# Pairing strategy

A **pairing strategy** for a player is specified by a set of pairs of empty cells. Each cell may belong to at most one pair, and not all empty cells need to be part of a pair. The player then plays as follows: whenever the opponent occupies one cell of a pair, the player occupies the other.

Pairing strategies are second-player strategies; they work even if the other player goes first.

Pairing strategies are useful because they are simple to describe and easy to carry out. If a game reaches a point where one of the players can win by a pairing strategy, the game can be considered to be won by that player (unless the player is so weak that they do not see the winning strategy).

A pairing strategy can be described succinctly by numbering the cells, giving the same number to both cells of a pair.

## Contents

## Examples

In the following examples, the pairing strategy is always for Red.

### Interior templates with pairing strategies

Most interior templates have pairing strategies. In fact, with the exception of the hammock, all of the templates listed on the interior template page have pairing strategies. The simplest template with a pairing strategy is the bridge:

In the bridge, there is only one pair, and if Blue plays in one cell, Red plays in the other. The following are more examples of interior templates with pairing strategies. Many of the templates admit more than one pairing strategy; we only give one for each.

**The wheel**

**The trapezoid**

**The crescent**

**The span**

**The scooter**

**The long crescent**

### Edge templates with pairing strategies

Most edge templates do not admit pairing strategies, but some of the smaller ones do. Here are some examples.

**The ziggurat**

### Non-square boards

As shown in more detail on the page on Hex theory, on boards of size *n*×*m* where *n* ≠ *m*, the player with the shorter distance between their edges can win by a simple pairing strategy, even if the other player goes first:

## Non-existence of pairing strategies

An example of an interior template that does not admit a pairing strategy is the hammock:

To see why, assume there was such a strategy. Then clearly *a* must be paired with *b*, or else Blue could get both *a* and *b* and disconnect the template's leftmost stone. For similar reasons, *c* must be paired with *d*, or else Blue could get *a*, *c*, and *d*. Now since Blue can get *b* and *c*, *e* must be paired with *f*. On the other hand, since Blue can get *a* and *c*, *e* must be paired with *g*. Thus, we arrived at a contradiction, showing that there is no pairing strategy.

## Pairing strategies in Reverse Hex

Pairing strategies can also be used in Reverse Hex to force the opponent to make a connection. See Reverse Hex for more details.