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= Proposed article: Bridge ladder =
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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 ''bridge ladder'' is a sequence of moves such as the following:
+
= Proposed page: Eric's move =
<hexboard size="7x8"
+
  edges="bottom"
+
  coords="none"
+
  contents="B a3 R c2 B 1:b4 R 2:d3 B 3:c5 R 4:e4 B 5:d6 R 6:f5 B 7:e7 R 8:g6"
+
  />
+
Here, Red is the ''attacker'', Blue is the ''defender'', and both players play a sequence of [[bridge]]s that approach the attacker's edge at a 30 degree angle, with the defender being closer to the edge than the attacker. Bridge ladders sometimes happen when the defender repeatedly tries to [[blocking|block]] the attacker with a [[Blocking#The_near_block|near block]], and the attacker repeatedly [[bridge]]s to one side.
+
  
In the above example, Red ''wins'' the bridge ladder (i.e., Red connects to the edge). However, if the ladder starts closer to a blue edge, the outcome can be different:
+
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.
<hexboard size="7x7"
+
  edges="bottom right"
+
  coords="none"
+
  contents="B a2 R c1 B 1:b3 R 2:d2 B 3:c4 R 4:e3 B 5:d5 R 6:f4 B 7:e6 R 8:g5 B 9:f7"
+
  />
+
This time Blue wins the ladder. Generally, when a bridge ladder moves towards an empty acute corner, whichever player is closer to the [[board#Diagonals|long diagonal]] wins the ladder. Also note that from the point of view of the red edge, Red is the attacker and Blue is the defender, but from the point of view of the blue edge, Blue is the attacker and Red is the defender. This is typical for bridge ladders approaching an acute corner.
+
  
== Bottlenecking from a bridge ladder ==
+
The move is named after Eric Demer, who discovered it.
  
Let us call the player who would lose a bridge ladder if it continued until the end the ''underdog''. So Blue is the underdog in the first example above, and Red is the underdog in the second example.
+
== Example ==
  
Since the underdog stands to lose the bridge ladder, the onus is usually on them to do something about it, typically by creating a [[bottleneck]].
+
Consider the following situation, with Red's 3rd row ladder approaching from the right, and Red to move.
 
+
<hexboard size="5x8"
=== Example ===
+
  coords="hide"
 
+
  edges="bottom left"
Consider a bridge ladder starting on the 6th row. Blue is the underdog.
+
  contents="B e2 f2 g2 g4 R h2 R g3 B f4 R f3 B e4"/>
<hexboard size="7x8"
+
There's not enough room for Red to [[ladder handling#Attacking|push]] one more time, as this will give Blue a 2nd row ladder:
   edges="bottom right"
+
<hexboard size="5x8"
   coords="none"
+
   coords="hide"
   contents="B b3 R d2 B 1:c4 R 2:e3 B 3:d5 R 4:f4 B 5:e6 R 6:g5 B 7:f7 R 8:h6"
+
   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"
 
   />
 
   />
Instead of continuing the ladder to the end, Blue has the choice to create a [[bottleneck]] on the 5th row, 4th row, or 3rd row:
+
The obvious solution is for Red to pivot immediately and hold Blue to a 5th row ladder:
 
+
<hexboard size="5x8"
'''5th row bottleneck:'''
+
   coords="hide"
<hexboard size="7x8"
+
   edges="bottom left"
   edges="bottom right"
+
   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"
   coords="none"
+
   contents="B b3 R d2 B 1:d3 R 2:c3 B 3:b5"
+
 
   />
 
   />
Red gets a pair of 4th row ladders.
+
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:
'''4th row bottleneck:'''
+
<hexboard size="5x8"
<hexboard size="7x8"
+
   coords="hide"
   edges="bottom right"
+
   edges="bottom left"
   coords="none"
+
   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)"
   contents="B b3 R d2 B 1:c4 R 2:e3 B 3:e4 R 4:d4 B 5:c6"
+
 
   />
 
   />
Red gets a pair of 3rd row ladders.
+
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.
  
'''3rd row bottleneck:'''
+
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="7x8"
+
<hexboard size="5x8"
   edges="bottom right"
+
   coords="hide"
   coords="none"
+
   edges="bottom left"
   contents="B b3 R d2 B 1:c4 R 2:e3 B 3:d5 R 4:f4 B 5:f5 R 6:e5 B 7:d7"
+
   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"
 
   />
 
   />
Red gets a pair of 2nd row ladders.
+
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.  
  
Blue must choose carefully when to bottleneck. One might think that it is good for Blue to bottleneck as soon as possible, because this results in a ladder further from the red edge. But on the other hand, especially when the bridge ladder is approaching an acute corner, bottlenecking sooner also keeps the Blue further from the blue edge. For example, in each of the above scenarios, Red may try to pivot as follows:
+
== Why it works ==
  
'''5th row bottleneck:'''
+
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:
<hexboard size="7x8"
+
<hexboard size="12x8"
   edges="bottom right"
+
   coords="hide"
   coords="none"
+
   edges="bottom left"
   contents="B b3 R d2 B d3 R c3 B b5 R 1:c4 B 2:c5 R 3:e4 B 4:d4 R 5:f2 B 6:e2"
+
   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"
 
   />
 
   />
Red pivots at 3. Assuming 3 connects to the bottom edge, Red gets a 4th row ladder along the bottom edge, and Blue gets a 4th row ladder along the right edge.
+
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.
  
'''4th row bottleneck:'''
+
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.
<hexboard size="7x8"
+
  edges="bottom right"
+
  coords="none"
+
  contents="B b3 R d2 B c4 R e3 B e4 R d4 B c6 R 1:d5 B 2:d6 R 3:f5 B 4:e5 R 5:g3 B 6:f3"
+
  />
+
Red pivots at 3. Red gets a 3rd row ladder along the bottom edge, and Blue gets a 3rd row ladder along the right edge.
+
  
'''3rd row bottleneck:'''
+
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="7x8"
+
  edges="bottom right"
+
  coords="none"
+
  contents="B b3 R d2 B c4 R e3 B d5 R f4 B f5 R e5 B d7 R 1:e6 B 2:e7 R 3:g6 B 4:f6 R 5:h4 B 6:g4"
+
  />
+
Red pivots at 3. Red gets a 2nd row ladder along the bottom edge, and Blue gets a 2nd row ladder along the right edge.
+
  
== Bridge ladder approaching an obtuse corner ==
 
  
When a bridge ladder approaches an obtuse corner, the situation is in principle similar, but there are some differences depending on who is the underdog.
+
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:
  
For example, consider the following:
 
 
<hexboard size="5x9"
 
<hexboard size="5x9"
 +
  edges="bottom"
 
   coords="none"
 
   coords="none"
   edges="bottom left"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="R h1 B g3 R 1:f2 B 2:e4 R 3:d3 B 4:c5 R 5:b4"
+
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2"
 
   />
 
   />
Here, Red wins the ladder, and Blue's last opportunity to [[bottleneck]] was move 2, which would have given Red a 2nd row ladder. On the other hand, when the bridge ladder starts further to the left, the situation is different:
+
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:
 
<hexboard size="5x9"
 
<hexboard size="5x9"
 +
  edges="bottom"
 
   coords="none"
 
   coords="none"
   edges="bottom left"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="R g1 B f3 R 1:e2 B 2:d4 R 3:c3 B 4:b5 R 5:a4 B 6:a5"
+
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 *:c4 S red:(d1,c3,b4,b5,c4,c5,d3)"
 
   />
 
   />
If the bridge ladder continues to the end, Blue connects. Red can't create a bottleneck, but Red can turn the ladder around, for example like this, resulting in a 2nd row ladder for Blue:
+
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"
 
<hexboard size="5x9"
 +
  edges="bottom"
 
   coords="none"
 
   coords="none"
   edges="bottom left"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="R g1 B f3 R 1:e2 B 2:d4 R 3:c3 B 4:b5 R 5:a5 B 6:b4"
+
   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"
 
   />
 
   />
or like this, resulting in a 4th row ladder for Blue:
+
The overlap consists of the cells marked "a", "b", and "c":
 
<hexboard size="5x9"
 
<hexboard size="5x9"
 +
  edges="bottom"
 
   coords="none"
 
   coords="none"
   edges="bottom left"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="R g1 B f3 R 1:e2 B 2:d4 R 3:c4 B 4:d3"
+
   contents="R arrow(12):c1,a4 B c2 a5 R 1:d2 E a:c3,d3 c:c4 b:b5,c5"
 
   />
 
   />
or even like this, resulting in no ladder for Blue:
+
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"
 
<hexboard size="5x9"
 +
  edges="bottom"
 
   coords="none"
 
   coords="none"
   edges="bottom left"
+
   visible="area(c1,a4,a5,i5,i3,g1)"
   contents="R g1 B f3 R 1:d2 B 2:e2 R 3:c2"
+
   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.
  
== Application: last opportunity to pivot from a ladder ==
+
== Connecting a 2-6 parallel ladder ==
  
Consider an (ordinary) [[ladder]] moving parallel to an edge. In the absence of a ladder escape, the attacker must at some point do something, typically [[Ladder_handling#Attacking|pivot]] or play a [[cornering]] move. One may ask when is the last possible opportunity to pivot. A useful heuristic is to consider the bridge ladder that would result if the defender yielded after the pivot. For example, consider a 4th row ladder approaching from the left. If Red pivots at 5, then Blue is the underdog in the resulting bridge ladder, so Blue has to do something else (like [[bottleneck|bottlenecking]]).
+
2-6 parallel ladders also connect. The required amount of space and the unique winning move within this space are shown:
<hexboard size="5x8"
+
<hexboard size="6x12"
   edges="bottom right"
+
   edges="bottom"
 
   coords="none"
 
   coords="none"
   contents="R a2 B a3 R b2 B b3 R 1:c2 B 2:c3 R 3:d2 B 4:d3 R 5:f2 B 6:e4 R 7:g3 B 8:f5 R 9:h4 B 9:g6"
+
  visible="area(d1,d3,b4,a5,a6,l6,l4,j2,f1)"
 +
   contents="R arrow(12):d1,a5 B d2,a6 R 1:f2"
 
   />
 
   />
On the other hand, if Red waits until 7 to pivot, ''Red'' ends up being the underdog, and cannot connect.
+
The basic idea is that this yields to 2-5, and then Red can use the previous trick.
<hexboard size="5x8"
+
 
   edges="bottom right"
+
== 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"
 
   coords="none"
 
   coords="none"
   contents="R a2 B a3 R b2 B b3 R 1:c2 B 2:c3 R 3:d2 B 4:d3 R 5:e2 B 6:e3 R 7:g2 B 8:f4 R 9:h3 B 10:g5"
+
  visible="area(c1,a4,a6,m6,m4,k2,g1)"
 +
   contents="R arrow(12):c1,a4 B c2,a5 R 1:d2"
 
   />
 
   />
Therefore, generally speaking, the last opportunity to pivot from a ladder approaching an [[Board#Corners|acute corner]] is before the ladder has reached the long diagonal. A similar analysis applies to ladders approaching an [[Board#Corners|obtuse corner]].
+
 
 +
== 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.

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.