User:Selinger

Proposed article: Flank

A flank is a sequence of friendly stones that are either adjacent or linked by bridges in a certain way, for example like this:

Apart from ladders, flanks are one of the most common "long-distance" patterns occuring in Hex. They are useful for climbing, and they can be used to form large interior and edge templates.

What makes a flank useful is that its owner can use it for climbing. For example, consider the following situation, and assume the stones "A" and "B" are connected to opposite edges.

Then Blue can climb all the way from A to the cell marked "*", by a sequence of forcing moves as follows:

Intruding into the flank's bridges does not help the opponent. The flank still works even if all the bridges have already been filled in:

Definition

A flank can belong to Red or to Blue, and it can be oriented in any of the 6 cardinal directions of the Hex board (a cardinal direction is parallel to an edge or to the short diagonal). In addition, it can be facing up or down (the side it is facing is the side where the empty space is). For simplicity, in the following, we will only consider blue flanks that are oriented left-to-right and facing upward.

We can define such flanks as follows:

• A single blue stone is a flank. The stone is both the starting point and the endpoint.

• A flank with endpoint x can be extended with any of the following patterns:
  
  
  Here, "−" denotes the previous endpoint, and "+" denotes the new endpoint.

Here is an example of a flank with starting point "A" and endpoint "B":

Capped flank

A flank is capped if it has been extended past its endpoint "B" with one of the following patterns:

Here, "B" denotes the original endpoint of the flank. The following are some examples of capped flanks. In each case, the flank's starting point "A" and original endpoint "B" are shown.

ADD SOME EXAMPLES HERE.

If Blue climbs along a capped flank, Blue will connect.

ADD EXAMPLES.

POINT OUT HOW THIS GENERALIZES A 2ND ROW LADDER, WITH THE FLANK GENERALIZING THE "EDGE" AND THE CAP GENERALIZING A LADDER ESCAPE.

Interior templates from capped flanks

Consider a capped flank with starting point "A", and suppose the hex marked "*" is also occupied by Blue:

Then, given the right amount of space, the hex marked "*" together with the capped flank forms an interior templates.

ADD SOME EXAMPLES HERE. ALSO EXPLAIN MORE CAREFULLY WHAT IS THE "RIGHT" AMOUNT OF SPACE.

Moreover, two capped flanks growing in opposite directions from an empty hex and facing the same way form an interior template.

ADD EXAMPLE.

Edge templates from capped flanks

ADD EXAMPLES.

Usage example

FROM A GAME.

3rd row ladders along flanks

Above, we pointed out that climbing along a flank is analogous to a 2nd row ladder. It is similarly possible to climb along a flank at a greater distance. In other words, there is an analog of a 3rd row ladder along a flank. This requires slightly more space, and if the ladder is to connect, it requires a different kind of cap (or ladder escape).

ADD EXAMPLE.