Book Club: Mathematical Go, Chilling Gets the last point by Elwyn Berlekamp and David Wolfe

So the first funny thing I suppose is the “Easy (?) endgame problem” Figure 1.3

which is to figure out if A or B is better as an endgame move.

The (surprising?) answer was that B is sometimes better but A is never better, and the emphasis is placed later in the explanation on “getting the last move is key”. In combinatorial games, this is actually the whole point (winning condition) of a family of games, so maybe that might not be surprising in some ways, but I guess the Go aspect can still be surprising and/or confusing.

The proof that b is sometimes better, is in an example like a symmetrical position:

Basically the top and bottom positions are identical but color swapped (if go back to just before the circled move). In CGT, when you do this, whatever value you can assign to the bottom position, say V, it means that the top position has value -V. So with all else equal (ignoring how that works on the board size), the two positions cancel, to give a value 0 position, which is a second player “win”. Ok win here, is that Black gets to play to last move if white goes first, and in this case it’s more like a draw on points.

So like in other games, in a mirrored position, there’s a strategy where you copy your opponents moves and that you guarantee and equal result and playing last. So this is the argument that b can sometimes be better.

But if Black answers A rather than copying with B they are supposed to lose by 1 point.

White’s strategy is said to be just to keep pushing into Black’s area and only blocking their own territory when necessary (which I suppose means, keeping the captured stones captured).

It’s funny then that “a” is never better, because you’d think that something similar where White plays “a” in the mirror position, Black could again just copy (they can), but I guess the point is that in that case black can do better. This is Figure 1.5

So this result is supposed to lead to a tie.


So I suppose the obvious things one might want to understand here are “why are the games played out the way they are” - which might be easier given the follow up sections on how some counting is done with corridors etc (or pulling in counting from elsewhere). Then understand the idea of how this sort of proof using the “difference” game as it’s called is supposed to work.

I think ways you can go about that are by counting the values of the moves each player takes. In a lot of ways both players make the same moves, so the values will cancel each other out, but it still might be useful to count some examples to see the difference.

Or one could move on, come back to it later, or maybe only some people are semi interested in it. It’s up for discussion anyway.

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