Difference between revisions of "User:Selinger"

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= Proposed article: Pivoting template =
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I have played Hex since early 2020, and I run the [[Hex clubs|Halifax Hex Club]]. I mostly use this user page for draft articles and other random bits and pieces that aren't yet ready to go into a real HexWiki article.
  
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.
+
= Proposed page: Eric's move =
  
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.
+
Eric's move is a trick that allows a player to make the best of a 3rd row [[ladder]] approaching an [[board|obtuse corner]]. It takes away the opponent's opportunity to get a 5th row ladder.
 +
 
 +
The move is named after Eric Demer, who discovered it.
  
 
== Example ==
 
== Example ==
  
The following is a pivoting template.
+
Consider the following situation, with Red's 3rd row ladder approaching from the right, and Red to move.
 +
<hexboard size="5x8"
 +
  coords="hide"
 +
  edges="bottom left"
 +
  contents="B e2 f2 g2 g4 R h2 R g3 B f4 R f3 B e4"/>
 +
There's not enough room for Red to [[ladder handling#Attacking|push]] one more time, as this will give Blue a 2nd row ladder:
 +
<hexboard size="5x8"
 +
  coords="hide"
 +
  edges="bottom left"
 +
  contents="B e2 f2 g2 g4 R h2 R g3 B f4 R f3 B e4 R 1:e3 B 2:d4 R 3:c3 B 4:b5 R 5:a5 B 6:b4 R 7:a4 B 8:b3"
 +
  />
 +
The obvious solution is for Red to pivot immediately and hold Blue to a 5th row ladder:
 +
<hexboard size="5x8"
 +
  coords="hide"
 +
  edges="bottom left"
 +
  contents="B e2 f2 g2 g4 R h2 R g3 B f4 R f3 B e4 R 1:d3 B 2:e3 R 3:d2 B 4:e1 E x:b4"
 +
  />
 +
However, this is not optimal, because Blue has a forcing move at x, giving Blue 2nd and 3rd row ladder escapes:
 +
A slightly better solution is the following:
 +
<hexboard size="5x8"
 +
  coords="hide"
 +
  edges="bottom left"
 +
  contents="B e2 f2 g2 g4 R h2 R g3 B f4 R f3 B e4 R 1:d4 B 2:e3 R 3:d2 B 4:e1 E x:b4 y:c3 S area(d2,a5,d5)"
 +
  />
 +
Note that Red has formed [[edge template IV2d]], still holding Blue to a 5th row ladder. Blue still has a forcing move at x, but because Red can respond at y, this only gives Blue a 2nd row ladder escape, rather than a 3rd row escape.
  
<hexboard size="5x9"
+
However, even this solution is not optimal for Red, as Blue still gets a 5th row ladder. It turns out that playing a different move 3 is generally even better for Red.
   coords="none"
+
<hexboard size="5x8"
   edges="bottom"
+
   coords="hide"
  visible="area(a5,i5,i3,h1,e1)-f1"
+
   edges="bottom left"
   contents="R A:e1 E B:g1"
+
   contents="B e2 f2 g2 g4 R h2 R g3 B f4 R f3 B e4 R 1:d4 B 2:e3 R 3:b2
 +
            E a:c2 b:d2 c:b3 d:c3 e:a4 f:b4"
 
   />
 
   />
 +
Move 3 is named '''Eric's move'''. If Red plays there, Blue in theory no longer gets a 5th row ladder, nor any kind of ladder. If Blue plays at a or c, Red responds at b; if Blue plays at b, Red responds at a; if Blue plays at d, Red responds at e; if Blue plays at f, Red responds at d.
  
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.
+
== Why it works ==
  
<hexboard size="5x9"
+
Eric's move prevents Blue from getting a 5th row ladder along the left edge. To see why, consider the following line of play, which is one of Blue's best attempts:
   coords="none"
+
<hexboard size="12x8"
   edges="bottom"
+
   coords="hide"
  visible="area(a5,i5,i3,h1,e1)-f1"
+
   edges="bottom left"
   contents="R A:e1 E B:g1 a:d2 b:e2 c:d3 d:c4 e:d4 f:b5 g:d5"
+
   contents="B e9 f9 g9 g11 R h9 R g10 B f11 R f10 B e11 R 1:d11 B 2:e10 R 3:b9
 +
            B 4:b10 R 5:d9 B 6:e8 R 7:d8 B 8:e7 R 9:c6 S red:f1--f8"
 
   />
 
   />
 +
If we imagine that the pink cells are occupied by a line of red stones, then Red's move 9 is actually [[Tom's move]], using that line of stones as its edge. In that case, Red would connect, proving that Blue cannot in general get a 5th row ladder. Even if the pink cells are not in fact occupied by Red, the situation is still typically good for Red.
  
If Blue plays at a, Red responds at b and connects outright by [[edge template IV1a]].
+
However, the use of Tom's move in this argument requires quite a bit of empty space. If there is less space, or if there are additional Blue stones in this area, then Blue might still be able to do something useful.
  
If Blue plays at b, Red responds with a 3rd row ladder escape fork:
+
The way in which Eric's move works is essentially the same way as blocking a 5th row ladder when there's not enough space under the ladder stone; see [[Theory_of_ladder_escapes#Definition_of_ladder_4|theory of ladder escapes]].
<hexboard size="5x9"
+
  coords="none"
+
  edges="bottom"
+
  visible="area(a5,i5,i3,h1,e1)-f1"
+
  contents="R A:e1 E B:g1 B 1:e2 R 2:d2 B 3:c4 R 4:d3 B 5:d4 R 6:f3 B 7:e3 R 8:g1"
+
  />
+
  
If Blue plays at c, d, or f, Red responds as follows and is connected by [[Fifth_row_edge_templates#V-2-f|edge template V2f]]. If Blue plays on the right instead of 3, Red responds as if defending template V2f.
 
<hexboard size="5x9"
 
  coords="none"
 
  edges="bottom"
 
  visible="area(a5,i5,i3,h1,e1)-f1"
 
  contents="R A:e1 E B:g1 B 1:d3 1:c4 1:b5 R 2:e3 B 3:e2 R 4:g1"
 
  />
 
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.
 
  
== Usage ==
+
etc.
  
Pivoting templates can be useful in many situations, but are especially useful in connection with [[flank]]s.
+
= Connecting parallel ladders =
  
[Todo: Add an example.]
+
== Connecting a 2-5 parallel ladder ==
  
== More examples ==
+
Like 2-4 and 3-5 parallel ladders, a 2-5 parallel ladder can also connect to the edge outright, given enough space. One way to do this is to yield to a 3-5 parallel ladder and then use [[Tom's move for 3rd and 5th row parallel ladders]]. However, there is a way to do it with much less space. In fact, the amount of space shown here is minimal:
  
<hexboard size="4x6"
+
<hexboard size="5x9"
  coords="none"
+
 
   edges="bottom"
 
   edges="bottom"
   visible="area(a4,f4,f1,e2,d2,d1)"
+
  coords="none"
   contents="R A:d1 E B:f1"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
 +
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2"
 
   />
 
   />
 +
The ladder stones are marked "↑", and Red's winning move is "1". It is Red's only winning move within this space.
  
<hexboard size="4x7"
+
Proof of connectedness: Red's main threats is "*", using the highlighted cells:
 +
<hexboard size="5x9"
 +
  edges="bottom"
 
   coords="none"
 
   coords="none"
 +
  visible="area(c1,a4,a5,i5,i3,g1)"
 +
  contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 *:c4 S red:(d1,c3,b4,b5,c4,c5,d3)"
 +
  />
 +
Red's other main threat is "*", connecting via [[edge template IV2b]], and only requiring 2 of the 3 cells x, y, z:
 +
<hexboard size="5x9"
 
   edges="bottom"
 
   edges="bottom"
   visible="area(b2,a4,g4,g2,f1,e1,d2,c1)"
+
  coords="none"
   contents="R A:c1 E B:e1"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
 +
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 *:c3 S red:c3,area(e2,b5,f5,f3) E x:b3 y:b4 z:d1"
 
   />
 
   />
 
+
The overlap consists of the cells marked "a", "b", and "c":
<hexboard size="3x5"
+
<hexboard size="5x9"
 +
  edges="bottom"
 
   coords="none"
 
   coords="none"
 +
  visible="area(c1,a4,a5,i5,i3,g1)"
 +
  contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 E a:c3,d3 c:c4 b:b5,c5"
 +
  />
 +
If Blue plays at "a", Red pushes the 2nd row ladder to "c" and then uses [[Tom's move]]. If Blue plays at "b", Red responds at "c" and then uses Tom's move. Finally, if Blue plays at "c", Red plays as follows:
 +
<hexboard size="5x9"
 
   edges="bottom"
 
   edges="bottom"
   visible="area(c1,a3,d3,e1)-d1"
+
  coords="none"
   contents="R A:c1 E B:e1"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
 +
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 B 2:c4 R 3:b4 B 4:b5 R 5:e3"
 
   />
 
   />
 +
This move isn't exactly a version of Tom's move, but it does for a 2-5 ladder what Tom's move does for a 2-4 ladder.
  
<hexboard size="5x7"
+
== Connecting a 2-6 parallel ladder ==
  coords="none"
+
 
 +
2-6 parallel ladders also connect. The required amount of space and the unique winning move within this space are shown:
 +
<hexboard size="6x12"
 
   edges="bottom"
 
   edges="bottom"
   visible="area(a5,g5,g1,d1,b3)-f1"
+
  coords="none"
   contents="R A:e1 E B:g1"
+
   visible="area(d1,d3,b4,a5,a6,l6,l4,j2,f1)"
 +
   contents="R arrow(12):d1,a5 B d2,a6 R 1:f2"
 
   />
 
   />
 +
The basic idea is that this yields to 2-5, and then Red can use the previous trick.
  
== Weak pivoting templates ==
+
== Connecting a 3-6 parallel ladder ==
  
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.
+
3-6 parallel ladders also connect. The required amount of space and the unique winning move within this space are shown:
  
The following are examples of weak pivoting templates:
+
<hexboard size="6x13"
 
+
<hexboard size="5x8"
+
  coords="none"
+
 
   edges="bottom"
 
   edges="bottom"
  visible="area(a5,h5,h1,f1,c2,b3)-g1"
 
  contents="R A:f1 c2 E B:h1"
 
  />
 
 
<hexboard size="5x10"
 
 
   coords="none"
 
   coords="none"
  edges="bottom"
+
   visible="area(c1,a4,a6,m6,m4,k2,g1)"
   visible="area(c2,c3,a5,j5,j3,h1,f1,e2)-d2"
+
   contents="R arrow(12):c1,a4 B c2,a5 R 1:d2"
   contents="R A:c2 E B:e2"
+
 
   />
 
   />
  
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:
+
== Remarks ==
<hexboard size="9x9"
+
  coords="show"
+
  edges="all"
+
  contents="R h2 g2 f3 f5 c6 B b5 d4 e5 i5 i8
+
            S area(a9,h9,h5,f5,c6,b7)-g5"
+
  />
+
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:
+
<hexboard size="9x9"
+
  coords="show"
+
  edges="all"
+
  contents="R h2 g2 f3 f5 B d5 b8 c6 e4 b7 i5 i8
+
            S area(b9,h9,h5,e5,c7)-g5"
+
  />
+
  
== See also ==
+
In all three cases, for the ladder to propagate, the top ladder should be one hex further ahead of the bottom ladder than shown above. (If the bottom ladder is already caught up, the top ladder can no longer be pushed). For the 3-5 and 3-6 parallel ladders, Red doesn't necessarily have to push the bottom ladder before playing 1. However, for the 2-6 ladder, Red ''does'' have to push the bottom ladder first.
  
* [[Climbing]]
+
Also, the fact that these ladders all connect means that they are not really "ladders" in the usual sense; they are basically just templates. Note that unlike Tom's move (2-4 and 3-5 ladders), the connection requires no space above the height of the ladder, so the space in which the ladder would normally travel is already enough space to connect it.
  
[[category:Edge templates]]
+
Also, as noted on the page [[Tom's move for 3rd and 5th row parallel ladders]], 3-5 parallel ladders can't always be pushed as 3-5 parallel ladders; the defender has the option to downgrade it to a 2nd row ladder with switchback threat. The resulting 2nd row ladder may not connect. Therefore, Tom's move may not be available if the ladder starts some distance from where there is space.
[[category:Advanced Strategy]]
+
[[category:Definition]]
+

Latest revision as of 12:00, 27 May 2024

I have played Hex since early 2020, and I run the Halifax Hex Club. I mostly use this user page for draft articles and other random bits and pieces that aren't yet ready to go into a real HexWiki article.

Proposed page: Eric's move

Eric's move is a trick that allows a player to make the best of a 3rd row ladder approaching an obtuse corner. It takes away the opponent's opportunity to get a 5th row ladder.

The move is named after Eric Demer, who discovered it.

Example

Consider the following situation, with Red's 3rd row ladder approaching from the right, and Red to move.

There's not enough room for Red to push one more time, as this will give Blue a 2nd row ladder:

83176254

The obvious solution is for Red to pivot immediately and hold Blue to a 5th row ladder:

4312x

However, this is not optimal, because Blue has a forcing move at x, giving Blue 2nd and 3rd row ladder escapes: A slightly better solution is the following:

43y2x1

Note that Red has formed edge template IV2d, still holding Blue to a 5th row ladder. Blue still has a forcing move at x, but because Red can respond at y, this only gives Blue a 2nd row ladder escape, rather than a 3rd row escape.

However, even this solution is not optimal for Red, as Blue still gets a 5th row ladder. It turns out that playing a different move 3 is generally even better for Red.

3abcd2ef1

Move 3 is named Eric's move. If Red plays there, Blue in theory no longer gets a 5th row ladder, nor any kind of ladder. If Blue plays at a or c, Red responds at b; if Blue plays at b, Red responds at a; if Blue plays at d, Red responds at e; if Blue plays at f, Red responds at d.

Why it works

Eric's move prevents Blue from getting a 5th row ladder along the left edge. To see why, consider the following line of play, which is one of Blue's best attempts:

987635421

If we imagine that the pink cells are occupied by a line of red stones, then Red's move 9 is actually Tom's move, using that line of stones as its edge. In that case, Red would connect, proving that Blue cannot in general get a 5th row ladder. Even if the pink cells are not in fact occupied by Red, the situation is still typically good for Red.

However, the use of Tom's move in this argument requires quite a bit of empty space. If there is less space, or if there are additional Blue stones in this area, then Blue might still be able to do something useful.

The way in which Eric's move works is essentially the same way as blocking a 5th row ladder when there's not enough space under the ladder stone; see theory of ladder escapes.


etc.

Connecting parallel ladders

Connecting a 2-5 parallel ladder

Like 2-4 and 3-5 parallel ladders, a 2-5 parallel ladder can also connect to the edge outright, given enough space. One way to do this is to yield to a 3-5 parallel ladder and then use Tom's move for 3rd and 5th row parallel ladders. However, there is a way to do it with much less space. In fact, the amount of space shown here is minimal:

1

The ladder stones are marked "↑", and Red's winning move is "1". It is Red's only winning move within this space.

Proof of connectedness: Red's main threats is "*", using the highlighted cells:

1

Red's other main threat is "*", connecting via edge template IV2b, and only requiring 2 of the 3 cells x, y, z:

z1xy

The overlap consists of the cells marked "a", "b", and "c":

1aacbb

If Blue plays at "a", Red pushes the 2nd row ladder to "c" and then uses Tom's move. If Blue plays at "b", Red responds at "c" and then uses Tom's move. Finally, if Blue plays at "c", Red plays as follows:

15324

This move isn't exactly a version of Tom's move, but it does for a 2-5 ladder what Tom's move does for a 2-4 ladder.

Connecting a 2-6 parallel ladder

2-6 parallel ladders also connect. The required amount of space and the unique winning move within this space are shown:

1

The basic idea is that this yields to 2-5, and then Red can use the previous trick.

Connecting a 3-6 parallel ladder

3-6 parallel ladders also connect. The required amount of space and the unique winning move within this space are shown:

1

Remarks

In all three cases, for the ladder to propagate, the top ladder should be one hex further ahead of the bottom ladder than shown above. (If the bottom ladder is already caught up, the top ladder can no longer be pushed). For the 3-5 and 3-6 parallel ladders, Red doesn't necessarily have to push the bottom ladder before playing 1. However, for the 2-6 ladder, Red does have to push the bottom ladder first.

Also, the fact that these ladders all connect means that they are not really "ladders" in the usual sense; they are basically just templates. Note that unlike Tom's move (2-4 and 3-5 ladders), the connection requires no space above the height of the ladder, so the space in which the ladder would normally travel is already enough space to connect it.

Also, as noted on the page Tom's move for 3rd and 5th row parallel ladders, 3-5 parallel ladders can't always be pushed as 3-5 parallel ladders; the defender has the option to downgrade it to a 2nd row ladder with switchback threat. The resulting 2nd row ladder may not connect. Therefore, Tom's move may not be available if the ladder starts some distance from where there is space.