Edge template VI2a

The edge template template VI2

Let us first see what possibilities Red has if he moves first.

There are two obvious options:

In both diagrams the possible intrusion points are marked by (+). So we only have to consider the intersection of the intrusion points. They are:

Intrusion at E5 and F5

If Blue blocks at E5 then Red plays F3, reducing to Template IVb

Likewise if blue blocks at F5:

Intrusion at E6

Red threatens to connect via D4. Blue must respond in one of the marked hexes.

The H4 piece is connected to the bottom with template III-1a, and is connected to the top in two non-overlapping ways:

and

Intrusion at F4

The Red piece at D4 is connected to the bottom. Blue has two direct attempts to block:

Block at F2

Red is now connected to the bottom via template III-1a. Note that neither of Red's threats overlapped.

Block at E3

And now Red can escape the ladder:

And now Red has connected. Attempts by Blue to block the use of the D4 piece as a ladder escape can be shown to not work.

Intrusion at G2

Blue has four options that don't immediately reduce to another edge template:

Block at E4

Red's G3 piece is connected to the top via F3 or H2.

Here Red has created a Ladder escape fork. If Blue blocks the ladder Red plays at D3.

Block at D5

And Red has connected. If blue choose to play at E6 instead of E5:

Block at C6

Block at E6

Play continues...

Intrusion at D6 or F6

The D6 case is shown here, but Red's responses work symmetrically for the F6 case.

Red's F5 piece is connected to the bottom. To prevent its connection to the top, Blue must move in one of the marked tiles.

Block at F4

Or, if for move three Blue played G3:

And Red is connected. This method can be used by Red in the symmetrical case of Blue intruding at F6.

Block at G2

Note that Red's F3 piece is connected via the two marked tiles. If Blue had played G3 for move three:

And Red connects via template III-1a.

Block at H2, G3, or G4

Red's responses are similar in all three cases:

Intrusion at C6 or G6

The G6 case is shown here, but Red's responses work symmetrically for the C6 case.

Red's F4 piece is connected to the bottom via template III-1a. The only move preventing the F4 piece from connecting to the top is G2. Intrusions into the template are met by Red with parallel moves which maintain the connection to the bottom while guaranteeing a connection to the top. F5, F6 and E5 are met by E4 while E4, E6, D5, D6 and C6 are met by G4, connecting Red to both the top and the bottom.

Block at G2

And Red cannot be stopped, the F4 piece being a valid ladder escape. If Blue had played E6 for move five:

And if Blue had played F5 for move five:

Red threatens to connect in two non-overlapping ways, while the E4 piece is connected with template III-1a.