# 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 zipper 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 been filled in:

## Contents

## 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:

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:

Note that climbing along a flank is a generalization of 2nd row ladders, with the cap acting as a ladder escape. Indeed, a board edge can be regarded as a straight row of stones, and is therefore a special kind of flank only made up of F1 pieces:

## Interior templates from capped flanks

There are several ways of constructing interior templates from capped flanks.

### Method 1

The simplest method 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.

### Method 2

Another way to construct interior templates from flanks is to combine a capped flank and the mirror image of a capped 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:

Here is a larger example:

### Method 3

A third way to construct interior templates from flanks is to combine a capped flank with a capped flank rotated by 180 degrees, schematically like this:

+ =or like this:

+ =If Blue plays at one of the hexes marked "J", Red can play at the other to keep the group connected.

Of the named interior templates, the parallelogram and the wide parallelogram are of this form:

But of course, it is again possible to construct infinitely many examples. Here is a larger example:

### Method 4

A fourth way to construct interior templates from flanks is to combine two capped flanks so that their points "J" overlap, with an additional red stone at "x", schematically like this:

+ + =Here again, the hex "J" remains empty. The idea is that if Blue plays in any of the 5 completely blank cells, Red responds at "J"; if Blue plays at "J", Red responds in the center between the two "A"s; in all other cases, Red defends the flanks. The following are examples:

Care must be taken that the carriers of the two flanks don't overlap; they could potentially overlap at the point marked "+". Note that the template guarantees that all red stones will be connected, i.e., not just the groups marked "A", but also "x".

## Edge templates from capped flanks

Not surprisingly, capped flanks (appropriately rotated and positioned) can also be used to construct edge templates. There are various schemas for doing so, and many of them are "flank versions" of standard edge templates. We give several examples. In each schema, we show the starting point "A" and the jumping-off point "J" of the capped flank, and we indicate by "*" the direction in which the flank continues. (Note that we are technically only claiming the validity of the constructed templates, not their minimality. They often do turn out to be minimal as well, but we do not claim that this is true in general.)

E1: E2: E3:

Note that E1, ..., E7 generalize edge templates III2-b, IV2-b, V2-k, V2-a, III2-a, IV2-a, and V2-b, respectively (in each case, the template is obtained from the corresponding schema by immediately capping the flank with cap C1). Here is an example using schema E3 and (an appropriately rotated and mirrored version of) the capped flank F0+F2+F1+F1+F2+C1:

More generally, one can attach a capped flank to any pivoting template to create an edge template.

## Usage example

The following example is from an actual game. Blue to move and win.

Note that Blue's central group is already connected to the left edge by double threat at e3 and c7. But how will Blue connect to the right edge? The problem is that h7 does not normally act as a 2nd row ladder escape. Blue starts at j1, then pushes the 2nd row ladder to j5 and pivots at j7. This forces Red to respond at j6.

Now the killer move is c10. This caps the blue flank, and the entire shaded area becomes an edge template. Blue is now connected by double threat at i6 and b10.