Difference between revisions of "Trapezoid"

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The '''mouth''' is a [[group]] with a wide base.
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The '''trapezoid''' is an [[interior template]] with a wide base. It is one of the most common interior templates after the [[bridge]]. The [[carrier]] of the template, i.e., the set of cells that must be empty for the template to be valid, is shown in the following diagram.
  
<hex>R5 C5 Vc2 Vd2 Vb4 Vd4</hex>
+
<hexboard size="3x3"
 +
  coords="none"
 +
  edges="none"
 +
  visible="-a1"
 +
  contents="R a3 b1 c1 c3"
 +
  />
 +
 
 +
An older version of this page called the trapezoid the "mouth", but that name does not seem to be widely used.
  
 
== Common patterns ==
 
== Common patterns ==
  
 
=== Gaining Ground ===
 
=== Gaining Ground ===
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The trapezoid can be used to steal [[territory]]. For example, in the following situation
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<hexboard size="6x7"
 +
  coords="none"
 +
  edges="bottom right"
 +
  visible="area(a5,a6,g6,g1,e1)"
 +
  contents="R e2 f2 B c4 d5"
 +
  />
 +
Red can play as follows:
 +
<hexboard size="6x7"
 +
  coords="none"
 +
  edges="bottom right"
 +
  visible="area(a5,a6,g6,g1,e1)"
 +
  contents="R e2 f2 B c4 d5 R 1:d4 B 2:c5 R 3:f4"
 +
  />
 +
=== Ladder Escapes ===
 +
The trapezoid also allows some [[ladder escape]] constructions.
 +
<hexboard size="6x11"
 +
  coords="none"
 +
  edges="bottom"
 +
  visible="area(e1,c3,a6,g6,h5,j1)"
 +
  contents="R e2 f2 h4 B h5 g5 R 1:g4 B 2:f5 R 3:f4 B 4:e5 R 5:d4"
 +
  />
  
<hex>R6 C7 Ve2 Vf2 Hc4 Hd5</hex>
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== Domination and Comparison ==
<hex>R6 C7 Ve2 Vf2 Hc4 Hd5  Vd4 Hc5 Vf4</hex>
+
  
=== Ladder Escapes ===
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Although Red is unlikely to begin play in the carrier (since this is a template),
 +
if Red does, then Red should play *, because this [[captured cell|captures]] the rest of the carrier.
 +
<hexboard size="3x3"
 +
  coords="none"
 +
  edges="none"
 +
  visible="-a1"
 +
  contents="R a3 b1 c1 c3 E *:a2"
 +
  />
  
<hex>R6 C10 Ve2 Vf2 Vd4 Vf4 Vg4 Vh4 He5 Hf5 Hg5 Hh5</hex>
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With the carrier empty, the region is no worse for Red than the following.
 +
<hexboard size="3x3"
 +
  coords="none"
 +
  edges="none"
 +
  visible="-a1"
 +
  contents="R a3 b1 b2 b3 c1 c2 c3 B a2"
 +
  />
  
 +
To see why, note that Red can play the following [[pairing strategy]].
 +
<hexboard size="3x3"
 +
  coords="none"
 +
  edges="none"
 +
  visible="-a1"
 +
  contents="R a3 b1 c1 c3 E a:a2 a:b2 b:c2 b:b3"
 +
  />
 +
If Blue plays in one of the cells marked "a", then Red plays in the other one. Similarly, if Blue plays in one of the cells marked "b", Red plays in the other one. The end result is that all of Red's pieces are connected, and any Blue pieces other than in "*" are dead. Thus Blue cannot achieve anything better than getting "*".
  
 
[[category:interior templates]]
 
[[category:interior templates]]
 +
[[category:basic Strategy]]

Latest revision as of 00:46, 8 January 2023

The trapezoid is an interior template with a wide base. It is one of the most common interior templates after the bridge. The carrier of the template, i.e., the set of cells that must be empty for the template to be valid, is shown in the following diagram.

An older version of this page called the trapezoid the "mouth", but that name does not seem to be widely used.

Common patterns

Gaining Ground

The trapezoid can be used to steal territory. For example, in the following situation

Red can play as follows:

132

Ladder Escapes

The trapezoid also allows some ladder escape constructions.

53142

Domination and Comparison

Although Red is unlikely to begin play in the carrier (since this is a template), if Red does, then Red should play *, because this captures the rest of the carrier.

With the carrier empty, the region is no worse for Red than the following.

To see why, note that Red can play the following pairing strategy.

aabb

If Blue plays in one of the cells marked "a", then Red plays in the other one. Similarly, if Blue plays in one of the cells marked "b", Red plays in the other one. The end result is that all of Red's pieces are connected, and any Blue pieces other than in "*" are dead. Thus Blue cannot achieve anything better than getting "*".