Bill: Another way of looking at tewari (2007-08-10 02:51) [#3786]

Instead of talking about permutation of sequences or removal of paired stones, I want to start with the idea of the effectiveness of plays.

One indication of the effectiveness of a play is how well it works with stones already on the board. If every one of my stones is perfectly placed, then if I remove one of them and then make a play, I can do no better than play a stone on the same spot where the removed stone was.

OC, there are different permutations of play that lead to the same position with different stones being the last one played. So the idea of permuting sequences of play arises naturally from the idea of the stones working together.

Next, suppose that I remove a stone that was played in sente, and, with the reply to it still on the board, there is now no need for that play. (E. g., I remove a stone that threatened to kill a group, and now that the group is safe, I can play that stone elsewhere.) Plainly we have to treat the removal of such a stone differently from the removal of other stones. One way to do that is to remove the stone that was played in response to it. Now, since making a (correct) sente does not alter the value of a position, I can do no better than to play the sente again, once both stones have been removed.

From this the idea of removing pairs of stones arises. We can generalize from removing a sente-gote pair to removing any pair of stones, as long as doing so leaves the value of the position unchanged. We can also generalize to pairing captured stones with stones inside the opponent's territory, as long as the only effect of the removal is to yield one more point of territory.

From this way of thinking we can see how the permutation of sequences and removal of paired stones rest upon the effectiveness of the stones.