# 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.]

# Proposed article: Mustplay region

Informally, a mustplay region for a player is a set of cells in which the player must move to avoid losing immediately. Mustplay analysis is an important tool for analyzing Hex positions, because it can help narrow down the number of possibilities a player must consider.

## Example

Consider the following position, with Red to move:

To determine Red's mustplay region, Red should consider the possible ways in which Blue could make a connection if it were Blue's turn. These are called Blue's threats. Blue has (at least) the following threats:

• If Blue plays at e5, then Blue is connected via two copies of edge template II and two bridges, as shown:
• Alternatively, if Blue plays at e5, Blue is also connected via edge template II and edge template III2e, as shown:
While the last two connections both use a Blue stone at e5, they have different carriers.
• If Blue plays at e4, Blue is connected via a 3rd row ladder, using f6 as a ladder escape. In this case, the carrier consists of the path the ladder will take, as well as the ladder escape template:

Red's mustplay region consists of those empty cells that are in the carriers of all of Blue's known threats. Therefore, Red's mustplay region consists of the cells a4, a5, e5, f5, g5, and g6.

Note that this does not mean that all of a4, a5, e5, f5, g5, and g6 are winning moves for Red, or even that Red has any winning moves at all. Rather, what it means is that all other moves are losing. In other words, if Red has any winning moves at all, they must be in Red's mustplay region. Red must now consider each of the six moves a4, a5, e5, f5, g5, and g6 and check if any of them are winning, or barring that, which one of them is least likely to be losing.

To help narrow down Red's choices even further, it helps to consider captured and dominated cells. In the above example, a4, a5, g5, and g6 are captured by Blue, and therefore, Red should not play there. This leaves Red with e5 and f5 as the only possible moves to consider. It so happens that e5 is winning and f5 is losing. Therefore, considering the mustplay region has helped Red identify the only possible winning move. Red will play e5 and win the game.

## Properties of the mustplay region

• All moves outside a player's mustplay region are losing. Moves within the mustplay region may be winning or losing.
• If a player's mustplay region is empty, the player is losing.
• If there are no winning moves in a player's mustplay region, the player is losing.
• The mustplay region is not unique. By considering more opponent threats, a player may arrive at a smaller mustplay region.

## Example: no winning move

If there are no winning moves in a player's mustplay region, the player is losing. To illustrate this, consider the following position, with Red to move.

Blue's main threats are:

The only empty cell in the carrier of all three threats is a4, hence Red's mustplay region consists of a4. This means that all moves except possibly a4 are losing for Red.

Unfortunately for Red, a4 is also losing, because if Red plays a4, Blue can win as follows:

Therefore Red has no winning moves at all and is losing the game.

TODO:

• application: foiling
• application: puzzle 20210618-5
• application: verification of templates
• example with ladder creation + escape.
• mustplay region in computer Hex