Difference between revisions of "Edge template"
m (added to cat.edge templates and stub) |
(destubbed) |
||
| Line 1: | Line 1: | ||
| − | An '''edge template''' is a [[pattern]] which guarantees a [[connection]] the [[edge]]. | + | An '''edge template''' is a [[pattern]] which guarantees a [[connection]] from a given stone to the [[edge]]. |
| − | Here is an example of | + | Here is an example of an edge template ([[template IIIa]], also known as the ''Ziggurat''): |
[[Image:Ziggurat.png]] | [[Image:Ziggurat.png]] | ||
| Line 7: | Line 7: | ||
<!-- <hex>R3 C4 Vc1 Sa1 Sb1 Sa2</hex> --> | <!-- <hex>R3 C4 Vc1 Sa1 Sb1 Sa2</hex> --> | ||
| − | The red stone has a certain connection to the bottom, using only the shaded hexagons, even if blue moves first. | + | The red stone on the third row has a certain connection to the bottom, using only the shaded hexagons, even if blue moves first. This is because (ignore the starred hexes, they are not part of the pattern) red is threatening either to play the piece marked 1 |
| + | <hexboard size="3x4" | ||
| + | coords="hide" | ||
| + | contents="E *:a1 E *:b1 R c1 E *:a2 R 1:b2 E +:a3 E +:b3" | ||
| + | /> | ||
| + | and to connect via one the two hexes marked with a plus in the above diagram, or to play the piece marked 2 | ||
| + | <hexboard size="3x4" | ||
| + | coords="hide" | ||
| + | contents="E *:a1 E *:b1 R c1 E +:d1 E *:a2 E +:c2 R 2:d2 E +:c3 E +:d3" | ||
| + | /> | ||
| + | and to go on to connect by taking two of the hexes marked with a plus in this second diagram. Because these two escape routes for red do not overlap at all, red is guaranteed to be able to take one of them whatever blue plays next, and thus connect. | ||
<!-- 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]]. --> | <!-- 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]]. --> | ||
| − | + | 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. | |
| + | |||
| + | An edge template is '''minimal''' if every hex in it is essential to the template -- in other words, if you remove a hex from the template then red is no longer guaranteed a connection. For example the Ziggurat is a minimal edge template; if we remove a hex from the template by marking it with an asterisk | ||
| + | |||
| + | <hexboard size="3x4" | ||
| + | coords="hide" | ||
| + | contents="E *:a1 E *:b1 R c1 E *:a2 E *:d3" | ||
| + | /> | ||
| + | |||
| + | then red can no longer connect if blue moves first in this area: | ||
| + | |||
| + | <hexboard size="3x4" | ||
| + | coords="hide" | ||
| + | contents="E *:a1 E *:b1 R c1 E *:a2 R 4:b2 R 2:d2 B 5:a3 B 1:b3 B 3:c3 E *:d3" | ||
| + | /> | ||
| + | |||
| + | 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. | ||
| + | |||
| + | An edge template is a [[Edge templates with one stone|single stone template]] if it comprises only of one stone. The Ziggurat is an example of a single-stone edge template. There are many examples of edge templates which are not single-stone templates. For example the below pattern | ||
| + | <hexboard size="4x6" | ||
| + | coords="hide" | ||
| + | contents="E *:a1 E *:b1 E *:c1 R 1:d1 E *:e1 E *:f1 E *:a2 E *:b2 R e2 E *:f2 E *:a3 R f3" | ||
| + | /> | ||
| + | is a multiple-stone template. Red can connect their piece marked 1 to the bottom edge, even if it is blue's move. As usual the hexes marked with an asterisk are not part of the template. 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" | ||
| + | contents="E *:a1 E *:b1 E *:c1 R d1 E *:e1 E *:f1 E *:a2 E *:b2 R 2:c2 B 1:d2 R e2 E *:f2 E *:a3 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" | ||
| + | /> | ||
| + | |||
| + | 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]]. | ||
| + | |||
| + | Other template pages on this wiki: | ||
* [[Edge templates everybody should know]] | * [[Edge templates everybody should know]] | ||
| Line 17: | Line 60: | ||
* [[Edge templates with two adjacent stones]] | * [[Edge templates with two adjacent stones]] | ||
* [[Edge templates with a bridge]] | * [[Edge templates with a bridge]] | ||
| + | * [[Open problems about edge templates]] | ||
| + | |||
[[category:edge templates]] | [[category:edge templates]] | ||
| − | |||
Revision as of 20:51, 20 May 2016
An edge template is a pattern which guarantees a connection from a given stone to the edge.
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 shaded hexagons, even if blue moves first. This is because (ignore the starred hexes, they are not part of the pattern) red is threatening either to play the piece marked 1
and to connect via one the two hexes marked with a plus in the above diagram, or to play the piece marked 2
and to go on to connect by taking two of the hexes marked with a plus in this second diagram. Because these two escape routes for red do not overlap at all, red is guaranteed to be able to take one of them whatever blue plays next, and thus connect.
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.
An edge template is minimal if every hex in it is essential to the template -- in other words, if you remove a hex from the template then red is no longer guaranteed a connection. For example the Ziggurat is a minimal edge template; if we remove a hex from the template by marking it with an asterisk
then red can no longer connect if blue moves first in this area:
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.
An edge template is a single stone template if it comprises only of one stone. The Ziggurat is an example of a single-stone edge template. There are many examples of edge templates which are not single-stone templates. For example the below pattern
is a multiple-stone template. Red can connect their piece marked 1 to the bottom edge, even if it is blue's move. As usual the hexes marked with an asterisk are not part of the template. Here is an example of an attempt to block red's connection by blue, plus red's route around the attempted block.
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.
Other template pages on this wiki:
