Playing Infinitesimals

Paths: <= CGT path =>   ·   <= Endgame =>
    Keywords: EndGame

The play in infinitesimals is all about getting the last play. Chilling produces infinitesimals from go positions that have miai values of 1.

Both {0 |tiny-1} (the lower corridor) and ^ (Up, the upper corridor) have atomic weights of 1. Which should White choose to play in?

[Diagram]
^ vs {0|tiny-1}  

Should White play at a or b?

The difference game provides the answer. Let's have White play at a in this position and Black play at b in the negative position.



(Note on the diagrams. The stones that frame the positions are, by convention, alive.)

[Diagram]
Difference game  

Each player has played the circled stone from an even position to set up the game. Does either player have an advantage?

[Diagram]
Difference game: Black first  

B1 - W2 is sente. After B5 the rest is miai.
Black to play gets the last play and wins.

[Diagram]
Difference game: White first  

After W1, B2 makes the rest miai. Black gets the last play when White plays first, too.


The difference game favors Black, so White should play in {0|tiny-1} rather than ^. In fact, White should prefer to play in any corridor ending in a tiny before one ending in *.

That seems reasonable, because White's eventual threat is larger in {0|tiny-1}.


Both 0(2)tiny-1 (upper corridor) and 0(2)tiny-2 (lower corridor) have atomic weights of 2. Which should White choose to play in?

[Diagram]
0(2)tiny-1 vs O(2)tiny-2  

Should White play at a or b?

Of course, we do the difference game. White plays at a, Black at b.

[Diagram]

Difference game (ii) Black first

If B1 is at 5, W2 at 8 gets the last play.

[Diagram]

Difference game (ii) Black first (cont)

After B1, White interposes W2 - B3 before W4. White still gets the last play.

[Diagram]

Difference game (ii) White first

Black interposes B4 - W5 before B6.

[Diagram]

Difference game (ii) White first (cont)

White gets the last play when she plays first, too.


Since White gets the last play, regardless of who has sente, the difference game favors White. That means that White should play at a, the corridor with the smaller sente at the end.

Now that's a surprise, isn't it?

The key, it seems, is to save the hotter sente for later. Much like saving a big ko threat. :-)

This is not a case where the opponent's play is my play. Black prefers b, removing the larger threat.


Black can block White at a or b. Here a is sente, with an atomic weight of 0, b has an atomic weight of 1. Which does Black prefer?[1]

[Diagram]
Which should Black prefer?  


Again, we play the difference game. Let Black play at a and White at b.

[Diagram]
Difference game Black first  

Black plays sente and then takes the last play.

[Diagram]
Difference game White first  

Here, too, Black plays sente and then gets the last play.
So Black should play a, taking away White's sente.

Note that the choice depends on the size of Black's threat versus the size of White's threat by playing at 4.

Also see

-- BillSpight


[1] Assuming Black is independently alive. --dnerra

I thought that was covered by assumptions of yose problems; but looking at that page, it could be made (more) explicit. -- Charles

Bill: I thought I covered that with the "note on the diagrams" above that said that the stones framing the diagrams were alive by convention.


Paths: <= CGT path =>   ·   <= Endgame =>
Playing Infinitesimals last edited by Bill on May 1, 2008 - 18:05
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