoting templates and flanks
Pivoting templates can be useful in many situations, but are especially useful in connection with flanks. Specifically, if we line up points A and B of any pivoting template with points A and J of a capped flank, we obtain a guaranteed connection to the edge. For example, consider the capped flank  Attaching this on top of one of the above pivoting templates, we get the following:  This guarantees that Red can connect A to the edge, because either A will connect outright, or else B connects to the edge and also to A via the flank.

Weak pivoting templates
There is another notion similar to a pivoting template, but slightly weaker. In a weak pivoting template, we merely require that the template's owner can guarantee to either connect A to the edge or occupy B and connect B to the edge, but we drop the requirement that the owner can "continuously threaten to connect A to the edge until" that point. Typically this means that after the player occupies B, the opponent can still choose whether to let the player connect A or B to the edge.

The following are examples of weak pivoting templates:





Weak pivoting templates are sufficient to form a connection when combined with a flank. However, there are some contexts where a proper pivoting template would connect, but a weak pivoting template does not. The following is an example of this: <hexboard size="9x9" coords="show" edges="all" contents="R h2 g2 f3 f5 c6 B b5 d4 e5 i5 i8           S area(a9,h9,h5,f5,c6,b7)-g5" /> The highlighted area is a weak pivoting template, but with Blue to move, the position is losing for Red. On the other hand, if we use a proper pivoting template in the analogous situation, the position is winning for Red: <hexboard size="9x9" coords="show" edges="all" contents="R h2 g2 f3 f5 B d5 b8 c6 e4 b7 i5 i8           S area(b9,h9,h5,e5,c7)-g5" />