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Infinitesimals

 

Other Infinitesimals
   

Yet more infinitesimals in chilled go!

[Diagram]
Diag.: Familiar

This infinitesimal is common. And it is one we already know. Which one is it?

If White fills, the result is 0, OC. If Black takes we get this.

[Diagram]
Diag.: Familiar (B)

And this we know to be * (STAR).

So our familiar infinitesimal is { * | 0 } = v (DOWN).

Let's check that. Since ^ + v = 0, this position should be miai.

[Diagram]
Diag.: Familiar miai

If B a then W b, and if W b then B a. Miai it is.


Somewhat less common is this position, which is also a familiar infinitesimal in disguise.

[Diagram]
Diag.: Friend in disguise


If Black plays first, we have a *,

[Diagram]
Diag.: Miny-2

While White can play to miny-2.
So what is our friend?

[Diagram]
Diag.: Reversal

Well, W 1 reverses to "zero" (ignoring the count, as usual).
This sequence is obvious to most go players who have not thought about infinitesimals. Well, just because W 1 leaves the local temperature the same does not keep it from reversing.

It's obvious, but looks can be deceiving. If W 1 really reverses, our friend is { * | 0 } = v. We can check that by adding it to ^.

[Diagram]
Diag.: Zero (Miai)?

If B a, then W b; and if W b, then B a. If White starts at a,


[Diagram]
Diag.: Zero (Miai)?

or if B 4, then W 1 - B 2, W 3.

So, indeed, our friend is v in disguise.


[Diagram]
Diag.: Monster

This monster looks formidable, but it is also a kind of familiar infinitesimal in disguise. :-)


[Diagram]
Diag.: Monster, Black first

B 1 moves to a Black sente position, a miny-8.


[Diagram]
Diag.: Monster White first

W 1 reverses to 0.
So the monster is { miny-8 | 0 }. Not so bad. :-)

That reverse for White is not so obvious. Let's check it.
{ miny-8 | 0 } + { 0 | tiny-8 } = 0 , so this should be miai.

[Diagram]
Diag.: Monster miai?

[Diagram]
Diag.: Monster miai? (B)

If Black goes first, W 4 gets tedomari.


[Diagram]
Diag.: Monster miai? (W)

And if White goes first, B 4 gets tedomari.

So the monster is a 0-miny. That's surprising. It doesn't look at all like this position:

[Diagram]
Diag.: 0-miny-8

The key is the eventual Black sente. Each Black play filling one of White's liberties is like a move in the corridor.

This kind of position can be very deceptive. That White sente is so tempting. ;-)

[Diagram]
Diag.: White to play


Where should White play? (OC, you have gotten a big hint. ;-))

[Diagram]
Diag.: So tempting!

How many people would jump at the sente?
But Black gets tedomari.


[Diagram]
Diag.: Nick of time

White should approach the tiny.
W 5 gets tedomari.

Real game example:

[Diagram]
Diag.: 'Group Rengo' after move 227

This position comes from our Ongoing Game. The large White group has 6 liberties (a - f). Filling the 5th one is Black's sente, so this infinitesimal is a 0(4)miny. With such a large atomic weight, White should have no trouble getting the tedomari at temperature 1. However, Black got the last 1 point move, because White took his sente.
It did not affect the final score, but it doesn't hurt to get that tedomari. :-)

Also see More Infinitesimals.

--BillSpight



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