# 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.

# Proposed article: Mustplay sets

Informally, a player's mustplay set is the set of cells in which the player must move to avoid losing immediately.

## Example

Consider the following position, with Red to move:

To determine Red's mustplay set, Red should consider the possible ways in which Blue could make a connection if it were Blue's turn. The two most obvious ways for Blue to connect are at x and y. These are called Blue's main threats.

If Blue plays at x, Blue is connected via edge template II, a bridges and a ziggurat. The carrier of Blue's connection, i.e., the set of cells that are part of Blue's connection, is shown:

Similarly, if Blue plays at y, Blue is connected with the following carrier:

TODO:

• complete this example
• create a better example
• explain what the mustplay set does and does not guarantee, and how it is used to narrow the possibilities to consider.
• explain what an empty mustplay set means.
• application: foiling
• application: puzzle 20210618-5
• explain "carrier"
• example with ladder creation + escape.