Defending against intrusions in template 1-Va

Here is the proof of Template 1-Va.

Contents 1 Red's threats 1.1 Reduction to template III-1-b 1.2 Reduction to template IVa 1.3 Reduction to template IVb 2 Blue's choice 2.1 Solution to the intrusion at f3 2.2 Solution to the intrusion at d5 2.3 Solution to the intrusion at f5

Red's threats

Reduction to template III-1-b

Reduction to template IVa

Reduction to template IVb

Blue's choice

There are only 3 ways for blue not to fall into one of the previous reduction, we can see it if we consider the intersection of every relevant move in previous diagrams.

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If Blue intrudes in the template at any hex besides the three marked '+', Red makes a move that reduces the situation to a closer template.

Solution to the intrusion at f3

The key move is the response d4. Yielding

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Red is threatening to play at the marked hex which would complete the connection of the g1 piece to the bottom. Blue must block by a playing at some hex between Red's two pieces. Red then plays h2 forcing Blue to block at h3 yielding

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Red has forced the most common pre-ladder formation. Red can get a second row ladder by squeezing through at g3 (Blue blocks at f5). Red's initial key d4 piece acts as a ladder escape which completes the connection. The final position is

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Solution to the intrusion at d5

Red's best response is to two-chain to h2. To stop the threatened immediate connection, Blue must block at h3 or play the forcing move g2 in the two-chain. The first play is defeated by the forcing sequence f4, g2, f2, f3, e3 yielding

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Red is now threatening to connect to the bottom in two non-overlapping ways, by playing e4 or by two-chaining to c4. Blue cannot stop both threats with a single move. The other play, g2 (after Red's h2) is defeated by the forcing sequence h1, h3, f4, g3, f2, f3, e3 yielding

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Red is again threatening to connect to the bottom in the same two non-overlapping ways: by playing at e4 or two-chaining to c4. Blue cannot stop both threats with a single move.

Solution to the intrusion at f5

Red's best response is play e4 which is connected to the bottom and forms a loose connection with the g1 piece. To stop the immediate connection, Blue must play in the middle of the loose connection at one of the hexes marked "+" in the following diagram.

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The move f2 is defeated by the forcing sequence g2, f3, g3 yielding

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Red is threatening to connect to the bottom in two non-overlapping ways: by playing at f4 or two-chaining to h4. The alternative response, f3, is defeated by the forcing sequence e2, d5, e5, e3, d3 yielding

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Red is threatening to connect to the bottom in two non-overlapping ways, by playing at d4 or c4.

Note that template Va occurs in a mirror-image form (in the mirror image form, the three hexes on the 5th row (from the bottom) are shifted over 1 hex to the G, H, and I columns). It may seem that this template is very strong because it reaches 5 rows into the board but it rarely occurs because of the huge size of the template; the template requires 31 empty hexes and 10 hexes along an edge — the entire edge on the 10x10 board!

Furthermore, the large perimeter makes it more vulnerable to encroaching adjacent plays and forcing moves. Additionally, template area surrounds the 5th row piece on both "shoulders" so that non-overlapping plays from the 5th row piece can occur in only two directions.