[hooitehb] Go cuts

Given two go 囲碁 chains (connected groups), can you connect them in 1 move?  If so, how many consecutive free (unanswered) moves beforehand by your opponent would eliminate your ability to connect in one move?  This gives a measure of how "almost connected" two chains are.

For example, the bamboo joint and diagonal connection are two examples of connections that require the opponent to make two free moves to cut.

Given two go 囲碁 chains, can you connect them in at most 2 free moves?  If so, how many consecutive free moves beforehand by your opponent would eliminate your ability to connect in at most two moves?  This gives another measure of how "almost connected" two chains are.

Obviously this can be generalized to more moves, yielding a table (specific to that initial position).  Can we then compute some interesting statistic over all the (infinite) entries in the table?

It feels a little like quantum mechanics, summing over all possible paths between two points.

Ko threats sometimes yield free moves.

Generalizing to some amount of alternation between moves: define a template of length N, and a color assigned to each number 1 through N.  There are 2^N possible templates of length N.  The template gives who gets to move on that move number (so not necessarily strictly alternating).  Given a template, determine whether two chains can be connected.  Figure out families of templates which result in connection or cut.

Unclear whether these exercises yield anything useful for the actual game of go 囲碁.  It might be more relevant to Hex, which ultimately tries to connect the two "groups" of opposite sides.