I’m going to suggest taking this a lot slower, doing things in a logical order, and getting really confident with each topic before moving onto the next. Starting with:

## Combinatorial games in general (Section 3.5 - 3.5.3)

I was going to summarize the chapter but really everyone should just read it, or read Winning Ways, and share what they are comfortable with. I will note that I discovered some useful Unicode symbols.

Two are even explicitly referenced as “Symbols used in game theory” in the Unicode standard!

≹ ⧾ ⧿

I’ll also add in the kinds of arrows @shinuito was using for reference:

↑ ↓ ⇑ ⇓ ⤊ ⤋

Definitions of the infinitesimals:

* = {0|0}

↑ = {0|*}

↓ = {*|0}

⇑ = ↑+↑

⤊ = ↑+↑+↑

⧾_{G} = { 0 | {0|-*G*} }, also written as {0 || 0 | -*G*}

⧿_{G} = { *G* | 0 || 0 }

I think this is worth clearing up. After re-reading the section a bit, my understanding is that usually we are talking about equivalence classes of games (*G* = *H* for all members of the class), since equivalent games have the same result and can be substituted for each other in all situations. So when we say “it’s a zero game”, we just mean that the second player wins and it doesn’t matter how.

This is a fun exercise. The point is to justify the notation by proving {0|1} + {0|1} = 1. And the way you do that kind of thing is by showing that {0|1} + {0|1} - 1 is a zero game; the second player wins.

So maybe we should talk about this some more.

If we are at *G* = * for example, Black can move to 0, which is the same as what Black would be able to do from *G* = 1. Effectively Black “rounded” it up to 1. Similarly, White playing on * would “round” it down to -1.

With 1/2 = {0|1}, Black “rounds” to 1 and White “rounds” to 0.

Is that something specific from the book?

Maybe a good way to think of it is to consider games related to *G* = 5 + 1/4 + *. Suppose for now Black is playing first. Black can then win *G* - 5 and White will win *G* - 6. These are the same outcomes as for a game *G* = 6, so it’s like Black rounded the score up to 6.

Similarly, if White plays first, you get the same outcomes as for a game *G* = 5, so it’s like White rounds it down to 5. The outcomes are actually the same as when Black plays first! But that is not the case for integer games, which is why the equivalent values are 5 or 6 depending on who goes first.

I hope I did that right…does it help? Anything else we should review?

Personally, in this section, I have been most confused by the “reversing reversible options” concept, which is ironic since supposedly Go players know it intuitively. I think I’m getting more comfortable with it now though.

Once we’re comfortable with games *in general*, I think we could move on to discussing Go as a game, cooling, and chilling.