Difference between revisions of "Edge template VI2a"

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m (→‎The edge template VI2: corrected an inner link (operlapping -> overlapping))
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== The [[edge template]] [[template VI2|VI2]] ==
+
Edge template IV2a is a 6th row [[edge template]] with two stones.
  
−
<hex>R6 C10 Vg1 Vh1 Sa1 Sa2 Sa3 Sa4 Sa5 Sb1 Sb2 Sb3 Sb4 Sc1 Sc2 Sc3 Sd1 Sd2 Se1 Sj1 Sj2 Sj3</hex>
+
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R ↑:g1 h1"
 +
/>
 +
 
 +
== Defending the template ==
  
 
Let us first see what possibilities [[Red (player)|Red]] has if he moves first.
 
Let us first see what possibilities [[Red (player)|Red]] has if he moves first.
Line 7: Line 14:
 
There are two obvious options:
 
There are two obvious options:
  
−
<hex>R6 C10 Vg1 Vh1 Sa1 Sa2 Sa3 Sa4 Sa5 Sb1 Sb2 Sb3 Sb4 Sc1 Sc2 Sc3 Sd1 Sd2 Se1 Sj1 Sj2 Sj3 Vf3 Pa6 Pb5 Pb6 Pc4 Pc5 Pc6 Pd4 Pd5 Pd6 Pe3 Pe4 Pe5 Pe6 Pf2 Pf4 Pf5 Pf6 Pg2 Pg4 Pg5 Pg6</hex>
+
<hexboard size="6x9"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R f3 g1 h1 E +:a6 +:b5 +:b6 +:c4 +:c5 +:c6 +:d4 +:d5 +:d6 +:e3 +:e4 +:e5 +:e6 +:f2 +:f4 +:f5 +:f6 +:g2 +:g4 +:g5 +:g6"
 +
/>
  
−
<hex>R6 C10 Vg1 Vh1 Sa1 Sa2 Sa3 Sa4 Sa5 Sb1 Sb2 Sb3 Sb4 Sc1 Sc2 Sc3 Sd1 Sd2 Se1 Sj1 Sj2 Sj3 Vg3 Pc6 Pd5 Pd6 Pe4 Pe5 Pe6 Pf4 Pf5 Pf6 Pg4 Pg5 Pg6 Ph3 Ph4 Ph5 Ph6 Pi4 Pi5 Pi6 Pg2 Ph2</hex>
+
<hexboard size="6x9"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R g1 g3 h1 E +:c6 +:d5 +:d6 +:e4 +:e5 +:e6 +:f4 +:f5 +:f6 +:g2 +:g4 +:g5 +:g6 +:h2 +:h3 +:h4 +:h5 +:h6 +:i4 +:i5 +:i6"
 +
/>
  
 
In both diagrams the possible [[Template intrusion|intrusion]] points are marked by (+). So we only have to consider the [[Overlapping templates|intersection of the intrusion points]]. They are:
 
In both diagrams the possible [[Template intrusion|intrusion]] points are marked by (+). So we only have to consider the [[Overlapping templates|intersection of the intrusion points]]. They are:
  
−
<hex>R6 C10 Vg1 Vh1 Sa1 Sa2 Sa3 Sa4 Sa5 Sb1 Sb2 Sb3 Sb4 Sc1 Sc2 Sc3 Sd1 Sd2 Se1 Sj1 Sj2 Sj3 Pc6 Pd5 Pd6 Pe4 Pe5 Pe6 Pf4 Pf5 Pf6 Pg4 Pg5 Pg6 Pg2</hex>
+
<hexboard size="6x9"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R g1 h1 E +:c6 +:d5 +:d6 +:e4 +:e5 +:e6 +:f4 +:f5 +:f6 +:g2 +:g4 +:g5 +:g6"
 +
/>
 +
 
 +
=== Intrusion at E5 and F5 ===
 +
 
 +
If Blue blocks at E5 then Red plays F3, reducing to [[Template IVb]]
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 2:f3 g1 h1 B 1:e5 E +:a6 +:b5 +:b6 +:c4 +:c5 +:c6 +:d4 +:d5 +:d6 +:e3 +:e4 +:e6 +:f4 +:f5 +:f6 +:g3 +:g4 +:g5 +:g6 +:h4 +:h5 +:h6"
 +
/>
 +
 
 +
Likewise if blue blocks at F5:
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R g1 2:g3 h1 B 1:f5 E +:b6 +:c5 +:c6 +:d4 +:d5 +:d6 +:e4 +:e5 +:e6 +:f3 +:f4 +:f6 +:g4 +:g5 +:g6 +:h3 +:h4 +:h5 +:h6 +:i4 +:i5 +:i6"
 +
/>
 +
 
 +
=== Intrusion at E6 ===
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 2:f3 g1 h1 B 1:e6 E +:a6 +:b5 +:b6 +:c4 +:c5 +:c6 +:d4 +:d5 +:d6 +:e3 +:e4"
 +
/>
 +
 
 +
Red threatens to connect via D4. Blue must respond in one of the marked hexes.
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 2:f3 g1 h1 4:h4 B 3:e4 1:e6"
 +
/>
 +
 
 +
The H4 piece is connected to the bottom with [[ziggurat|template III-1a]], and is connected to the top in two non-overlapping ways:
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 2:f3 g1 h1 4:h4 i2 B 3:e4 1:e6 E +:h2 +:h3 +:i1 +:i3"
 +
/>
 +
 
 +
and
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 2:f3 g1 g4 h1 4:h4 B 3:e4 1:e6 E +:f2 +:f4 +:g2 +:g3"
 +
/>
 +
 
 +
=== Intrusion at F4 ===
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 2:d4 g1 h1 B 1:f4"
 +
/>
 +
 
 +
The Red piece at D4 is connected to the bottom. Blue has two direct attempts to block:
 +
 
 +
==== Block at F2 ====
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R d4 g1 2:g3 h1 4:h4 B 1:f2 3:f3 f4"
 +
/>
 +
 
 +
Red is now connected to the bottom via [[ziggurat|template III-1a]]. Note that neither of Red's threats overlapped.
 +
 
 +
==== Block at E3 ====
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R d4 2:f3 g1 4:g4 h1 10:h4 6:i2 8:i3 B 1:e3 3:e4 f4 5:g3 7:h3 9:h5"
 +
/>
 +
 
 +
And now Red can escape the ladder:
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R d4 6:d5 4:e5 f3 2:f5 g1 g4 h1 h4 i2 i3 B 5:d6 e3 e4 3:e6 f4 1:f6 g3 h3 h5"
 +
/>
 +
 
 +
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 ===
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 2:f2 g1 h1 B 1:g2"
 +
/>
 +
 
 +
Blue has four options that don't immediately reduce to another edge template:
 +
==== Block at E4 ====
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R f2 g1 2:g3 h1 B 1:e4 g2"
 +
/>
 +
 
 +
Red's G3 piece is connected to the top via F3 or H2.
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 4:c5 f2 2:f4 g1 g3 h1 B e4 3:e6 g2 1:g4"
 +
/>
 +
 
 +
Here Red has created a [[Ladder escape fork]]. If Blue blocks the ladder Red plays at D3.
 +
 
 +
==== Block at D5 ====
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 2:e4 f2 g1 4:g4 h1 6:h2 B 1:d5 3:e5 5:f4 g2"
 +
/>
 +
 
 +
And Red has connected. If blue choose to play at E6 instead of E5:
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R e4 2:e5 f2 g1 6:g4 4:g5 h1 8:h2 B d5 3:d6 1:e6 7:f4 5:f5 g2"
 +
/>
 +
 
 +
==== Block at C6 ====
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 2:e4 f2 g1 4:g4 h1 6:h2 B 1:c6 3:e5 5:f4 g2"
 +
/>
 +
 
 +
==== Block at E6 ====
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 2:e4 4:e5 f2 g1 6:g5 h1 B 3:d5 5:d6 1:e6 g2"
 +
/>
 +
 
 +
Play continues...
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R e4 e5 f2 g1 2:g4 g5 h1 4:h2 B d5 d6 e6 3:f4 1:f5 g2"
 +
/>
 +
 
 +
=== Intrusion at D6 or F6 ===
 +
 
 +
The D6 case is shown here, but Red's responses work symmetrically for the F6 case.
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 2:f5 g1 h1 B 1:d6 E +:f4 +:g2 +:g3 +:g4 +:h2"
 +
/>
 +
 
 +
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 ====
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 8:c5 6:e4 4:f3 f5 g1 2:g4 h1 B d6 7:e5 1:f4 5:g3 3:h2"
 +
/>
 +
 
 +
Or, if for move three Blue played G3:
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R f5 g1 g4 10:g5 h1 8:h5 2:i2 4:i3 6:i4 B d6 f4 1:g3 9:g6 3:h3 5:h4 7:h6"
 +
/>
 +
 
 +
And Red is connected. This method can be used by Red in the symmetrical case of Blue intruding at F6.
 +
 
 +
==== Block at G2 ====
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 8:c5 6:e4 4:f3 f5 g1 h1 2:h2 B d6 7:e5 5:f4 1:g2 3:g4 E +:f2 +:g3"
 +
/>
 +
 
 +
Note that Red's F3 piece is connected via the two marked tiles. If Blue had played G3 for move three:
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R f5 g1 h1 2:h2 4:h3 6:h4 B d6 1:g2 3:g3 5:g4"
 +
/>
 +
 
 +
And Red connects via [[ziggurat|template III-1a]].
 +
 
 +
==== Block at H2, G3, or G4 ====
 +
 
 +
Red's responses are similar in all three cases:
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 6:c5 4:e4 2:f3 f5 g1 h1 B d6 5:e5 3:f4 1:h2"
 +
/>
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 6:c5 4:e4 2:f3 f5 g1 h1 B d6 5:e5 3:f4 1:g3"
 +
/>
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 6:c5 4:e4 2:f3 f5 g1 h1 B d6 5:e5 3:f4 1:g4"
 +
/>
 +
 
 +
=== Intrusion at C6 or G6 ===
 +
 
 +
The G6 case is shown here, but Red's responses work symmetrically for the C6 case.
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 2:f4 g1 h1 B 1:g6 E +:c6 +:d5 +:d6 +:e4 +:e5 +:e6 +:f5 +:f6"
 +
/>
 +
 
 +
Red's F4 piece is connected to the bottom via [[ziggurat|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 ====
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R f4 g1 h1 2:h2 4:h3 6:h4 B 1:g2 3:g3 5:g4 g6"
 +
/>
 +
 
 +
And Red cannot be stopped, the F4 piece being a valid ladder escape. If Blue had played E6 for move five:
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 8:d5 f4 g1 6:g4 h1 2:h2 4:h3 B 5:e6 7:f6 1:g2 3:g3 g6"
 +
/>
 +
 
 +
And if Blue had played F5 for move five:
 +
 
 +
<hexboard size="6x9"
 +
  coords="bottom right"
 +
  edges="bottom"
 +
  visible="area(a6,f1,i1,i6)"
 +
  contents="R 6:e4 f4 g1 h1 2:h2 4:h3 B 5:f5 1:g2 3:g3 g6 E +:e3 +:f2 +:f3 +:g4"
 +
/>
 +
 
 +
Red threatens to connect in two non-overlapping ways, while the E4 piece is connected with [[ziggurat|template III-1a]].
  
−
''More on this later...''
+
[[Category:Edge templates]]

Latest revision as of 14:36, 11 May 2023

Edge template IV2a is a 6th row edge template with two stones.

abcdefghi123456

Defending the template

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

abcdefghi12345621

Likewise if blue blocks at F5:

abcdefghi12345621

Intrusion at E6

abcdefghi12345621

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

abcdefghi1234562341

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

abcdefghi1234562341

and

abcdefghi1234562341

Intrusion at F4

abcdefghi12345621

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

Block at F2

abcdefghi1234561324

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

Block at E3

abcdefghi12345661257834109

And now Red can escape the ladder:

abcdefghi123456642531

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

abcdefghi12345621

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

Block at E4

abcdefghi12345621

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

abcdefghi1234562143

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

Block at D5

abcdefghi123456625413

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

abcdefghi12345687625431

Block at C6

abcdefghi123456625431

Block at E6

abcdefghi123456234651

Play continues...

abcdefghi1234564321

Intrusion at D6 or F6

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

abcdefghi12345621

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

abcdefghi12345634561287

Or, if for move three Blue played G3:

abcdefghi12345621345610897

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

Block at G2

abcdefghi12345612465387

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

abcdefghi123456123456

And Red connects via template III-1a.

Block at H2, G3, or G4

Red's responses are similar in all three cases:

abcdefghi123456124365
abcdefghi123456214365
abcdefghi123456243165

Intrusion at C6 or G6

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

abcdefghi12345621

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

abcdefghi123456123456

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

abcdefghi12345612346857

And if Blue had played F5 for move five:

abcdefghi123456123465

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

Retrieved from "https://www.hexwiki.net/index.php?title=Edge_template_VI2a&oldid=8173"