I have been thinking about various mathsy topics in tsumego problems.
One was the refutation graph. Given a tsumego problem, draw a graph.
Let the vertices be possible valid first two moves in the problem.
Draw a directed edge from A to B if A as a first move is refuted by B on the second move.
Consider the relation "y follows x" xRy with x,y vertices if y can be reached from x by following these directed edges. This is transitive but not symmetric or reflexive.
Turn this into an equivalence relation by saying x =(R) y if and only if
1) xRy and yRx
or
2) x=y
Then consider the equivalence classes.
There are three types
1) Single nodes {x} where xRx is false, and x is not the right first move, so xRy is true for some y.
2) Single nodes {x} with zero outdegree so xRy is false for all y. This is because x is a correct first move (with no refutation).
3) Clusters with more than one node.
Does anyone think this is interesting/useful?
Has anyone thought about this?
For example
- How does the structure of such graphs depend on the board graph (does 2D cartesian make it special)?
- How does the structure of such graphs relate to good shapes, reading, and strategy?
- What structures are possible?
To add context, I was thinking about this in relation to a discussion between John F and Bill S about strategic ideas such as
"If A is your opponent's good move, try playing there yourself."
"If you cannot win in two moves after your opponent plays A, then to have a chance of killing, you must play A first." This actually requires that A doesn't capture any stones for the commutativity proof to work.
- Click Here To Show Diagram Code
[go]$$ Black to play
$$ --------------
$$ | a b c d . . . . .
$$ | e f g h . . . . .
$$ | i j X O O . O O .
$$ | k l X . . . . . .
$$ | m X O . O . . . .
$$ | . O O . O . . . .
$$ | . O . . . . . . .
$$ | . . . . . . . . .[/go]
As an example, this is a classic problem that is very tricky for its size.
f is the unique correct first move, though I think m is ko.
The refutations I could find were
a->f
b->h
c->h
d->h
e->h
g->l
g->m
h->f
i->m
i->f
i->d
j->d
k->f
k->d
l->f
l->d
l->h
m->f
there are plenty more pairs that I haven't mentioned for all the bad moves. It looks like there aren't any clusters, in which case the equivalence relation above is useless. But the graph may still be useful for something.
Instead the key lesson seems to be that g is a bad move for both sides due to missing the key points all around it. It seems that the defender shouldn't play too solidly as that fills in their own eyespace, while the attacker shouldn't leave overconcentrated cutting points as they can be exploited.
d,h,m seem to be key refutation points, on a similar level.
Of course the asymmetry of putting black's moves on the left and white's moves on the right is quite clear. In this shape, stones can get captured, so a good move for one side is not necessarily a good move for the other side.