Other Infinitesimals

    Keywords: EndGame

Yet more infinitesimals in chilled go!

(Note on the diagrams: The stones framing the positions are considered alive by convention.)


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

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

Familiar (Black)  

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.

Familiar miai  

If Black a then White b, and if White b then Black a. Miai it is.

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

Friend in disguise  

If Black plays first, we have a *:


While White can play to miny-2.


So what is our friend?


Well, W1 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 White 1 leaves the local temperature the same does not keep it from reversing.

It's obvious, but looks can be deceiving. If W1 really reverses, our friend is { * | 0 } = v. We can check that by adding it to ^.

Zero (Miai)?  

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

Zero (Miai)?  

or if B4, then W1 - B2, W3.

So, indeed, our friend is v in disguise.


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

Monster, Black first  

B1 moves to a Black sente position, a miny-8.

Monster White first  

W1 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.

Monster miai?  
Monster miai? (Black)  

If Black goes first, W4 gets tedomari.

Monster miai? (W)  

And if White goes first, B4 gets tedomari.

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


The key is the eventual sente play for Black. 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. ;-)

White to play  

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

So tempting!  

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

Nick of time  

White should approach the tiny.
W5 gets tedomari.

Real game example:

'Group Rengo' after move 227  

This position comes from our Ongoing Game. The large White group has six liberties (a - f). Filling the fifth 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 one-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


Other Infinitesimals last edited by BillSpight on October 15, 2003 - 03:07
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