Difference between revisions of "Edge template"

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An '''edge template''' is a [[pattern]] which guarantees a [[connection]] the [[edge]].
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An '''edge template''' is a minimal [[pattern]] that guarantees a [[virtual connection]] from a given stone, or sometimes several stones, to the [[edge]].
  
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Here is an example of a [[Third row|third-row]] edge template ([[template IIIa]], also known as the ''Ziggurat''):
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== Example ==
  
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[[Image:Ziggurat.png]]
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Here is an example of an edge template ([[template IIIa]], also known as the [[ziggurat]]):
  
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<!-- <hex>R3 C4 Vc1 Sa1 Sb1 Sa2</hex> -->
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<hexboard size="3x4"
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  visible="area(a3,d3,d1,c1)"
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  edges="bottom"
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  coords="none"
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  contents="R c1"
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/>
  
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The red stone has a certain connection to the bottom, using only the shaded hexagons, even if blue moves first.
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The red stone on the third row has a certain connection to the bottom, using only the hexes that are shown, even if Blue moves first. This is because Red is threatening either to play at "A" and connect via the shaded area
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<hexboard size="3x4"
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  coords="hide"
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  edges="bottom"
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  visible="area(a3,d3,d1,c1)"
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  contents="R c1 R A:b2 S a3 b3 c1 b2"
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  />
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or to play at "B" and connect via the shaded area
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<hexboard size="3x4"
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  coords="hide"
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  edges="bottom"
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  visible="area(a3,d3,d1,c1)"
 +
  contents="R c1 R B:d2 S c1 d1 c2 d2 c3 d3"
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  />
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Since these two escape routes for Red do not overlap at all, Red is guaranteed to be able to take one of them, no matter where Blue plays. Thus, Red is guaranteed a connection to the bottom edge.
  
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<!-- The [[Hex (board element)|cells]] marked by a star are irrelevant to the template, and can be occupied by either [[player]], or can be [[Empty hex|empty]]. Even if [[Blue (player)|Blue]] moves first, there is no way he can prevent from connecting to the [[bottom edge|bottom]]. -->
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== Minimality ==
  
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Countless edge templates can be constructed, all of them of varying degrees of usefulness and frequency of occurance.  
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An edge template is by definition '''minimal'''. This means that every hex in it is essential to the template &mdash; in other words, if you remove any hex from the template then Red is no longer guaranteed a connection. For example, the ziggurat is a minimal edge template; if we remove any hex from the template, for example the hex marked "*"
 +
<hexboard size="3x4"
 +
  coords="hide"
 +
  edges="bottom"
 +
  visible="area(a3,d3,d1,c1)"
 +
  contents="R c1 E *:d3"
 +
  />
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then Red can no longer connect if Blue moves first in this area:
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<hexboard size="3x4"
 +
  coords="hide"
 +
  edges="bottom"
 +
  visible="area(a3,d3,d1,c1)"
 +
  contents="R c1 R 4:b2 R 2:d2 B 5:a3 B 1:b3 B 3:c3 E *:d3"
 +
  />
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Blue has established a line across the bottom edge and clearly Red can no longer connect. There is no way through for Red, if Blue plays correctly.
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== Classification of edge templates ==
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''Main article: [[Naming_of_templates#Edge_templates|Naming of templates]].''
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Edge templates are classified by how many stones the template contains, and how far the furthest of these stones is from the edge.
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An edge template is a [[Edge templates with one stone|single stone template]] if it comprises only one stone. The ziggurat is an example of a single-stone edge template. There are many examples of edge templates that are not single-stone templates. For example the below pattern is a 3-stone template.
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<hexboard size="4x6"
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  coords="hide"
 +
  edges="bottom"
 +
  visible="area(d1 a4 f4 f3)"
 +
  contents="R arrow(12):d1 R e2 R f3"
 +
  />
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Red can connect their piece marked "↑" to the bottom edge, even if it is Blue's move. Here is an example of an attempt to block Red's connection by Blue, plus Red's route around the attempted block.
 +
<hexboard size="4x6"
 +
  coords="hide"
 +
  edges="bottom"
 +
  visible="area(d1 a4 f4 f3)"
 +
  contents="R arrow(12):d1 R 2:c2 B 1:d2 R e2 R 4:c3 R 6:d3 R 8:e3 R f3 B 3:b4 B 5:c4 B 7:d4 B 9:e4 R 10:f4"
 +
  />
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== More edge templates ==
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Many edge templates are known. Some are very important and show up in many games; others are so big and unwieldy that they are of essentially no practical use and are just of interest to people such as those analysing the mathematical theory of Hex.  
 +
 
 +
Other template pages on this wiki:
  
 
* [[Edge templates everybody should know]]
 
* [[Edge templates everybody should know]]
Line 17: Line 79:
 
* [[Edge templates with two adjacent stones]]
 
* [[Edge templates with two adjacent stones]]
 
* [[Edge templates with a bridge]]
 
* [[Edge templates with a bridge]]
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* [[Ladder creation template]]
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* [[Open problems about edge templates]]
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== See also ==
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 +
Unsurprisingly, edge templates play a very important role in hex. Players who want to learn more of the basics about edge templates should now read [[Edge templates everybody should know]].
 +
 +
On a theoretical level edge templates are interesting too. Here are some [[open problems about edge templates]].
 +
 +
 +
[[category:edge templates]]

Latest revision as of 01:55, 12 June 2021

An edge template is a minimal pattern that guarantees a virtual connection from a given stone, or sometimes several stones, to the edge.

Example

Here is an example of an edge template (template IIIa, also known as the ziggurat):

The red stone on the third row has a certain connection to the bottom, using only the hexes that are shown, even if Blue moves first. This is because Red is threatening either to play at "A" and connect via the shaded area

A

or to play at "B" and connect via the shaded area

B

Since these two escape routes for Red do not overlap at all, Red is guaranteed to be able to take one of them, no matter where Blue plays. Thus, Red is guaranteed a connection to the bottom edge.

Minimality

An edge template is by definition minimal. This means that every hex in it is essential to the template — in other words, if you remove any hex from the template then Red is no longer guaranteed a connection. For example, the ziggurat is a minimal edge template; if we remove any hex from the template, for example the hex marked "*"

then Red can no longer connect if Blue moves first in this area:

42513

Blue has established a line across the bottom edge and clearly Red can no longer connect. There is no way through for Red, if Blue plays correctly.

Classification of edge templates

Main article: Naming of templates.

Edge templates are classified by how many stones the template contains, and how far the furthest of these stones is from the edge.

An edge template is a single stone template if it comprises only one stone. The ziggurat is an example of a single-stone edge template. There are many examples of edge templates that are not single-stone templates. For example the below pattern is a 3-stone template.

Red can connect their piece marked "↑" to the bottom edge, even if it is Blue's move. Here is an example of an attempt to block Red's connection by Blue, plus Red's route around the attempted block.

21468357910

More edge templates

Many edge templates are known. Some are very important and show up in many games; others are so big and unwieldy that they are of essentially no practical use and are just of interest to people such as those analysing the mathematical theory of Hex.

Other template pages on this wiki:

See also

Unsurprisingly, edge templates play a very important role in hex. Players who want to learn more of the basics about edge templates should now read Edge templates everybody should know.

On a theoretical level edge templates are interesting too. Here are some open problems about edge templates.

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