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, and with a certain amount of space on one side, 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 "B" and "C" are connected to opposite edges.

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

It is not actually necessary for Red to play moves 6, 12, and 16; Red could also skip these moves. However, they usually do not hurt and may be useful to Red by solidifying Red's position below the flank.

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, the following definition refers to red flanks that are oriented left-to-right and facing upward.

We can define such flanks inductively as follows:

• Base case: A single red stone, together with the indicated space, is a flank. The stone marked "+" is both the starting point and the endpoint of the flank.
  F0:

• Induction step: A flank can be extended with any of the following patterns:
  F1:
  F2:
  F3:
  
  Here, "−" denotes the previous endpoint, and "+" denotes the new endpoint. The orientation of these patterns matters, i.e., they cannot be rotated.

Here is an example of the flank obtained by starting with F0 and then extending with F1, F1, F3, F1, F2, F3, and F1. We always use "A" to denote the starting point and "B" to denote the endpoint of the flank:

We can also use algebraic notation to denote flanks. For example, we write F0+F1+F1+F3+F1+F2+F3+F1 for the above flank.

Capped flank

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

C1:

C2:

C3:

C4:

Here, "B" denotes the original endpoint of the flank. Other cap patterns are also possible; the above C1–C4 are just some common examples of caps.

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

F0+F1+C1:

F0+F2+C2:

F0+F2+F2+F3+F2+C1:

For any flank, let "C" be the hex that is positioned relative to the flank's starting point "A" as follows:

If Red plays at "C" and climbs along a capped flank, Red will connect. Consequently, any capped flank, with a red stone added at position "C", is a connected group.

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.