Difference between revisions of "Pivoting template"

Latest revision as of 02:51, 27 June 2026

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.

List of pivoting templates

2nd row

3rd row

4th row

5th row

6th row

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:

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:

Weak pivoting templates with an extra semi-connection

Suppose apart from the (weak) pivoting template, there is an extra semi-connection between A and B that does not overlap with the template, then the following holds when taking the semi-connection into consideration.

Connecting B with sente

If Blue decides to block Red from connecting A, then Red not only gets to connect B, but also gets a sente. This is because once the connection from Red B to the edge is ensured, Blue will have to spend a move blocking B from connecting back to A through the semi-connection.

Deriving pre-templates

Placing a red stone at B results into a pre-template connecting A to the edge. This is because, by definition, if Blue tries to block A from the edge, Red will play B at some moment from the strategy of the original (weak) pivoting template. However, if Red already has a stone at B, they can spend that move connecting B back to A through the semi-connection, and the rest of the strategy will connect either A or B to the edge, which means A is gauranteed to be connected to the edge.

It is not true that the pre-template is always a template. For example, if we consider the following pivoting template with a red stone at B and with an extra cell at *, then the result is not a template; the cells with + can be removed.

Weak pivoting templates with an extra virtual connection

Suppose apart from the (weak) pivoting template, there is an extra virtual connection between A and B that does not overlap with the template, then they form a pre-template of A connecting to the edge. This is because if Blue tries to block A from the edge, Red will play B at some moment. Now A and B are connected through the virtual connection, and the rest of the strategy will connect either A or B to the edge, which means A is gauranteed to be connected to the edge.

Weak pivoting templates and capped flanks

Specifically, if we line up points A and B of any (weak) 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 (weak) pivoting templates, we get the following:

This guarantees that Red can connect A to the edge. These pre-templates make up a subset of a larger group of edge templates formed with capped flanks.

Pivoting ladder creation templates

Sometimes, the pivoting property can be combined with other properties of templates. For example, a ladder creation template guarantees that Red can get at least a specified ladder. A pivoting ladder creation template is a template with a red stone A and an empty hex B, such that Red can continuously threaten to get a ladder from A, until the point where Red either gets the ladder or occupies and connects B to a ladder. When combined with an appropriate ladder escape, a pivoting ladder creation template becomes a pivoting template.

For example, the following is a pivoting 3rd-row-ladder creation template.

The only way for Blue to prevent A from connecting outright or getting the ladder is to play at 1. In this case, Red responds as follows:

See also

• Climbing
• Flank