User:Selinger

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 sometimes happens when one player repeatedly tries to block the other from the edge, and the other player repeatedly bridges to one side, like this:

In the above example, Red wins the bridge ladder (i.e., Red connects to the edge). However, if the ladder starts in a slightly different place, 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. We call the winning player the attacker and the losing player the defender of the bridge ladder. Thus, in the first example above, Red is the attacker and Blue is the defender, and in the second example, Blue is the attacker and Red is the defender.

Bottlenecking from a bridge ladder

In the case of an ordinary ladder, if the ladder continues until the end, it is the defender who connects. Therefore, it is up to the attacker to figure out how to disrupt the ladder, usually by pivoting or cornering.

For bridge ladders, the situation is reversed. If the ladder continues until the end, it is the attacker who connects, and therefore, the onus is on the defender to do something about it. The defender usually does this by creating a bottleneck.

Example

Consider a bridge ladder starting on the 6th row, with Red as the attacker.

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.

The defender must choose carefully when to bottleneck. One might think that it is good for the defender to bottleneck as soon as possible, because this results in a ladder further from the attacker's edge. But on the other hand, especially when the bridge ladder is approaching an acute corner, bottlenecking sooner also keeps the defender further from their 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 the bridge ladder approaches an obtuse corner, the situation is similar to the examples above.

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. Rather than being able to create a bottleneck, 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 Red is the attacker 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 defender in the resulting bridge ladder, and cannot connect.

Therefore generally speaking, the last opportunity to pivot from a ladder is before the ladder has reached the long diagonal (in case of a ladder approaching an acute corner). In case the ladder approaches an obtuse corner, a similar heuristic applies.