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(Added "D5 does not escape a 4th row ladder")
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= Proposed page: Near ladder escapes =
+
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.
  
There are a number of [[ladder]] situations where a player does not technically have a [[ladder escape]], but in practice often ends up escaping the ladder anyway. This usually happens because the opponent must play extremely precisely in order to prevent the ladder from escaping, and can easily miss the correct move. In such cases, we may speak of a '''near ladder escape'''.
+
= Proposed page: Eric's move =
  
This pages lists some common near ladder escapes, and how to thwart them.
+
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.
  
== C4 does not escape a 5th row ladder ==
+
The move is named after Eric Demer, who discovered it.
  
A single stone at c4 (or the equivalent cell on the opposite side of the board) does not escape a 5th row ladder, even when there is a certain amount space on the 6th row as shown here:
+
== Example ==
<hexboard size="6x11"
+
  edges="bottom right"
+
  coords="none"
+
  visible="area(a6,k6,k1,f1)"
+
  contents="R i3 1:e2 B d3 f1 g1"
+
  />
+
  
However, there is only one way to prevent the ladder from connecting. Blue must play as follows.
+
Consider the following situation, with Red's 3rd row ladder approaching from the right, and Red to move.
<hexboard size="6x11"
+
<hexboard size="5x8"
   edges="bottom right"
+
  coords="hide"
   coords="none"
+
   edges="bottom left"
   visible="area(a6,k6,k1,f1)"
+
  contents="B e2 f2 g2 g4 R h2 R g3 B f4 R f3 B e4"/>
   contents="R i3 1:e2 B d3 f1 g1 B 2:e3 R 3:f2 B 4:h4"
+
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"
 
   />
 
   />
In this situation, 2 followed by 4 is the only winning sequence for Blue. The best Red can do is the following, which is not sufficient to connect Red's ladder:
+
The obvious solution is for Red to pivot immediately and hold Blue to a 5th row ladder:
<hexboard size="6x11"
+
<hexboard size="5x8"
  edges="bottom right"
+
   coords="hide"
   coords="none"
+
   edges="bottom left"
   visible="area(a6,k6,k1,f1)"
+
   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"
   contents="R i3 e2 B d3 f1 g1 B e3 R f2 B h4 R 5:h3 B 6:i4 R 7:g4 B 8:g3 R 9:f3 B 10:e5 R 11:j2 B 12:f4 R 13:h1 B 14:g2"
+
  />
 +
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 gets a 5th-to-3rd row [[foldback]], so if Red escapes a 3rd row ladder moving left, Red connects.
+
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.
  
Also note that Red would be able to connect if the stone to the left of 13 were not occupied. Therefore, with slightly more space on the 6th row, a single stone at c4 actually does escape a 5th row ladder:
+
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.
<hexboard size="6x11"
+
<hexboard size="5x8"
  edges="bottom right"
+
   coords="hide"
   coords="none"
+
   edges="bottom left"
   visible="area(a6,k6,k1,f1)"
+
   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
   contents="R i3 1:e2 B d3 f1"
+
            E a:c2 b:d2 c:b3 d:c3 e:a4 f:b4"
 
   />
 
   />
Conversely, if there is less space on the 6th row, Blue has additional ways of blocking the ladder, such as this:
+
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.
<hexboard size="6x11"
+
 
  edges="bottom right"
+
== Why it works ==
   coords="none"
+
 
   visible="area(a6,k6,k1,f1)"
+
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:
   contents="R i3 1:e2 B d3 f1 g1 h1 B 2:e3 R 3:f2 B 4:f3 R 5:g2 B 6:h3 R 7:g3 B 8:f5 R 9:g4 B 10:g6"
+
<hexboard size="12x8"
 +
   coords="hide"
 +
   edges="bottom left"
 +
   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.
  
== D5 does not escape a 4th row ladder ==
+
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.
  
A single stone at D5 (or the equivalent cell on the opposite side of the board) does not escape a 4th row ladder, even when the 6th row is empty as shown here. However, the situation is still very threatening. Red gets both a [[foldback]] and a [[switchback]].  
+
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="6x12"
+
 
   edges="bottom right"
+
 
 +
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:
 +
 
 +
<hexboard size="5x9"
 +
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="area(a6,l6,l1,f1)"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="R i2 1:d3 B c4"
+
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2"
 
   />
 
   />
In the above situation, Blue's only winning move is to [[ladder handling|push]].
+
The ladder stones are marked "↑", and Red's winning move is "1". It is Red's only winning move within this space.
<hexboard size="6x12"
+
 
   edges="bottom right"
+
Proof of connectedness: Red's main threats is "*", using the highlighted cells:
 +
<hexboard size="5x9"
 +
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="area(a6,l6,l1,f1)"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="R i2 1:d3 B c4 B 2:d4 R 3:e3 E x:e4 y:f5 z:d6"
+
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 *:c4 S red:(d1,c3,b4,b5,c4,c5,d3)"
 
   />
 
   />
For move 4, Blue has three possible choices: x, y, or z. If Blue plays moves 4 and 6 at y and z (in either order), Red gets a foldback and a switchback, but does not connect outright:
+
Red's other main threat is "*", connecting via [[edge template IV2b]], and only requiring 2 of the 3 cells x, y, z:
<hexboard size="6x12"
+
<hexboard size="5x9"
   edges="bottom right"
+
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="area(a6,l6,l1,f1)"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="R i2 1:d3 B c4 B 2:d4 R 3:e3 B 4:f5 R 5:f4 B 6:d6 R 7:e5 B 8:e6 R 9:d5 B 10:c6 R 11:h4 B 12:g4 R 13:h2 B 14:f3 R 15: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"
 
   />
 
   />
Note that Blue cannot play move 6 on the 2nd row, or else Red gets a forcing move that allows Red to connect outright:
+
The overlap consists of the cells marked "a", "b", and "c":
<hexboard size="6x12"
+
<hexboard size="5x9"
   edges="bottom right"
+
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="area(a6,l6,l1,f1)"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="R i2 1:d3 B c4 B 2:d4 R 3:e3 B 4:f5 R 5:f4 B 6:e5 R 7:e4 B 8:d5 R 9:h4 B 10:g4 R 11:h2"
+
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 E a:c3,d3 c:c4 b:b5,c5"
 
   />
 
   />
If Blue plays move 4 at x, then on the next move, Red again has three possiblities:
+
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="6x12"
+
<hexboard size="5x9"
   edges="bottom right"
+
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="area(a6,l6,l1,f1)"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="R i2 1:d3 B c4 B 2:d4 R 3:e3 B 4:e4 R 5:f3 E x:g4 y:g5 z:e6"
+
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 B 2:c4 R 3:b4 B 4:b5 R 5:e3"
 
   />
 
   />
If Blue plays moves 6 and 8 at y and z (in either order), Red gets a foldback and a switchback:
+
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:
 
<hexboard size="6x12"
 
<hexboard size="6x12"
   edges="bottom right"
+
   edges="bottom"
 
   coords="none"
 
   coords="none"
   visible="area(a6,l6,l1,f1)"
+
   visible="area(d1,d3,b4,a5,a6,l6,l4,j2,f1)"
   contents="R i2 1:d3 B c4 B 2:d4 R 3:e3 B 4:e4 R 5:f3 B 6:g5 R 7:g4 B 8:e6 R 9:f5 B 10:f6 R 11:e5 B 12:d6 R 13:i4 B 14:h4 R 15:j3 B 16:g3 R 17:h1"
+
   contents="R arrow(12):d1,a5 B d2,a6 R 1:f2"
 
   />
 
   />
If Blue plays move 6 at x, Red also gets a foldback and switchback:
+
The basic idea is that this yields to 2-5, and then Red can use the previous trick.
<hexboard size="6x12"
+
  edges="bottom right"
+
  coords="none"
+
  visible="area(a6,l6,l1,f1)"
+
  contents="R i2 1:d3 B c4 B 2:d4 R 3:e3 B 4:e4 R 5:f3 B 6:g4 R 7:f4 B 8:e6 R 9:e5 B 10:d6 R 11:f5 B 12:f6 R 13:h5 B 14:g5 R 15:h3 B 16:g3 R 17:h1"
+
  />
+
In all other cases, Red connects outright.
+
  
 +
== 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:
  
 
+
<hexboard size="6x13"
 
+
   edges="bottom"
= Unused draft material for "Question" =
+
   coords="none"
 
+
   visible="area(c1,a4,a6,m6,m4,k2,g1)"
The following was a draft example for the page [[Question]], but it turned out to be too complicated to and not have a good answer.
+
   contents="R arrow(12):c1,a4 B c2,a5 R 1:d2"
 
+
== Example: Template intrusion ==
+
 
+
Consider the following position, with Blue to move:
+
<hexboard size="11x11"
+
  coords="left bottom right"
+
   edges="left bottom right"
+
  visible="area(a7,a11,k11,k7)"
+
  contents="R f7 g7 B d8 h9 S area(f7,c9,a11,g11,g7)"
+
  />
+
Note that Red is connected to the edge by [[Fifth_row_edge_templates#V-2-b|edge template V2b]], as highlighted. Blue would like to intrude into this template to gain strength either on the left or on the right.
+
 
+
Blue would like a 4th row ladder escape on the left. But the problem is that if Blue plays at d9 or c10, Red can reconnect by playing a [[minimax|minimaxing move]] at h7, which strenghtens Red's position.
+
<hexboard size="11x11"
+
   coords="left bottom right"
+
  edges="left bottom right"
+
   visible="area(a7,a11,k11,k7)"
+
  contents="R f7 g7 B d8 h9 B 1:d9 R 2:h7 S area(f7,e8,c11,g11,h8,h7)"
+
  />
+
Blue would also like a 4th row ladder escape on the right. But again, the problem is that if Blue moves at g10 or g9, Red can reconnect at g8, or by playing a minimaxing move, say at b10:
+
<hexboard size="11x11"
+
  coords="left bottom right"
+
  edges="left bottom right"
+
  visible="area(a7,a11,k11,k7)"
+
   contents="R f7 g7 B d8 h9 B 1:g10 R 2:b10 S area(f7,c9,a11,f11,g8,g7)"
+
  />
+
Neither of these outcomes is great for Blue. Instead, what Blue can do is ask the template a question:
+
<hexboard size="11x11"
+
  coords="left bottom right"
+
  edges="left bottom right"
+
  visible="area(a7,a11,k11,k7)"
+
  contents="R f7 g7 B d8 h9 B 1:f8"
+
  />
+
Basically, the question is: "How do you want to reconnect?" And based on the answer, Blue will be able to gain some strength on the left or on the right, without giving Red quite as much territory as would otherwise have been the case.
+
 
+
For example, if Red reconnects at e8, then Blue can play e10:
+
<hexboard size="11x11"
+
  coords="left bottom right"
+
  edges="left bottom right"
+
  visible="area(a7,a11,k11,k7)"
+
  contents="R f7 g7 B d8 h9 B 1:f8 R 2:e8 B 3:e10 S red:(d9 d10 f9 f10 g8 g10)"  
+
 
   />
 
   />
  
Now Red's [[mustplay region]] consists of the 6 highlighted cells. If Red plays at d9, Blue gets a forcing move at b10, giving Blue a 4th row escape on the left, without Red getting g8. If Red plays at d10, g8, or g10, Blue gets a forcing move at d9, giving Blue a 4th row escape on the left without Red getting g8. If Red plays at g9, Blue defends the bridge at f9 and then plays as before. Finally, if Red plays at f10, Blue can respond at g8, getting a 4th row escape on the right. Although Red can still reconnect at b10, taking away Blue's ladder escape on the left, Red does not have the option of getting g8.
+
== Remarks ==
  
To be continued... and simplified?
+
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.
  
== To do ==
+
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.
  
Add other illustrative examples, such as a template intrusion that forces the player to  trade-off between a stronger connection and letting the opponent get a ladder escape, etc.
+
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.

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.