Difference between revisions of "Pivoting template"
(New article "Pivoting template") |
(Add new pivoting template) |
||
| (13 intermediate revisions by 3 users not shown) | |||
| Line 88: | Line 88: | ||
coords="none" | coords="none" | ||
edges="bottom" | edges="bottom" | ||
| − | visible="-a1 a2 c1" | + | visible="-(a1 a2 a3 c1)" |
contents="R A:b1 b3 E B:d1" | contents="R A:b1 b3 E B:d1" | ||
| − | /> | + | /> |
=== 5th row === | === 5th row === | ||
| Line 106: | Line 106: | ||
visible="area(a5,i5,i3,h1,e1)-f1" | visible="area(a5,i5,i3,h1,e1)-f1" | ||
contents="R A:e1 E B:g1" | contents="R A:e1 E B:g1" | ||
| + | /> | ||
| + | |||
| + | <hexboard size="6x9" | ||
| + | coords="none" | ||
| + | edges="bottom" | ||
| + | visible="area(f1,i4,i6,a6,c3,d2)" | ||
| + | contents="R A:d2 c3 E B:f1" | ||
| + | /> | ||
| + | |||
| + | <hexboard size="6x7" | ||
| + | coords="none" | ||
| + | edges="bottom" | ||
| + | visible="area(g1,g6,i6,a6,c3,d2)" | ||
| + | contents="R A:d2 c3 E B:g1" | ||
| + | /> | ||
| + | |||
| + | === 6th row === | ||
| + | |||
| + | <hexboard size="6x10" | ||
| + | coords="none" | ||
| + | edges="bottom" | ||
| + | visible="area(a6,j6,j2,i1,f1)-h1" | ||
| + | contents="R A:g1 E B:i1" | ||
| + | /> | ||
| + | |||
| + | <hexboard size="6x8" | ||
| + | coords="none" | ||
| + | edges="bottom" | ||
| + | visible="area(f1,a6,h6,h1)-g1" | ||
| + | contents="R A:f1 d3 E B:h1" | ||
| + | /> | ||
| + | |||
| + | <hexboard size="6x9" | ||
| + | coords="none" | ||
| + | edges="bottom" | ||
| + | visible="area(f1,a6,i6,i1)-h1" | ||
| + | contents="R A:f1 g1 B d3 E B:i1" | ||
| + | /> | ||
| + | |||
| + | <hexboard size="6x9" | ||
| + | coords="none" | ||
| + | edges="bottom" | ||
| + | visible="area(f1,a6,i6,i1)-h1" | ||
| + | contents="R A:f1 g1 B i3 E B:i1" | ||
| + | /> | ||
| + | |||
| + | <hexboard size="6x7" | ||
| + | coords="none" | ||
| + | edges="bottom" | ||
| + | visible="area(e1,d2,d3,a6,f6,f3,g2,g1)-f1" | ||
| + | contents="R A:e1 E B:g1 R f3" | ||
| + | /> | ||
| + | |||
| + | <hexboard size="6x8" | ||
| + | coords="none" | ||
| + | edges="bottom" | ||
| + | visible="area(e1,e2,a6,h6,h2,g1)-f1" | ||
| + | contents="R A:e1 E B:g1 R f3" | ||
| + | /> | ||
| + | |||
| + | <hexboard size="6x9" | ||
| + | coords="none" | ||
| + | edges="bottom" | ||
| + | visible="area(a6,f1,i1,i2,g6)-h1" | ||
| + | contents="R A:g1 E B:i1 R g3" | ||
| + | /> | ||
| + | |||
| + | <hexboard size="6x10" | ||
| + | coords="none" | ||
| + | edges="bottom" | ||
| + | visible="area(a6,c4,e3,f3,h1,j1,j2,h6)-i1" | ||
| + | contents="R A:h1 E B:j1 R h3" | ||
/> | /> | ||
| Line 126: | Line 198: | ||
/> | /> | ||
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. | 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. | ||
| + | |||
| + | These templates make up a subset of a larger group of [https://www.hexwiki.net/index.php/Flank#Edge_templates_from_capped_flanks edge templates formed with capped flanks]. | ||
| + | |||
| + | == If A and B are semi-connected == | ||
| + | |||
| + | The following arguments only apply when the [https://www.hexwiki.net/index.php/AND_and_OR_rules#Connections_and_semi-connections semi-connection] does not overlap with the template. If there is an overlap, these properties may or may not hold. | ||
| + | |||
| + | === Connecting B with sente === | ||
| + | |||
| + | If A and B are semi-connected and Blue decides to block Red from connecting A, Red not only gets to connect B, but also gets [https://www.hexwiki.net/index.php/Initiative sente]. This is because once Red connects B, Blue will have to spend a move blocking B from connecting back to A. | ||
| + | |||
| + | === Deriving normal templates === | ||
| + | |||
| + | A normal template for connecting A can be derived from a pivoting template by placing a red stone in B, and adding sufficient space for A and B to be semi-connected. This is because, by definition, if Blue tries to block A from the edge, Red can connect B. However, if Red already has a stone at B, they can spend that move connecting back to A. | ||
== Weak pivoting templates == | == Weak pivoting templates == | ||
| Line 160: | Line 246: | ||
contents="R h2 g2 f3 f5 B d5 b8 c6 e4 b7 i5 i8 | contents="R h2 g2 f3 f5 B d5 b8 c6 e4 b7 i5 i8 | ||
S area(b9,h9,h5,e5,c7)-g5" | S area(b9,h9,h5,e5,c7)-g5" | ||
| + | /> | ||
| + | |||
| + | == Pivoting ladder creation templates == | ||
| + | |||
| + | Sometimes, the pivoting property can be combined with other properties of templates. For example, a [[ladder creation template]] guarantees that Red can get at least a specified ladder. A ''pivoting ladder creation template'' is a template with a red stone A and an empty hex B, such that Red can continuously threaten to get a ladder from A, until the point where Red either gets the ladder or occupies and connects B to a ladder. When combined with an appropriate ladder escape, a pivoting ladder creation template becomes a pivoting template. | ||
| + | |||
| + | For example, the following is a pivoting 3rd-row-ladder creation template. | ||
| + | <hexboard size="4x6" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(d1,a4,f4,f1)" | ||
| + | contents="R A:d1 E B:f1 B e3 E arrow(3):f2--f4 S red:f2--f4" | ||
| + | /> | ||
| + | The only way for Blue to prevent A from connecting outright or getting the ladder is to play at 1. In this case, Red responds as follows: | ||
| + | <hexboard size="4x6" | ||
| + | edges="bottom" | ||
| + | coords="none" | ||
| + | visible="area(d1,a4,f4,f1)" | ||
| + | contents="R A:d1 E B:f1 B e3 E arrow(3):f2--f4 S red:f2--f4 B 1:d2 R 2:c2 B 3:b4 R 4:d3 B 5:c3 R 6:f1" | ||
/> | /> | ||
Latest revision as of 08:02, 1 March 2024
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.
Contents
Example
The following is a pivoting template.
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.
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:
If Blue plays at c, d, or f, Red responds as follows and is connected by edge template V2f. If Blue plays on the right instead of 3, Red responds as if defending template V2f.
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
3rd row
4th row
5th row
6th row
Pivoting templates and flanks
Pivoting templates can be useful in many situations, but are especially useful in connection with flanks. 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
Attaching this on top of one of the above pivoting templates, we get the following:
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.
These templates make up a subset of a larger group of edge templates formed with capped flanks.
If A and B are semi-connected
The following arguments only apply when the semi-connection does not overlap with the template. If there is an overlap, these properties may or may not hold.
Connecting B with sente
If A and B are semi-connected and Blue decides to block Red from connecting A, Red not only gets to connect B, but also gets sente. This is because once Red connects B, Blue will have to spend a move blocking B from connecting back to A.
Deriving normal templates
A normal template for connecting A can be derived from a pivoting template by placing a red stone in B, and adding sufficient space for A and B to be semi-connected. This is because, by definition, if Blue tries to block A from the edge, Red can connect B. However, if Red already has a stone at B, they can spend that move connecting back to A.
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:
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:
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:
Pivoting ladder creation templates
Sometimes, the pivoting property can be combined with other properties of templates. For example, a ladder creation template guarantees that Red can get at least a specified ladder. A pivoting ladder creation template is a template with a red stone A and an empty hex B, such that Red can continuously threaten to get a ladder from A, until the point where Red either gets the ladder or occupies and connects B to a ladder. When combined with an appropriate ladder escape, a pivoting ladder creation template becomes a pivoting template.
For example, the following is a pivoting 3rd-row-ladder creation template.
The only way for Blue to prevent A from connecting outright or getting the ladder is to play at 1. In this case, Red responds as follows:
