Difference between revisions of "Trapezoid"

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(Dominated cells: Added a caveat.)
(mouth -> trapezoid)
 
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The '''mouth''' or '''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.
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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.
  
 
<hexboard size="3x3"
 
<hexboard size="3x3"
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   contents="R a3 b1 c1 c3"
 
   contents="R a3 b1 c1 c3"
 
   />
 
   />
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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 ===
The mouth can be used to steal [[territory]]. For example, in the following situation
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The trapezoid can be used to steal [[territory]]. For example, in the following situation
 
<hexboard size="6x7"
 
<hexboard size="6x7"
 
   coords="none"
 
   coords="none"
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   />
 
   />
 
=== Ladder Escapes ===
 
=== Ladder Escapes ===
The mouth also allows some [[ladder escape]] constructions.
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The trapezoid also allows some [[ladder escape]] constructions.
 
<hexboard size="6x11"
 
<hexboard size="6x11"
 
   coords="none"
 
   coords="none"
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   />
 
   />
  
== Dominated cells ==
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== Domination and Comparison ==
  
If Blue wants to intrude into the trapezoid, and assuming that Red wants to defend it, the only possible intrusion point is the hex marked "*".  
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Although Red is unlikely to begin play in the carrier (since this is a template),
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if Red does, then Red should play *, because this [[captured cell|captures]] the rest of the carrier.
 
<hexboard size="3x3"
 
<hexboard size="3x3"
 
   coords="none"
 
   coords="none"
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   contents="R a3 b1 c1 c3 E *:a2"
 
   contents="R a3 b1 c1 c3 E *:a2"
 
   />
 
   />
To see why, note that Red can play the following pairing strategy.
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With the carrier empty, the region is no worse for Red than the following.
 
<hexboard size="3x3"
 
<hexboard size="3x3"
 
   coords="none"
 
   coords="none"
 
   edges="none"
 
   edges="none"
 
   visible="-a1"
 
   visible="-a1"
   contents="R a3 b1 c1 c3 E a:a2 a:b2 b:c2 b:b3"
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   contents="R a3 b1 b2 b3 c1 c2 c3 B a2"
 
   />
 
   />
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. Since Blue cannot achieve anything better than to get "*", playing at "*" immediately is at least as good as playing anywhere else in the template. In other words, from Blue's point of view, "*" [[dominated cell|dominates]] all other possible moves in the trapezoid's carrier.
 
  
The same holds from Red's point of view as well. If Red is to play anywhere in the trapezoid's carrier, the only sensible move is at "*", because this [[captured cell|captures]] the rest of the carrier.
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To see why, note that Red can play the following [[pairing strategy]].
 +
<hexboard size="3x3"
 +
  coords="none"
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  edges="none"
 +
  visible="-a1"
 +
  contents="R a3 b1 c1 c3 E a:a2 a:b2 b:c2 b:b3"
 +
  />
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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]]
 
[[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 "*".