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= Proposed article: Pivoting template =
 
  
A ''pivoting template'' is a kind of edge template that guarantees that the template's owner can either connect the template's stone(s) to the edge, or else can occupy a specified empty hex and connect it to the edge.
 
 
More precisely, a pivoting template is a pattern that has a stone A and an empty hex B, such that the template's owner can continuously threaten to connect A to the edge until the point where the template's owner either connects A to the edge or occupies B and connects B to the edge. To be considered a "template", its [[carrier]] should moreover be minimal with this property.
 
 
== Example ==
 
 
The following is a pivoting template.
 
 
<hexboard size="5x9"
 
  coords="none"
 
  edges="bottom"
 
  visible="area(a5,i5,i3,h1,e1)-f1"
 
  contents="R A:e1 E B:g1"
 
  />
 
 
Proof: Red's main [[threat]] is to [[bridge]] to c and connect to the edge by [[ziggurat]] or [[edge template III1b]]. Therefore, to prevent Red from connecting to the edge outright, Blue must play in one of the cells a,...,g.
 
 
<hexboard size="5x9"
 
  coords="none"
 
  edges="bottom"
 
  visible="area(a5,i5,i3,h1,e1)-f1"
 
  contents="R A:e1 E B:g1 a:d2 b:e2 c:d3 d:c4 e:d4 f:b5 g:d5"
 
  />
 
 
If Blue plays at a, Red responds at b and connects outright by [[edge template IV1a]].
 
 
If Blue plays at b, Red responds with a 3rd row ladder escape fork:
 
<hexboard size="5x9"
 
  coords="none"
 
  edges="bottom"
 
  visible="area(a5,i5,i3,h1,e1)-f1"
 
  contents="R A:e1 E B:g1 B 1:e2 R 2:d2 B 3:c4 R 4:d3 B 5:d4 R 6:f3 B 7:e3 R 8:g1"
 
  />
 
 
If Blue plays at c, d, or f, Red responds as follows and is connected by [[Fifth_row_edge_templates#V-2-f|edge template V2f]]. If Blue plays on the right instead of 3, Red responds as if defending template V2f.
 
<hexboard size="5x9"
 
  coords="none"
 
  edges="bottom"
 
  visible="area(a5,i5,i3,h1,e1)-f1"
 
  contents="R A:e1 E B:g1 B 1:d3 1:c4 1:b5 R 2:e3 B 3:e2 R 4:g1"
 
  />
 
If Blue plays at e or g, Red responds at c and gets a 2nd or 3rd row ladder, which can reach B by ladder escape fork.
 
 
== List of pivoting templates ==
 
 
=== 2nd row ===
 
 
<hexboard size="2x3"
 
  coords="none"
 
  edges="bottom"
 
  visible="-b1"
 
  contents="R A:a1 E B:c1"
 
/>
 
 
=== 3rd row ===
 
 
<hexboard size="3x5"
 
  coords="none"
 
  edges="bottom"
 
  visible="area(c1,a3,d3,e1)-d1"
 
  contents="R A:c1 E B:e1"
 
  />
 
 
<hexboard size="3x5"
 
  coords="none"
 
  edges="bottom"
 
  visible="area(b1,a3,e3,e1)-c1"
 
  contents="R A:b1 E B:d1"
 
/>
 
 
=== 4th row ===
 
 
<hexboard size="4x6"
 
  coords="none"
 
  edges="bottom"
 
  visible="area(a4,f4,f1,e2,d2,d1)"
 
  contents="R A:d1 E B:f1"
 
  />
 
 
<hexboard size="4x7"
 
  coords="none"
 
  edges="bottom"
 
  visible="area(b2,a4,g4,g2,f1,e1,d2,c1)"
 
  contents="R A:c1 E B:e1"
 
  />
 
 
<hexboard size="4x4"
 
  coords="none"
 
  edges="bottom"
 
  visible="-a1 a2 c1"
 
  contents="R A:b1 b3 E B:d1"
 
/>
 
 
=== 5th row ===
 
 
<hexboard size="5x7"
 
  coords="none"
 
  edges="bottom"
 
  visible="area(a5,g5,g1,d1,b3)-f1"
 
  contents="R A:e1 E B:g1"
 
  />
 
 
<hexboard size="5x9"
 
  coords="none"
 
  edges="bottom"
 
  visible="area(a5,i5,i3,h1,e1)-f1"
 
  contents="R A:e1 E B:g1"
 
  />
 
 
== Pivoting templates and flanks ==
 
 
Pivoting templates can be useful in many situations, but are especially useful in connection with [[flank]]s.
 
Specifically, if we line up points A and B of any pivoting template with points A and J of a capped flank, we obtain a guaranteed connection to the edge. For example, consider the capped flank
 
<hexboard size="4x4"
 
  edges="none"
 
  coords="none"
 
  visible="-a1 a2 b1 d4"
 
  contents="R A:a4 b2 c1 E J:c4"
 
  />
 
Attaching this on top of one of the above pivoting templates, we get the following:
 
<hexboard size="8x8"
 
  edges="bottom"
 
  coords="none"
 
  visible="area(d4,b6,a8,g8,g4,h3,h1,g1)"
 
  contents="R A:e4 f2 g1 S area(d4,b6,a8,g8,g4)-f4 E B:g4"
 
  />
 
This guarantees that Red can connect A to the edge, because either A will connect outright, or else B connects to the edge and also to A via the flank.
 
 
== Weak pivoting templates ==
 
 
There is another notion similar to a pivoting template, but slightly weaker. In a ''weak pivoting template'', we merely require that the template's owner can guarantee to either connect A to the edge or occupy B and connect B to the edge, but we drop the requirement that the owner can "continuously threaten to connect A to the edge until" that point. Typically this means that after the player occupies B, the opponent can still choose whether to let the player connect A or B to the edge.
 
 
The following are examples of weak pivoting templates:
 
 
<hexboard size="5x8"
 
  coords="none"
 
  edges="bottom"
 
  visible="area(a5,h5,h1,f1,c2,b3)-g1"
 
  contents="R A:f1 c2 E B:h1"
 
  />
 
 
<hexboard size="5x10"
 
  coords="none"
 
  edges="bottom"
 
  visible="area(c2,c3,a5,j5,j3,h1,f1,e2)-d2"
 
  contents="R A:c2 E B:e2"
 
  />
 
 
Weak pivoting templates are sufficient to form a connection when combined with a [[flank]]. However, there are some contexts where a proper pivoting template would connect, but a weak pivoting template does not. The following is an example of this:
 
<hexboard size="9x9"
 
  coords="show"
 
  edges="all"
 
  contents="R h2 g2 f3 f5 c6 B b5 d4 e5 i5 i8
 
            S area(a9,h9,h5,f5,c6,b7)-g5"
 
  />
 
The highlighted area is a weak pivoting template, but with Blue to move, the position is losing for Red. On the other hand, if we use a proper pivoting template in the analogous situation, the position is winning for Red:
 
<hexboard size="9x9"
 
  coords="show"
 
  edges="all"
 
  contents="R h2 g2 f3 f5 B d5 b8 c6 e4 b7 i5 i8
 
            S area(b9,h9,h5,e5,c7)-g5"
 
  />
 
 
== See also ==
 
 
* [[Climbing]]
 
* [[Flank]]
 
 
[[category:Edge templates]]
 
[[category:Advanced Strategy]]
 
[[category:Definition]]
 

Revision as of 22:42, 6 September 2021