Difference between revisions of "User:Selinger"

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== Proposed article: Tidiness ==
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= Proposed article: Flank =
  
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'''Tidiness''', or '''tidying up''', means not leaving loose ends that could benefit the opponent later. It is not a reaction to a specific imminent threat, but a basic preventative measure that decreases the likelihood of bad things happening in the future. If a player can make a move that carries little cost or risk, but takes away some opportunity from the opponent, it is untidy not to make the move.
+
A '''flank''' is a sequence of [[friendly]] [[stone]]s that are either adjacent or linked by [[bridge]]s in a certain way, for example like this:
−
 
+
<hexboard size="6x11"
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== Examples ==
+
  edges="none"
−
 
+
  coords="none"
−
=== Acute corner example ===
+
  visible="area(a6,d6,g5,k3,i3,e5,d5)"
 +
  contents="B 1:a6 2:b6 3:c6 4:e5 5:f5 6:h4 7:j3 8:k3"
 +
  />
 +
Apart from [[ladder]]s, flanks are one of the most common "long-distance" patterns occuring in Hex. They are useful for [[climbing]], and they can be used to form large [[interior template|interior]] and [[edge template]]s.
  
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Consider the following situation, where Red has just played 1 to connect her group to the bottom edge.
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What makes a flank useful is that its owner can use it for [[climbing]]. For example, consider the following situation, and assume the stones "A" and "B" are connected to opposite edges.  
−
<hexboard size="4x5"
+
<hexboard size="6x12"
 +
  edges="none"
 
   coords="none"
 
   coords="none"
−
   edges="bottom right"
+
   visible="area(a6,d6,g5,k3,k1,i1,e3,d3,a4)"
−
   contents="B c2 R d1 1:d2"
+
   contents="B A:a4 a6 b6 c6 e5 f5 h4 j3 B:k3 E *:k1"
 
   />
 
   />
−
Blue would like to play elsewhere on the board. However, this would leave Red with a 2nd row [[ladder escape]] along the bottom edge. While this ladder escape may not look immediately threatening to Blue, it would be untidy to just leave it unattended. Instead, Blue tidies up by first playing 2, which forces Red to reconnect, say at 3.
+
Then Blue can [[climbing|climb]] all the way from A to the cell marked "*", by a sequence of forcing moves as follows:
−
<hexboard size="4x5"
+
<hexboard size="6x12"
 +
  edges="none"
 
   coords="none"
 
   coords="none"
−
   edges="bottom right"
+
   visible="area(a6,d6,g5,k3,k1,i1,e3,d3,a4)"
−
   contents="B c2 R d1 1:d2 B 2:c3 R 3:d3"
+
   contents="B a6 b6 c6 e5 f5 h4 j3 B:k3
 +
            B A:a4 2:b4 4:c4 6:d6 8:e3 10:f3 12:g5 14:h2 16:i4 18:j1 20:k1
 +
            R 1:a5 3:b5 5:d5 7:c5 9:e4 11:g4 13:f4 15:i3 17:h3 19:j2 21:k2"
 
   />
 
   />
−
Now Blue has taken away Red's ladder escape and is free to move elsewhere. In fact, Blue also gained a small amount of [[territory]].
+
Intruding into the flank's bridges does not help the opponent. The flank still works even if all the bridges have already been filled in:
−
 
+
<hexboard size="6x12"
−
To illustrate that this can make a difference, consider the following position, with Blue to move. In this situation, "a" is winning, but "b" and all other moves are losing.
+
  edges="none"
−
<hexboard size="6x6"
+
 
   coords="none"
 
   coords="none"
−
   edges="all"
+
   visible="area(a6,d6,g5,k3,k1,i1,e3,d3,a4)"
−
   contents="R b3 e3 e4 B d3 d4 E a:d5 b:e2"
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   contents="B A:a4 a6 b6 c6 e5 f5 h4 j3 B:k3 E *:k1
 +
            R d5 B d6 R g4 B g5 R i3 B i4"
 
   />
 
   />
  
 +
== Definition ==
  
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=== Obtuse corner example ===
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A flank can belong to Red or to Blue, and it can be oriented in any of the 6 cardinal directions of the Hex board (a cardinal direction is parallel to an edge or to the short diagonal). In addition, it can be facing up or down (the side it is facing is the side where the empty space is). For simplicity, in the following, we will only consider blue flanks that are oriented left-to-right and facing upward.
  
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Red has just played 1, threatening to connect to the bottom edge. Blue responds at 2. This will result in a 2nd row [[ladder]] along the bottom edge.
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We can define such flanks as follows:
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<hexboard size="5x5"
+
 
 +
* A single blue stone is a flank. The stone is both the starting point and the endpoint.
 +
 
 +
* A flank with endpoint ''x'' can be extended with any of the following patterns:<br><hexboard size="1x2"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  contents="B -:a1 +:b1"
 +
  /><hexboard size="2x3"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  visible="area(a2,b2,c1,b1)"
 +
  contents="B -:a2 +:c1"
 +
  /><hexboard size="2x3"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  visible="area(a2,b2,c1,b1)"
 +
  contents="B -:a2 +:c1 R b1 B b2"
 +
  /><br>Here, "−" denotes the previous endpoint, and "+" denotes the new endpoint.
 +
 
 +
Here is an example of a flank with starting point "A" and endpoint "B":
 +
<hexboard size="6x11"
 +
  edges="none"
 
   coords="none"
 
   coords="none"
−
   edges="bottom left"
+
   visible="area(a6,d6,g5,k3,i3,e5,d5)"
−
   contents="R d2 e2 B d3 e3 R 1:c3 B 2:b5"
+
   contents="B A:a6 b6 c6 R d5 B d6 e5 f5 h4 R i3 B i4 j3 B:k3"
 
   />
 
   />
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Let's assume that Red has a [[ladder escape]], so that Red will "win" the ladder. Red could start the ladder right away, but it is better to first tidy up the corner by playing 3 and 4:
+
 
−
<hexboard size="5x5"
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== Capped flank ==
 +
 
 +
A flank is '''capped''' if it has been extended past its endpoint "B" with one of the following patterns:
 +
<hexboard size="2x2"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  visible="a2 b1"
 +
  contents="B B:a2 b1"
 +
  /><hexboard size="2x1"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  contents="B B:a2 a1"
 +
  /><hexboard size="3x2"
 +
  float="inline"
 +
  edges="none"
 +
  coords="none"
 +
  visible="a2,a3,b1,b2"
 +
  contents="B B:a3 b1"
 +
  /><hexboard size="3x2"
 +
  float="inline"
 +
  edges="none"
 
   coords="none"
 
   coords="none"
−
   edges="bottom left"
+
   visible="a2,a3,b1,b2"
−
   contents="R d2 e2 B d3 e3 R 1:c3 B 2:b5 R 3:b3 B 4:a5"
+
   contents="B B:a3 b1 R a2 B b2"
 
   />
 
   />
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This gains a bit of territory for Red. Note that 3 is immediately forcing: if Blue does not respond, Red connects to the edge. On the other hand, if Red has already played and connected the ladder, 3 is no longer forcing.
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Here, "B" denotes the original endpoint of the flank. The following are some examples of capped flanks. In each case, the flank's starting point "A" and original endpoint "B" are shown.
  
−
To illustrate that this can make a difference, consider the following position, with Red to move. In this situation, "a" is winning, but "b" and all other moves are losing.
+
ADD SOME EXAMPLES HERE.
−
<hexboard size="5x5"
+
 
 +
If Blue climbs along a capped flank, Blue will connect.
 +
 
 +
ADD EXAMPLES.
 +
 
 +
POINT OUT HOW THIS GENERALIZES A 2ND ROW LADDER, WITH THE FLANK GENERALIZING THE "EDGE" AND THE CAP GENERALIZING A LADDER ESCAPE.
 +
 
 +
== Interior templates from capped flanks ==
 +
 
 +
Consider a capped flank with starting point "A", and suppose the hex marked "*" is also occupied by Blue:
 +
<hexboard size="3x1"
 +
  edges="none"
 
   coords="none"
 
   coords="none"
−
  edges="all"
+
   contents="B *:a1 A:a3"
−
   contents="B b1 d3 b5 R c3 d5 E a:b3 b:c4"
+
 
   />
 
   />
 +
Then, given the right amount of space, the hex marked "*" together with the capped flank forms an interior templates.
 +
 +
ADD SOME EXAMPLES HERE. ALSO EXPLAIN MORE CAREFULLY WHAT IS THE "RIGHT" AMOUNT OF SPACE.
 +
 +
Moreover, two capped flanks growing in opposite directions from an empty hex and facing the same way form an interior template.
 +
 +
ADD EXAMPLE.
 +
 +
== Edge templates from capped flanks ==
 +
 +
ADD EXAMPLES.
 +
 +
== Usage example ==
  
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=== Incomplete joseki ===
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FROM A GAME.
  
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Failing to correctly complete a [[joseki]] often results in an untidy situation.
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== 3rd row ladders along flanks ==
  
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== Related concepts ==
+
Above, we pointed out that climbing along a flank is analogous to a 2nd row ladder. It is similarly possible to climb along a flank at a greater distance. In other words, there is an analog of a 3rd row ladder along a flank. This requires slightly more space, and if the ladder is to connect, it requires a different kind of cap (or ladder escape).
  
−
In Go, there is the concept of [https://senseis.xmp.net/?Aji aji], which means something like "the possibilities left in a position".
+
ADD EXAMPLE.

Revision as of 03:00, 2 March 2021

Proposed article: Flank

A flank is a sequence of friendly stones that are either adjacent or linked by bridges in a certain way, for example like this:

78645123

Apart from ladders, flanks are one of the most common "long-distance" patterns occuring in Hex. They are useful for climbing, and they can be used to form large interior and edge templates.

What makes a flank useful is that its owner can use it for climbing. For example, consider the following situation, and assume the stones "A" and "B" are connected to opposite edges.

BA

Then Blue can climb all the way from A to the cell marked "*", by a sequence of forcing moves as follows:

18201419218101715BA2491311161375126

Intruding into the flank's bridges does not help the opponent. The flank still works even if all the bridges have already been filled in:

BA

Definition

A flank can belong to Red or to Blue, and it can be oriented in any of the 6 cardinal directions of the Hex board (a cardinal direction is parallel to an edge or to the short diagonal). In addition, it can be facing up or down (the side it is facing is the side where the empty space is). For simplicity, in the following, we will only consider blue flanks that are oriented left-to-right and facing upward.

We can define such flanks as follows:

  • A single blue stone is a flank. The stone is both the starting point and the endpoint.
  • A flank with endpoint x can be extended with any of the following patterns:

    Here, "−" denotes the previous endpoint, and "+" denotes the new endpoint.

Here is an example of a flank with starting point "A" and endpoint "B":

BA

Capped flank

A flank is capped if it has been extended past its endpoint "B" with one of the following patterns:

B
B
B
B

Here, "B" denotes the original endpoint of the flank. The following are some examples of capped flanks. In each case, the flank's starting point "A" and original endpoint "B" are shown.

ADD SOME EXAMPLES HERE.

If Blue climbs along a capped flank, Blue will connect.

ADD EXAMPLES.

POINT OUT HOW THIS GENERALIZES A 2ND ROW LADDER, WITH THE FLANK GENERALIZING THE "EDGE" AND THE CAP GENERALIZING A LADDER ESCAPE.

Interior templates from capped flanks

Consider a capped flank with starting point "A", and suppose the hex marked "*" is also occupied by Blue:

A

Then, given the right amount of space, the hex marked "*" together with the capped flank forms an interior templates.

ADD SOME EXAMPLES HERE. ALSO EXPLAIN MORE CAREFULLY WHAT IS THE "RIGHT" AMOUNT OF SPACE.

Moreover, two capped flanks growing in opposite directions from an empty hex and facing the same way form an interior template.

ADD EXAMPLE.

Edge templates from capped flanks

ADD EXAMPLES.

Usage example

FROM A GAME.

3rd row ladders along flanks

Above, we pointed out that climbing along a flank is analogous to a 2nd row ladder. It is similarly possible to climb along a flank at a greater distance. In other words, there is an analog of a 3rd row ladder along a flank. This requires slightly more space, and if the ladder is to connect, it requires a different kind of cap (or ladder escape).

ADD EXAMPLE.

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