# Infinitesimals Discussion

__Keywords__: EndGame

Here are some infinitesimals in chilled go.

Can you tell what they are? What is this position worth?

(Answers in Corridor Infinitesimals, further discussion here. :-))

Holigor:

The upper one is 2 points for Black if Black moves first and 0 points if White moves first.

So, it is a one-point move, value one point.

The second is 3 (Black)/reduces to the first one (White). As we know that the first was one point then the score is 2 points, and a move is worth one point.

The third is 4 points (Black)/reduces to previous (White). The previous was two points. Thus the score is three points, a move value is one point.

Bill: Yes, indeed. :-)

JanDeWit: Let's see if I have been paying attention :-)

Top position is {2 | 0}, mast value 1, temperature 1, and chills to * (star), I think.

Bill: Right. :-)

Second one is { 2 , 3 | {4 | 0} , {2 | 0} } which is equal to {3 | {2 | 0}} (both first options are dominated by the second). Left stop is 3 with Right to play; right stop is 2 with Right to play. Cooling by one gives {2| 1+*}, so I think the mast value is 5/2 and temperature is 1/2.

Bill: Chilling gives {2 | 2*}. Remember to include the effect of a white play to 1*.

Bottom position is (crunch crunch) { {3|1}, 3 , 5 | {4,5|{6|0},{4|0}} , { {4|2},5 | {6|0},{2|0} } , {2,3 | {4|0} ,{2|0}} } which simplifies to

{ 5 | {5|{4|0}} , { 5 | {2|0}} , {3 | {2 | 0}} which in turn simplifies to { 5 | { 3 | { 2 | 0 }}}. (Modulo the early-morning fudge factor of course!)

Bill: A little too early! ;-) The game simplifies to

{ 4 ||| 3 || 2 | 0 } which chills to { 3 | 3^} = 3^^*

Jan: Indeed! Memo to self : 3 + 1 captured stone is not equal to 5; I think I counted the captured stone three times. The rest is wrong as well now... :-(

Juuitchan: I have seen this kind of weird math on several pages so far. How exactly does it work? How do these "infinitesimals" work?

Bill: Infinitesimals are games in Combinatorial Game Theory in which the payoff is getting the last move, or tedomari. In their book, Mathematical Go, Berlekamp and __Wolfe__? show how certain go positions with a miai value of 1, such as the ones at the top of this page, act like infinitesimals. When tedomari is the issue, knowing something about infinitesimals may help.

I am trying to figure out the one above. Is it 2 + {0|2*}?

Slarty: That one is {5|{2|0}}. I think it chills to {4|1+{1|1}} = 2 + {2|*}.

2* is mentioned in Analysis of Composite Corridors and looks like -

The room above chills to 1/2 according to Mathematical Go (or 3/2 if you ignore the marking above b). This makes sense to me if black plays b, but if back plays a, it appears to me to chill to {1|1/2}|1. If black a, black moves the game to 2|3/2, if b, black moves to 1. White moves to 2 starting from any of a, b or c. Have I missed something?

Bill: Yes, you have missed the miai.

Slarty: not sure about those 1/2's. here's my initial take after black *a*.
... So, black *a* makes the position {3||1+*|*+*}, which chills to {2||1+*|2+*+*} ignore the infinitesimals, and we see this is {2|1} (edit: wrong, just 2|3/2). whose mean is 3/2, but black paid a tax point for playing *a*, giving 1/2. I don't know what marking is good for in this example. You missed white's followup.

I don't understand why 2|{1|2} = {2|1}. I think {1|2} = 3/2, so I have 2|{1|2} = 2|3/2. Secondly you have {2|1} = 3/2, but I'm pretty sure that's not correct as {2|1} is not a number (the left is larger than the right). Your argument (if correct) shows that the original diagram chills to {1/2|?} (if black plays a), but what about the right (white) option?

Slarty: OK, I was not being very consistent with bookkeeping the tax among other things. For the white option, in chilled Go, it's clear that white only has the option to move to 0. (If white *a* or *c*, all that's left are one point sized rooms.) And if black moves first, black gets 1. So eyeballing it (maybe just repeating my mistake), the chilled game is {1-1|0+1} = {0|1} = 1/2. Here is the more ugly derivation... G = {<Black plays *a*>|<White plays *a*>} = {3||1+*|*+* ||| {1|*}+{1|*}}
{1|*}+{1|*} = {1+{1|*} || *+{1|*}} ( cgsuite says this is 1* = 1+{0|0}. I don't know an elegant way in notation to show it, but any way it must chill to 0.)
Now chill G. G chilled = {2|3/2 -1 || 0 + 1 } = {1|1/2||1} = 1/2 ? I don't have enough definitions in front of me to do this properly. I think you got an alternative form of 1/2. What do you make of it.

I think you could add some linebreaks for clarity. The first diagram chills to {2|3/2}|2 (?), cgsuite gives {2|3/2}|2 = 3/2. Putting aside for them moment the question of whether {2|3/2}|2 is correct, why (or via what process) does {2|3/2}|2 = 3/2?

Stumped.