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Here are some infinitesimals in chilled go where at least one player has more than one live option. In the other cases we have seen, one option dominated or options were equivalent.

[Diagram]
Diag.: Beast #1

[Diagram]
Diag.: Beast #1 (B)

Black 1 moves to 0 (with the usual adjustments in chilling).


[Diagram]
Diag.: Beast #1 (White i)

White 1 moves to *.


[Diagram]
Diag.: Beast #1 (White ii)

White 1 - 3 reverses to 0.

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

     { 0 | 0, * }

In fact, it equals v + *, and is written v* (DOWN STAR).

To see that, let's add it to ^ and *. We should get a miai position.

[Diagram]
Diag.: Miai?

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


[Diagram]
Diag.: Miai (Black i)

If Black 1, White 2 gets tedomari.


[Diagram]
Diag.: Miai (Black ii)

Black 1 offers White the hardest choice. But White 2 plays in the ^ and gets tedomari.


[Diagram]
Diag.: Miai (White i)

If White 1, then Black 2 gets tedomari. Conversely, if White 2, then Black 1.


[Diagram]
Diag.: Miai (White ii)

Finally, if White 1, then Black still gets tedomari.

So we do have miai, confirming that this beast is v*.

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


[Diagram]
Diag.: Beast #2

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


[Diagram]
Diag.: Beast #2 (White)

White 1 - 3 plays to 0.


[Diagram]
Diag.: Beast #2 (Black i)

Black 1 - 3 also plays to 0.


[Diagram]
Diag.: Beast #2 (Black ii)

But Black also has a sente option, with Black 1. 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 ^. In fact, White will prefer to play in this game to playing in ^.

[Diagram]
Diag.: 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.

[Diagram]
Diag.: Beast #2 vs. ^ Difference game


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

[Diagram]
Diag.: Beast #2 vs. ^ Difference game (B)

Black a is no worse than Black 4, and plainly Black 1 is at least as good as Black a. Through White 4, White gets tedomari.


[Diagram]
Diag.: Beast #2 vs. ^ Difference game (W)

White plays sente with White 1 and then gets tedomari with White 3.

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

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


[Diagram]
Diag.: *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, * | 0, * } = *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 *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:

[Diagram]
Diag.: Nim, anyone?

This is the sum of *2 + *, like a game of Nim with one heap with two tokens and one heap with one token.

[Diagram]
Diag.: Nim, anyone? (Black)

Black 1 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. :-)


[Diagram]
Diag.: Nim error

Black 1 loses. White 2 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 *3 + *5 + *6 = 0 :-)

Bill: Yes, who would have thought it? ;-)
As for *3 + *5 + *6, for starters, how about finding *3 that is not the sum of * and *2? :-)



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