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 "J" are connected to opposite edges.

Then Red can climb all the way from J 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.

Each flank has three distinguished points: a starting point, which we usually mark "A", an endpoint, which we usually mark "B", and a jumping-off point, which we mark "J". We can define flanks inductively as follows:

• Base case: A single red stone, together with the indicated space, is a flank. In this case, the stone marked "B" is both the starting point and the endpoint. The jumping-off point "J" is also shown.
  F0:

• Induction step: A flank can be extended with any of the following patterns:
  F1:
  F2:
  F3:
  
  Here, "−" denotes the previous endpoint, and "B" 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:

The point of capped flanks is that if Red plays at the jumping-off point "J" of any capped flank, Red can connect:

Todo: 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

There are several ways of constructing interior templates from capped flanks. The simplest is to add a Red piece to the jumping-off point "J". Since this connects to the rest of the flank. Such a group can be viewed as a (potentially very large) interior template.

Many of the named interior templates are of this form. This is the case for the crescent, trapezoid (in more than one way), scooter, bicycle, as well as the long crescent and various long trapezoids.

Here is a larger template constructed by the same method.

Another way to construct interior templates from capped flanks is to combine a flank and the mirror image of a flank so that they overlap at the point "J", schematically like this:

+

=

Here, the hex "J" remains empty. The point is that if Blue plays at "x", Red plays at "J", and vice versa.

Several of the named interior templates are of this form. This is the case for the span, the box, the shopping cart, and the long span:

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.