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Playing Infinitesimals
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The play in infinitesimals is all about getting tedomari. Chilling produces infinitesimals in 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 stone that frame the positions are, by convention, alive.)

[Diagram]
Difference 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 tedomari and wins.


[Diagram]
Difference game: White first

After W1, B2 makes the rest miai. Black gets tedomari 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 tedomari.


[Diagram]
Difference game (ii) Black first (cont)

After B1, White interposes W2 - B3 before W4. White still gets tedomari.


[Diagram]
Difference game (ii) White first

Black interposes B4 - W5 before B6.


[Diagram]
Difference game (ii) White first (cont)

White gets tedomari when she plays first, too.



Since White gets tedomari, 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?

[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 tedomari.


[Diagram]
Difference game White first

Here, too, Black plays sente and then gets tedomari.
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 Other Infinitesimals and More Infinitesimals.

-- BillSpight



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This is a copy of the living page "Playing Infinitesimals" at Sensei's Library.
(OC) 2003 the Authors, published under the OpenContent License V1.0.