Revision as of 22:57, 8 September 2021

Proposed article: Bridge ladder

To do: figure out a useful way to define "attacker" and "defender". I currently define this in terms of who will "win" the ladder, but that only makes sense when the ladder approaches an acute corner. Maybe it should be defined with reference to an edge instead, or not at all.

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

Here, Red is the attacker, Blue is the defender, and both players play a sequence of bridges 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 block the attacker with a near block, and the attacker repeatedly bridges 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:

This time Blue wins the ladder. Generally, when a bridge ladder moves towards an empty acute corner, whichever player is closer to the 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.

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:

Red gets a pair of 4th row ladders.

4th row bottleneck:

Red gets a pair of 3rd row ladders.

3rd row bottleneck:

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:

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:

Red gets a 3th row ladder and 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:

Red gets a 3th row ladder and 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:

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:

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:

or like this, resulting in a 4th row ladder for Blue:

or even like this, resulting in no ladder for Blue:

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 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 bottlenecking).

On the other hand, if Red waits until 7 to pivot, Red ends up being the underdog, and cannot connect.

Therefore generally speaking, the last opportunity to pivot from a ladder approaching an acute corner is before the ladder has reached the long diagonal. A similar analysis applies to ladders approaching an obtuse corner.