View source for Bridge ladder

A ''bridge ladder'' is a sequence of moves such as the following:
<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:
<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 ==

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.

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]].

=== Example ===

Consider a bridge ladder starting on the 6th row. Blue is the underdog.
<hexboard size="7x8"
  edges="bottom right"
  coords="none"
  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"
  />
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:

'''5th row bottleneck:'''
<hexboard size="7x8"
  edges="bottom right"
  coords="none"
  contents="B b3 R d2 B 1:d3 R 2:c3 B 3:b5"
  />
Red gets a pair of 4th row ladders.

'''4th row bottleneck:'''
<hexboard size="7x8"
  edges="bottom right"
  coords="none"
  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.

'''3rd row bottleneck:'''
<hexboard size="7x8"
  edges="bottom right"
  coords="none"
  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"
  />
Red gets a pair of 2nd row ladders.

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:

'''5th row bottleneck:'''
<hexboard size="7x8"
  edges="bottom right"
  coords="none"
  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"
  />
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.

'''4th row bottleneck:'''
<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:'''
<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.

For example, consider the following:
<hexboard size="5x9"
  coords="none"
  edges="bottom left"
  contents="R h1 B g3 R 1:f2 B 2:e4 R 3:d3 B 4:c5 R 5:b4"
  />
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:
<hexboard size="5x9"
  coords="none"
  edges="bottom left"
  contents="R g1 B f3 R 1:e2 B 2:d4 R 3:c3 B 4:b5 R 5:a4 B 6:a5"
  />
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:
<hexboard size="5x9"
  coords="none"
  edges="bottom left"
  contents="R g1 B f3 R 1:e2 B 2:d4 R 3:c3 B 4:b5 R 5:a5 B 6:b4"
  />
or like this, resulting in a 4th row ladder for Blue:
<hexboard size="5x9"
  coords="none"
  edges="bottom left"
  contents="R g1 B f3 R 1:e2 B 2:d4 R 3:c4 B 4:d3"
  />
or even like this, resulting in no ladder for Blue:
<hexboard size="5x9"
  coords="none"
  edges="bottom left"
  contents="R g1 B f3 R 1:d2 B 2:e2 R 3:c2"
  />

== Application: last opportunity to pivot from a 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]]).
<hexboard size="5x8"
  edges="bottom right"
  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"
  />
On the other hand, if Red waits until 7 to pivot, ''Red'' ends up being the underdog, and cannot connect.
<hexboard size="5x8"
  edges="bottom right"
  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"
  />
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]].

[[category:Intermediate Strategy]]
[[category:Definition]]