More Infinitesimals

Beast #1 (B)  

moves to 0 (with the usual adjustments in chilling).

Beast #1 (White i)  

moves to `` ast `` (STAR).

Beast #1 (White ii)  

- reverses to 0.

The games `` 0 `` and `` ast `` are confused, written 0 <> *, neither dominating the other, and here `` ast `` does not reverse. So the game looks like this:

`` { 0 | 0, ast } ``

In fact, it equals `` darr + ast `` (DOWN + STAR), which is written “``"↓"ast``” and referred to as “DOWN STAR”.

To see that, let's add it to `` uarr `` (UP) and `` ast ``. We should get a miai position.

Miai?  

The miai is not so obvious. Let's look at some lines of play.

Miai (Black i)  

If , gets tedomari.

Miai (Black ii)  

offers White the hardest choice. But plays in the `` uarr `` and gets tedomari.

Miai (White i)  

If , then gets tedomari. Conversely, if , then .

Miai (White ii)  

Finally, if , then Black still gets tedomari.

So we do have miai, confirming that this beast is `` "↓"ast ``.

How to play it? It is confused with ``darr``, and has an atomic weight of -1. Treat it as the sum of two positions, `` darr + ast ``. If there is an ``uarr`` on the board, their sum is `` ast ``. If there are an odd number of `` ast ``s on the board, their sum with this beast is ``darr``. Play accordingly.

Beast #2  

This beast crops up occasionally. In fact, one like it is in the top left corner of the Ongoing Game. It is an ambiguous position.

Beast #2 (White)  

- plays to 0.

Beast #2 (Black i)  

- also plays to 0.

Beast #2 (Black ii)  

But Black also has a sente option, with . Black threatens the sagari at a, saving his stone.

The chilled game looks like this:

`` { 0, { 3.75 | 0 } " || " 0 } ``

Of course, the extra option for Black gives him an advantage. Somewhat surprisingly, this game has an atomic weight of 1, just like ``uarr``. In fact, White will prefer to play in this game to playing in ``uarr``.

Beast #2 vs. ↑  

In other words, White should play at a rather than b.

To check that, let's play the difference game. On the left, White plays at a; in the negative game on the right, Black plays at b.

Beast #2 vs. ↑ Difference game  

If White's play at a is correct, White should get tedomari regardless of who plays first.

Beast #2 vs. ↑ Difference game (B)  

If , plays to a negative position, which is a White win.

Beast #2 vs. ↑ Difference game (B)  

Plainly is at least as good as Black a. But White still gets tedomari.

Beast #2 vs. ↑ Difference game (W)  

White plays sente with and then gets tedomari with .

So White should prefer to play in the beast over playing in ``uarr``, or in a longer such corridor.

That illustrates the power of the extra option. White would prefer to play in ``uarr`` over a play in either `` ast `` (gote) or a miny (reverse sente).

*2 (Star-2)  

This infinitesimal was only recently found in chilled go. Each player has the choice of a or b. A move to a produces a 0, a move to b produces a *.
The game looks like this:

`` { 0, ast | 0, ast } = ast"2" ``

At first, the idea of a play at b seems absurd. Why make a play that allows your opponent to return to even, when you can just gain a point?

It happens that ``ast"2"`` is well known. It is the infinitesimal for a Nim heap of ``2``. If you have played Nim, you know that you can take either one or two tokens from such a heap.

Here is a Nim position in go:

Nim, anyone?  

This is the sum of `` {:ast"2":} + ast ``, like a game of Nim with one heap with two tokens and one heap with one token.

Nim, anyone? (Black)  

wins (gets tedomari) by the equivalent of taking one token from the heap with two tokens. The rest is miai.
White wins in the same way. :-)

Nim error  

loses. gets tedomari. Now a and b are miai.

– BillSpight

Jan: Bill, this is quite interesting! Nim positions in Go, who would have thought it? I would like to see a position showing that `` {:ast"3":} + {:ast"5":} + {:ast"6":} = 0 `` :-)

Beast #3  

Bill: Assuming, as usual, Black territory to the left, it appears to chill to `` { ast | 0^2 -_4 } ``. See BQM222.

Beast #3, White starts  

White plays the hane-tsugi, leaving a favorable corridor on the right.

Beast #3, Black starts  

Black plays the hane, leaving a STAR.