Second order template

A second order template is a template that guarantees a connection even if the opponent starts with two moves in the template. Put another way, a second order template is a pattern in which an intrusion is not a forcing move. A pattern can be proved to be a second order template by showing that every possible intrusion preserves at least one first order template. To qualify as a second order template, the pattern should also be minimal.

Phrases such as "Blue cannot even threaten to disconnect Red" and "if Blue moves in this area, Red can just ignore it" are often used to indicate that Red has a second order template.

Examples

Third row

Red has three non-overlapping threats:

So no matter where Blue plays in the template, Red will still have at least two non-overlapping threats, and therefore a first-order template, remaining. Or to put it another way: If Blue makes two moves in the template, Blue can disable at most two of Red's threats, so Red can still use the third one to reconnect.

Here is another second order template on the third row:

Fourth row

The following is a second order template:

Usage

In play

It is usually not a good idea to create a second order template on purpose, as this tends to waste a move that would be better spent elsewhere. However, it is still useful to recognize second order templates in case they form accidentally.

When the opponent intrudes into a first order template, it is usually necessary to defend the template to preserve the connection. The opponent can take advantage of this by playing template intrusions that will later be useful to the opponent, for example as ladder escapes or to gain territory. Such moves belong to the category of double threats.

On the other hand, when the opponent intrudes into a second order template, no immediate response is necessary; the template's owner can simply ignore the intrusion and is free to move elsewhere, thereby gaining the initiative. Recognizing second order templates helps to know whether an area is safe or might be subject to threats.

In mustplay analysis

Second order templates can sometimes be useful in the analysis of Hex positions, such as mustplay analysis. For example, suppose we want to prove the correctness of the following 6th row (first order) edge template:

We can reason as follows: if Blue plays anywhere in the template except a, b, or c, then Red can play at c, forming the second order template

Since Blue has at most one stone in this template, the result is still a first-order template, so that Red is connected to the edge. Therefore, the only intrusions we need to consider are a, b, and c. This greatly simplifies the analysis, as we must now only consider 3 possible intrusions, rather than all 22 of them. (The intrusions at a or c are easily dealt with, since Red can simply respond at b to connect via edge template IV2a. The final intrusion at b has a few further cases to consider, but is relatively straighforward).

Second order ladder creation templates

The following is a second order ladder creation template.

What this means is that even if Blue is allowed two moves in the template, Red can still guarantee to either connect to the edge or get the indicated 2nd row ladder.