Thermography

[Bill Spight]

I begin to understand the interest of chilling-go but I need to know another result. At the end of the chilled-go, after each infinitesimal has been played, each of the remaining areas has a miai value ≤1.

In terms of regular go, chilled go infinitesimals have a miai value of 1. So the remaining plays have a miai value strictly less than 1. (Miai value is a go term, not a CGT term. It applies to plays, not areas. In a way, we have too much terminology. )

One important lesson of Mathematical Go is that the fight for the last play of the game in go is actually about the fight for the last play at temperature 1; i.e., for the last play in the chilled go game. Hence the subtitle: Chilling gets the last point. That discovery was new to go theorists, and many of them may still be unaware of it, unless they have read Mathematical Go. AFAICT, the bots haven't learned that, either. It is not exactly obvious.

Let's call G1, G2, G3 ... all these remaining areas and let's call s1, s2, s3 .. the score of these areas.

In chilled go they are scores, in regular go they are counts or territorial values. (Too much terminology. )

The remaining game is G = G1 + G2 + G3 + ....
Is it true that, at the end of chilling-go we have:
1) G score = s1 + s2 + s3 + ....
2) G miai value ≤1

Well, the chilled score for G will be the sum of the scores of its components. Which will be their combined count in regular go.

In regular go the effective miai value of the sum of games will be less than 1. For instance, although the miai value of each play in the following diagram is 1, the effective miai value of their sum is -1.

$$ Miai
$$ -----------------
$$ | O . O . . . . |
$$ | X X O . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | X X O . . . . |
$$ | O . O . . . . |
$$ -----------------

Click Here To Show Diagram Code

[go]$$ Miai
$$ -----------------
$$ | O . O . . . . |
$$ | X X O . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | X X O . . . . |
$$ | O . O . . . . |
$$ -----------------[/go]

In case of miai, the miai value of the sum is 0, isn't it?

Well, the mast is purple up to temperature 1, which means that either player can choose to play at or below that temperature. It also means that the miai value of a play in the sum is ambiguous. Either player could play at temperature 1, but neither player needs to play above temperature 0, or, if the players agree to the score, at all (temperature -1).

I think that most go players would say that the miai value of this combination is 1. To score the game, most rules require play at temperature 0, so in practice that is the effective miai value. Thanks to Berlekamp's subterranean thermography, we can also say that the effective miai value is -1. At area scoring the players could wait until the end of the game to play in the sum.

[Gérard TAILLE]

Well, the chilled score for G will be the sum of the scores of its components. Which will be their combined count in regular go.

In regular go the effective miai value of the sum of games will be less than 1. For instance, although the miai value of each play in the following diagram is 1, the effective miai value of their sum is -1.

$$ Miai
$$ -----------------
$$ | O . O . . . . |
$$ | X X O . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | X X O . . . . |
$$ | O . O . . . . |
$$ -----------------

Click Here To Show Diagram Code

[go]$$ Miai
$$ -----------------
$$ | O . O . . . . |
$$ | X X O . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | X X O . . . . |
$$ | O . O . . . . |
$$ -----------------[/go]

Well, the mast is purple up to temperature 1, which means that either player can choose to play at or below that temperature. It also means that the miai value of a play in the sum is ambiguous. Either player could play at temperature 1, but neither player needs to play above temperature 0, or, if the players agree to the score, at all (temperature -1).

I think that most go players would say that the miai value of this combination is 1. To score the game, most rules require play at temperature 0, so in practice that is the effective miai value. Thanks to Berlekamp's subterranean thermography, we can also say that the effective miai value is -1. At area scoring the players could wait until the end of the game to play in the sum.

Don't we have G = {2|0}+{2|0} = 2
Unless you handle G as a ko threat for each side the miai value of G is the miai value of 2 isn't it?
Well maybe it looks like a feeling, you may decide to choose whatever miai value you want.

[Bill Spight]

Well, the chilled score for G will be the sum of the scores of its components. Which will be their combined count in regular go.

In regular go the effective miai value of the sum of games will be less than 1. For instance, although the miai value of each play in the following diagram is 1, the effective miai value of their sum is -1.

$$ Miai
$$ -----------------
$$ | O . O . . . . |
$$ | X X O . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | X X O . . . . |
$$ | O . O . . . . |
$$ -----------------

Click Here To Show Diagram Code

[go]$$ Miai
$$ -----------------
$$ | O . O . . . . |
$$ | X X O . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | X X O . . . . |
$$ | O . O . . . . |
$$ -----------------[/go]

Well, the mast is purple up to temperature 1, which means that either player can choose to play at or below that temperature. It also means that the miai value of a play in the sum is ambiguous. Either player could play at temperature 1, but neither player needs to play above temperature 0, or, if the players agree to the score, at all (temperature -1).

I think that most go players would say that the miai value of this combination is 1. To score the game, most rules require play at temperature 0, so in practice that is the effective miai value. Thanks to Berlekamp's subterranean thermography, we can also say that the effective miai value is -1. At area scoring the players could wait until the end of the game to play in the sum.

Don't we have G = {2|0}+{2|0} = 2
Unless you handle G as a ko threat for each side the miai value of G is the miai value of 2 isn't it?
Well maybe it looks like a feeling, you may decide to choose whatever miai value you want.

The effective miai value of the sum is the miai value of 2. But the miai value is not just a feeling. Either player could play in the sum at temperature 1. So you will have players who consider the miai value to be 1. It is hard to disprove that claim. That is the reason for the term, effective miai value, to indicate the temperature at the bottom of the mast. (Although that is not how non-thermographers would explain the term. )

[Gérard TAILLE]

Well, the chilled score for G will be the sum of the scores of its components. Which will be their combined count in regular go.

In regular go the effective miai value of the sum of games will be less than 1. For instance, although the miai value of each play in the following diagram is 1, the effective miai value of their sum is -1.

$$ Miai
$$ -----------------
$$ | O . O . . . . |
$$ | X X O . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | X X O . . . . |
$$ | O . O . . . . |
$$ -----------------

Click Here To Show Diagram Code

[go]$$ Miai
$$ -----------------
$$ | O . O . . . . |
$$ | X X O . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | X X O . . . . |
$$ | O . O . . . . |
$$ -----------------[/go]

Well, the mast is purple up to temperature 1, which means that either player can choose to play at or below that temperature. It also means that the miai value of a play in the sum is ambiguous. Either player could play at temperature 1, but neither player needs to play above temperature 0, or, if the players agree to the score, at all (temperature -1).

I think that most go players would say that the miai value of this combination is 1. To score the game, most rules require play at temperature 0, so in practice that is the effective miai value. Thanks to Berlekamp's subterranean thermography, we can also say that the effective miai value is -1. At area scoring the players could wait until the end of the game to play in the sum.

Don't we have G = {2|0}+{2|0} = 2
Unless you handle G as a ko threat for each side the miai value of G is the miai value of 2 isn't it?
Well maybe it looks like a feeling, you may decide to choose whatever miai value you want.

The effective miai value of the sum is the miai value of 2. But the miai value is not just a feeling. Either player could play in the sum at temperature 1. So you will have players who consider the miai value to be 1. It is hard to disprove that claim. That is the reason for the term, effective miai value, to indicate the temperature at the bottom of the mast. (Although that is not how non-thermographers would explain the term. )

I see here a contradiction in the theory between general result of CGT and particular result of thermography.
In one hand CGT tells us that {2|0}+{2|0} = 2. In this context this equality is very strong indeed. It means that in all cases (OC always excluding ko considerations) you can replace {2|0}+{2|0} by 2.
In the other hand thermography shows a difference between game {2|0}+{2|0} and game 2 due to the color purple between temperature 0 and 1.

How can we handle such issue?
1) we can decide to change thermography and to draw the vertical purple mast in black color
2) we do not accept to change thermography and we add a warning to the equality G=H by saying that you can always replace G by H providing you do not use then miai values.

I am reluctant taking the second solution because I fear that could add many warnings on various results of the theory. For example what about infinitesimals which use miai values : if G=H and H is an infinitesimal then we cannot imply that G is also an infinitesimal can we?

[Bill Spight]

Well, the chilled score for G will be the sum of the scores of its components. Which will be their combined count in regular go.

In regular go the effective miai value of the sum of games will be less than 1. For instance, although the miai value of each play in the following diagram is 1, the effective miai value of their sum is -1.

$$ Miai
$$ -----------------
$$ | O . O . . . . |
$$ | X X O . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | X X O . . . . |
$$ | O . O . . . . |
$$ -----------------

Click Here To Show Diagram Code

[go]$$ Miai
$$ -----------------
$$ | O . O . . . . |
$$ | X X O . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | X X O . . . . |
$$ | O . O . . . . |
$$ -----------------[/go]

Well, the mast is purple up to temperature 1, which means that either player can choose to play at or below that temperature. It also means that the miai value of a play in the sum is ambiguous. Either player could play at temperature 1, but neither player needs to play above temperature 0, or, if the players agree to the score, at all (temperature -1).

I think that most go players would say that the miai value of this combination is 1. To score the game, most rules require play at temperature 0, so in practice that is the effective miai value. Thanks to Berlekamp's subterranean thermography, we can also say that the effective miai value is -1. At area scoring the players could wait until the end of the game to play in the sum.

Don't we have G = {2|0}+{2|0} = 2
Unless you handle G as a ko threat for each side the miai value of G is the miai value of 2 isn't it?
Well maybe it looks like a feeling, you may decide to choose whatever miai value you want.

The effective miai value of the sum is the miai value of 2. But the miai value is not just a feeling. Either player could play in the sum at temperature 1. So you will have players who consider the miai value to be 1. It is hard to disprove that claim. That is the reason for the term, effective miai value, to indicate the temperature at the bottom of the mast. (Although that is not how non-thermographers would explain the term. )

I see here a contradiction in the theory between general result of CGT and particular result of thermography.
In one hand CGT tells us that {2|0}+{2|0} = 2. In this context this equality is very strong indeed. It means that in all cases (OC always excluding ko considerations) you can replace {2|0}+{2|0} by 2.

It means that you can substitute {2|0} + {2|0} by 2 in equations.

In the other hand thermography shows a difference between game {2|0}+{2|0} and game 2 due to the color purple between temperature 0 and 1.

All games that equal 2 have a vertical mast at a score of 2, at or above a temperature of -1. It does not matter what color the mast is.

Originally, thermographs started from temperature 0, and may still do so, because subzero temperature seldom matter. Berlekamp came up with the idea of subterranean thermography, which gave more information about games in their thermographs.

Originally, thermographs were not colored, and they still do not need to be. The color of thermographic lines is not an essential feature of them. I came up with the simple idea of coloring the masts to give more information about games in their thermographs. These colors do not affect the equations.

How can we handle such issue?
1) we can decide to change thermography and to draw the vertical purple mast in black color
2) we do not accept to change thermography and we add a warning to the equality G=H by saying that you can always replace G by H providing you do not use then miai values.

G = H

says nothing about miai values, which is a go concept, not a CGT concept. Now it is true that

{2|0} + {2|0}

and

2

have the same effective miai values associated with them. That is true for all games that are equal to 2.

I am reluctant taking the second solution because I fear that could add many warnings on various results of the theory. For example what about infinitesimals which use miai values : if G=H and H is an infinitesimal then we cannot imply that G is also an infinitesimal can we?

Yes, we can. Infinitesimals are not defined by their associated miai values. They are defined by their stops. All infinitesimals have stops of 0, but are not equal to 0.

{2|0|||0|0||-2}

is an infinitesimal, for instance.

The associated effective miai values of infinitesimals are equal to 0. That is so for chilled go, as well. OC, if a move has an effective miai value of 0 in chilled go, it has an effective miai value of 1 in regular go. If I said that go infinitesimals have miai values of 1, that was informal talk.

[Gérard TAILLE]

G = H

says nothing about miai values, which is a go concept, not a CGT concept. Now it is true that

{2|0} + {2|0}

and

2

have the same effective miai values associated with them. That is true for all games that are equal to 2.

https://senseis.xmp.net/?Thermography:
The thermograph of a game is a graphical representation of the value of playing in it at different temperatures. The axes of the thermograph are the temperature (vertical) and the count (horizontal). At each temperature, the left wall of the thermograph shows what Left (Black) gets by playing first, and the right wall shows what Right (White) gets by playing first. At high enough temperatures the thermograph is topped by a vertical mast, which coincides with the left and right walls, and represents the count of the game. The temperature at the base of this vertical mast is the temperature of the game. In go it corresponds to miai value.

Three points concerning this thermograph definition:
1) the thermograph is not colored
2) the temperature of the game is the temperature at the base of the vertical mast
3) "miai value" is not defined in thermography, it is only a notion in go world, corresponding (?) to the temperature of the game in thermography

It is what you said in your post and I agree with you.
In practice the problem is that in many articles about thermography the wording miai value is used instead of temperature of the game.
I understand you proposed also the wording effective miai value to avoid misunderstanding. Isn'it simplier to keep the wording temperature of the game?

In any case what is the defintion of what you call "miai value"?
In the game G = {2|0}+{2|0} the temperature of the game is 0 according to the defintion but you seem to claim that the "miai value" is different. What is the defintion of miai value? How do you calculate it with this game G? Due to the purple color it seems we have here a tally = 0 and it seems we are facing a double sente area which does not exist does it?

[Bill Spight]

G = H

says nothing about miai values, which is a go concept, not a CGT concept. Now it is true that

{2|0} + {2|0}

and

2

have the same effective miai values associated with them. That is true for all games that are equal to 2.

https://senseis.xmp.net/?Thermography:
The thermograph of a game is a graphical representation of the value of playing in it at different temperatures. The axes of the thermograph are the temperature (vertical) and the count (horizontal). At each temperature, the left wall of the thermograph shows what Left (Black) gets by playing first, and the right wall shows what Right (White) gets by playing first. At high enough temperatures the thermograph is topped by a vertical mast, which coincides with the left and right walls, and represents the count of the game. The temperature at the base of this vertical mast is the temperature of the game. In go it corresponds to miai value.

Three points concerning this thermograph definition:
1) the thermograph is not colored
2) the temperature of the game is the temperature at the base of the vertical mast
3) "miai value" is not defined in thermography, it is only a notion in go world, corresponding (?) to the temperature of the game in thermography

Guilty as charged.

Color is not a defining property of thermographs, but they may be colored. I came up with that idea.

I introduced the term, miai value, to English speaking go players on rec.games.go in the 1990s, in an effort to clarify our thinking about move values. Western players had learned what is called in Japanese deiri values, but most of them assumed that they meant the same thing as miai values. My efforts almost completely failed. Instead, the new term for move values caused more confusion than it alleviated. Too much terminology!

I don't know if I was the first to introduce the idea of temperature on rec.games.go, but I certainly used it. However, go players online adopted the term, not to refer the temperature of a position (game) but to the temperature of the whole board. Fine. Language at work. But that means that the technical meaning of temperature in CGT is different, and using it would cause confusion. So I don't. I have enough trouble with the technical meaning of sente.

It is what you said in your post and I agree with you.
In practice the problem is that in many articles about thermography the wording miai value is used instead of temperature of the game.

Yes, that's probably my attempt to communicate with regular go players, at least those who use the term, miai value. That is my aim on SL. As such, it is informal language. Most people who do use miai value use it to refer to both plays and positions, and to use it where in CGT we would use temperature.

I understand you proposed also the wording effective miai value to avoid misunderstanding. Isn'it simplier to keep the wording temperature of the game?

By the time of that writing temperature was used differently by go players. No point in creating confusion. IMX, a surprisingly large number of people want a word to mean only one thing. Most words have multiple meanings, but ambiguity does cause misunderstanding and confusion.

In any case what is the defintion of what you call "miai value"?

Well, as a go term, it is not exactly unambiguously defined. It seems to me that it means how much a gote play or gote sequence as a unit gains, on average. Whether it can be used for how much the first play in a sente sequence gains is unclear. I have not seen it used for that. OC, it can be used that way, I suppose. I have had discussions about whether it can be used for mistakes. Again, I have never seen it used that way in professional writing. At one time I argued against such usage, but now I think it can be useful to do so, as long as you say the play is a mistake, or may be.

In the game G = {2|0}+{2|0} the temperature of the game is 0 according to the defintion but you seem to claim that the "miai value" is different.

In my early studies of CGT what I read about the temperature of a game is the temperature at the base of its vertical mast. However, in discussing a miai pair like {2|0} + {2|0}, Berlekamp, who certainly knew what temperature means, said that it was ambiguous, that you could consider the temperature of the sum to be any number up to 1.

Anyway, the miai value of a play in {2|0} is 1. {2|0} is an independent game. So in the sum, {2|0} + {2|0}, which exists only in the abstract, why is it not the same? This argument also applies to temperature. The CGT text I had read said that the temperature of a sum of games is less than or equal to the maximum temperature of each game in the sum. That does not exactly resolve the ambiguity.

What is the defintion of miai value?

One meaning is that it is the average gain of a gote play or sequence of play, as a unit.

It probably has broader meanings, as well.

How do you calculate it with this game G? Due to the purple color it seems we have here a tally = 0 and it seems we are facing a double sente area which does not exist does it?

AFAIK, I am the only go writer who calls this miai a double sente. Anyway, we do not talk about the miai value of a sente play sequence, except when we really mean the miai value of the corresponding reverse sente play or sequence. As I said, it's an informal term.

[Gérard TAILLE]

G = H

says nothing about miai values, which is a go concept, not a CGT concept. Now it is true that

{2|0} + {2|0}

and

2

have the same effective miai values associated with them. That is true for all games that are equal to 2.

https://senseis.xmp.net/?Thermography:
The thermograph of a game is a graphical representation of the value of playing in it at different temperatures. The axes of the thermograph are the temperature (vertical) and the count (horizontal). At each temperature, the left wall of the thermograph shows what Left (Black) gets by playing first, and the right wall shows what Right (White) gets by playing first. At high enough temperatures the thermograph is topped by a vertical mast, which coincides with the left and right walls, and represents the count of the game. The temperature at the base of this vertical mast is the temperature of the game. In go it corresponds to miai value.

Three points concerning this thermograph definition:
1) the thermograph is not colored
2) the temperature of the game is the temperature at the base of the vertical mast
3) "miai value" is not defined in thermography, it is only a notion in go world, corresponding (?) to the temperature of the game in thermography

Guilty as charged.

Color is not a defining property of thermographs, but they may be colored. I came up with that idea.

I introduced the term, miai value, to English speaking go players on rec.games.go in the 1990s, in an effort to clarify our thinking about move values. Western players had learned what is called in Japanese deiri values, but most of them assumed that they meant the same thing as miai values. My efforts almost completely failed. Instead, the new term for move values caused more confusion than it alleviated. Too much terminology!

I don't know if I was the first to introduce the idea of temperature on rec.games.go, but I certainly used it. However, go players online adopted the term, not to refer the temperature of a position (game) but to the temperature of the whole board. Fine. Language at work. But that means that the technical meaning of temperature in CGT is different, and using it would cause confusion. So I don't. I have enough trouble with the technical meaning of sente.

It is what you said in your post and I agree with you.
In practice the problem is that in many articles about thermography the wording miai value is used instead of temperature of the game.

Yes, that's probably my attempt to communicate with regular go players, at least those who use the term, miai value. That is my aim on SL. As such, it is informal language. Most people who do use miai value use it to refer to both plays and positions, and to use it where in CGT we would use temperature.

I understand you proposed also the wording effective miai value to avoid misunderstanding. Isn'it simplier to keep the wording temperature of the game?

By the time of that writing temperature was used differently by go players. No point in creating confusion. IMX, a surprisingly large number of people want a word to mean only one thing. Most words have multiple meanings, but ambiguity does cause misunderstanding and confusion.

In any case what is the defintion of what you call "miai value"?

Well, as a go term, it is not exactly unambiguously defined. It seems to me that it means how much a gote play or gote sequence as a unit gains, on average. Whether it can be used for how much the first play in a sente sequence gains is unclear. I have not seen it used for that. OC, it can be used that way, I suppose. I have had discussions about whether it can be used for mistakes. Again, I have never seen it used that way in professional writing. At one time I argued against such usage, but now I think it can be useful to do so, as long as you say the play is a mistake, or may be.

In the game G = {2|0}+{2|0} the temperature of the game is 0 according to the defintion but you seem to claim that the "miai value" is different.

In my early studies of CGT what I read about the temperature of a game is the temperature at the base of its vertical mast. However, in discussing a miai pair like {2|0} + {2|0}, Berlekamp, who certainly knew what temperature means, said that it was ambiguous, that you could consider the temperature of the sum to be any number up to 1.

Anyway, the miai value of a play in {2|0} is 1. {2|0} is an independent game. So in the sum, {2|0} + {2|0}, which exists only in the abstract, why is it not the same? This argument also applies to temperature. The CGT text I had read said that the temperature of a sum of games is less than or equal to the maximum temperature of each game in the sum. That does not exactly resolve the ambiguity.

What is the defintion of miai value?

One meaning is that it is the average gain of a gote play or sequence of play, as a unit.

It probably has broader meanings, as well.

How do you calculate it with this game G? Due to the purple color it seems we have here a tally = 0 and it seems we are facing a double sente area which does not exist does it?

AFAIK, I am the only go writer who calls this miai a double sente. Anyway, we do not talk about the miai value of a sente play sequence, except when we really mean the miai value of the corresponding reverse sente play or sequence. As I said, it's an informal term.

I do not have any experience in the work of convincing go community to a new theory but I imagine it could be very frustrating. Very few people are really open to study new ideas.
Yes for gote area the miai value seems clear.
For sente or double sente (I mean miai gote areas) we have to live with ambiguity haven't we?

Why I prefer using a miai value = 0 for miai gote points? The reason is quite simple.
Let's take a game G = G1 + G2 + G3 + ... each G_i having a miai value m_i<1½ and a score s_i.
Let's suppose score(G) = s1 + s2 + s3 + ... = 2¼
Because the real result of the game is an ordinal number and because the result of the game lies between 2¼ and 2¼+1½ we know for sure that the result of the game should be 3.
Let's now take the game G + H with H = {10|-10} + {10|-10}. If you take as miai value of H the value 10 then you can no more predict the result of the game and that sounds for me as a loss.
What kind of advantage can you see by not taking the miai value 0 for H?

[Bill Spight]

Why I prefer using a miai value = 0 for miai gote points? The reason is quite simple.
Let's take a game G = G1 + G2 + G3 + ... each G_i having a miai value m_i<1½ and a score s_i.
Let's suppose score(G) = s1 + s2 + s3 + ... = 2¼
Because the real result of the game is an ordinal number and because the result of the game lies between 2¼ and 2¼+1½ we know for sure that the result of the game should be 3.

Yes, the left (Black) stop of G will be 3, and the right (White) stop will be 1 or 2.

Let's now take the game G + H with H = {10|-10} + {10|-10}. If you take as miai value of H the value 10 then you can no more predict the result of the game and that sounds for me as a loss.
What kind of advantage can you see by not taking the miai value 0 for H?

Well, yes, for the purpose of prediction recognizing the miai pair is very important. But there are perhaps practical reasons for remembering the temperature of each of the miai pair is 10 and playing one of them now.

A lot of players, when they first learn about evaluating plays, think that making the largest play at each turn is correct. With a little learning and experience they find out that that is not necessarily so. But it is close enough to the truth that it is a good heuristic, which is known in CGT as hotstrat, the strategy of playing the hottest play. When reading a position, then, we usually start off playing hotstrat and I was doing that with Berlekamp one time. I suggested playing the hottest gote, which was the hottest play, but he said to take a sente instead. OC, the sente raised the global temperature, and was answered, but it removed a potential ko threat. At the same time, it did not risk allowing the opponent to play the reverse sente. Berlekamp did not interpret hotstrat literally.

Now, while recognizing the miai is important for analysis, I have little doubt that Berlekamp would have played next in the miai, breaking it, unless there were a sente for him with a sufficiently large threat. Except in that case, besides the practical matter of avoiding a possible error, the difference in temperature between each of the miai pair and the third hottest play is so great that surely playing one of them dominates all other plays.

[Gérard TAILLE]

Corridors from UP or TINY

$$ White to play
$$ -----------------------
$$ | X O . . O O . . . . |
$$ | X X X X X O . . . . |
$$ | X O . . . O O . . . |
$$ | X X X X X X O . . . |
$$ | X O . . . . O O . . |
$$ | X X X X X X X O . . |
$$ | X O . . . . . O . . |
$$ | X X X X X X X X . . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------

Click Here To Show Diagram Code

[go]$$ White to play
$$ -----------------------
$$ | X O . . O O . . . . |
$$ | X X X X X O . . . . |
$$ | X O . . . O O . . . |
$$ | X X X X X X O . . . |
$$ | X O . . . . O O . . |
$$ | X X X X X X X O . . |
$$ | X O . . . . . O . . |
$$ | X X X X X X X X . . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------[/go]

I understood the infinitesimals of the corridors ending with an UP form an arithmetic serie, the adding value being ↑* (the atomic weight)

$$ White to play
$$ -----------------------
$$ | X . O . . O O . . . |
$$ | X X X X X X O . . . |
$$ | X . O . . . O O . . |
$$ | X X X X X X X O . . |
$$ | X . O . . . . O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . . O . |
$$ | X X X X X X X X X . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------

Click Here To Show Diagram Code

[go]$$ White to play
$$ -----------------------
$$ | X . O . . O O . . . |
$$ | X X X X X X O . . . |
$$ | X . O . . . O O . . |
$$ | X X X X X X X O . . |
$$ | X . O . . . . O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . . O . |
$$ | X X X X X X X X X . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------[/go]

What about corridors ending with a TINY? Do we have here also an arithmetic serie?
I tried the atomic weight but it does not work.

[Bill Spight]

$$ White to play
$$ -----------------------
$$ | X . O . a O O . . . |
$$ | X X X X X X O . . . |
$$ | X . O . . b O O . . |
$$ | X X X X X X X O . . |
$$ | X . O . . . c O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . d O . |
$$ | X X X X X X X X X . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------

Click Here To Show Diagram Code

[go]$$ White to play
$$ -----------------------
$$ | X . O . a O O . . . |
$$ | X X X X X X O . . . |
$$ | X . O . . b O O . . |
$$ | X X X X X X X O . . |
$$ | X . O . . . c O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . d O . |
$$ | X X X X X X X X X . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------[/go]

What about corridors ending with a TINY? Do we have here also an arithmetic serie?
I tried the atomic weight but it does not work.

(Labels added by me.)

No, it's not an arithmetic series. OC, White should play at a first. Mathematical Go says that if the final tinies are equal, then White should play in the shortest corridor first.

As they are not an arithmetic series, the question arises of the comparison between d - c and c - b. The difference between the two is d - 2c + b.

$$B Difference game
$$ -----------------------
$$ | X . O . . b O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . d O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X c . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X c . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------

Click Here To Show Diagram Code

[go]$$B Difference game
$$ -----------------------
$$ | X . O . . b O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . d O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X c . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X c . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------[/go]

$$B Black first
$$ -----------------------
$$ | X . O . . . O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . 2 O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X 1 . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X . . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------

Click Here To Show Diagram Code

[go]$$B Black first
$$ -----------------------
$$ | X . O . . . O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . 2 O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X 1 . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X . . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------[/go]

After White makes mirror go and wins jigo.

$$W White first
$$ -----------------------
$$ | X . O 4 3 1 O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . 5 O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X 2 6 7 . X . O . |
$$ | . X O O O O O O O . |
$$ | . X . . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------

Click Here To Show Diagram Code

[go]$$W White first
$$ -----------------------
$$ | X . O 4 3 1 O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . 5 O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X 2 6 7 . X . O . |
$$ | . X O O O O O O O . |
$$ | . X . . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------[/go]

White wins.

So c - b > d - c.

Edited for correctness.

[Gérard TAILLE]

$$ White to play
$$ -----------------------
$$ | X . O . a O O . . . |
$$ | X X X X X X O . . . |
$$ | X . O . . b O O . . |
$$ | X X X X X X X O . . |
$$ | X . O . . . c O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . d O . |
$$ | X X X X X X X X X . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------

Click Here To Show Diagram Code

[go]$$ White to play
$$ -----------------------
$$ | X . O . a O O . . . |
$$ | X X X X X X O . . . |
$$ | X . O . . b O O . . |
$$ | X X X X X X X O . . |
$$ | X . O . . . c O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . d O . |
$$ | X X X X X X X X X . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------[/go]

What about corridors ending with a TINY? Do we have here also an arithmetic serie?
I tried the atomic weight but it does not work.

(Labels added by me.)

No, it's not an arithmetic series. OC, White should play at a first. Mathematical Go says that if the final tinies are equal, then White should play in the shortest corridor first.

As they are not an arithmetic series, the question arises of the comparison between d - c and c - b. The difference between the two is d - 2c + b.

$$B Difference game
$$ -----------------------
$$ | X . O . . b O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . d O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X c . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X c . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------

Click Here To Show Diagram Code

[go]$$B Difference game
$$ -----------------------
$$ | X . O . . b O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . d O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X c . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X c . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------[/go]

$$B Black first
$$ -----------------------
$$ | X . O . . . O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . 2 O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X 1 . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X . . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------

Click Here To Show Diagram Code

[go]$$B Black first
$$ -----------------------
$$ | X . O . . . O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . 2 O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X 1 . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X . . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------[/go]

After White makes mirror go and wins jigo.

$$W White first
$$ -----------------------
$$ | X . O 4 3 1 O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . 5 O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X 2 6 7 . X . O . |
$$ | . X O O O O O O O . |
$$ | . X . . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------

Click Here To Show Diagram Code

[go]$$W White first
$$ -----------------------
$$ | X . O 4 3 1 O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . 5 O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X 2 6 7 . X . O . |
$$ | . X O O O O O O O . |
$$ | . X . . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------[/go]

White wins.

So c - b > d - c.

Edited for correctness.

Fine proof Bill !

BTW I asked me the following problem: what are the solutions of the equation
G + G = 0
Of course we have the two obvious solutions G = 0 and G = *
Does it exist other solutions or do you have a proof that we have only these two solutions?
For the equation
G + G + G = 0
does it exist another solution than G = 0 ?
This arithmetic is not so easy to handle is it?

[Bill Spight]

$$ White to play
$$ -----------------------
$$ | X . O . a O O . . . |
$$ | X X X X X X O . . . |
$$ | X . O . . b O O . . |
$$ | X X X X X X X O . . |
$$ | X . O . . . c O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . d O . |
$$ | X X X X X X X X X . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------

Click Here To Show Diagram Code

[go]$$ White to play
$$ -----------------------
$$ | X . O . a O O . . . |
$$ | X X X X X X O . . . |
$$ | X . O . . b O O . . |
$$ | X X X X X X X O . . |
$$ | X . O . . . c O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . d O . |
$$ | X X X X X X X X X . |
$$ | . . . . . . . . . . |
$$ | . . . . . . . . . . |
$$ -----------------------[/go]

What about corridors ending with a TINY? Do we have here also an arithmetic serie?
I tried the atomic weight but it does not work.

(Labels added by me.)

No, it's not an arithmetic series. OC, White should play at a first. Mathematical Go says that if the final tinies are equal, then White should play in the shortest corridor first.

As they are not an arithmetic series, the question arises of the comparison between d - c and c - b. The difference between the two is d - 2c + b.

$$B Difference game
$$ -----------------------
$$ | X . O . . b O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . d O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X c . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X c . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------

Click Here To Show Diagram Code

[go]$$B Difference game
$$ -----------------------
$$ | X . O . . b O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . d O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X c . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X c . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------[/go]

$$B Black first
$$ -----------------------
$$ | X . O . . . O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . 2 O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X 1 . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X . . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------

Click Here To Show Diagram Code

[go]$$B Black first
$$ -----------------------
$$ | X . O . . . O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . 2 O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X 1 . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X . . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------[/go]

After White makes mirror go and wins jigo.

$$W White first
$$ -----------------------
$$ | X . O 4 3 1 O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . 5 O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X 2 6 7 . X . O . |
$$ | . X O O O O O O O . |
$$ | . X . . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------

Click Here To Show Diagram Code

[go]$$W White first
$$ -----------------------
$$ | X . O 4 3 1 O O O . |
$$ | X X X X X X X X O . |
$$ | X . O . . . . 5 O . |
$$ | X X X X X X X X O . |
$$ | . X O O O O O O O . |
$$ | . X 2 6 7 . X . O . |
$$ | . X O O O O O O O . |
$$ | . X . . . . X . O . |
$$ | . X O O O O O O O . |
$$ | . X O . . . . . . . |
$$ -----------------------[/go]

White wins.

So c - b > d - c.

Edited for correctness.

Fine proof Bill !

BTW I asked me the following problem: what are the solutions of the equation
G + G = 0
Of course we have the two obvious solutions G = 0 and G = *
Does it exist other solutions or do you have a proof that we have only these two solutions?

Combinatorial games form a group, so for every game G there exists a game, -G. In go, you just switch the colors of the stones.

For the equation
G + G + G = 0
does it exist another solution than G = 0 ?

I don't know, but I don't think so. G has to be fuzzy, and it cannot be impartial, since every impartial game is symmetric, and thus its own negative. And since G must be fuzzy, then G + G must also be fuzzy. Let each G be in its simplest form. If G + G in its simplest form is not exactly as simple as G, then it cannot be equal to -G. So G has to thread the needle.

Hmmm. Without loss of generality, let Black play first. Suppose that Black plays to Gb. If White plays in Gb to Gbw, then Gbw = G, which is impossible if G is in its simplest form. So White must play in a different G to Gw. Then Gb + Gw = -G. That is so even if White plays first. So for every Black option in G there is a White option such that the sum of the two equals -G, and vice versa. So each player must have the same number of options. And the options for each player must be incomparable.

If G has depth 0, the only game that meets those criteria is 0.
If G has depth 1, the only possible fuzzy game is {0|0} = *, which is impartial.
If G has depth 2, the only possible fuzzy games are {0,*|0,*} = *2, which is impartial, {1|-1}, {1|0}, and {0|-1}. Doubling each of the last three yields a number, which is not fuzzy.
* * *

I would be surprised if Conway or someone hasn't found a proof, one way or the other.

This arithmetic is not so easy to handle is it?

[Gérard TAILLE]

BTW I asked me the following problem: what are the solutions of the equation
G + G = 0
Of course we have the two obvious solutions G = 0 and G = *
Does it exist other solutions or do you have a proof that we have only these two solutions?

Combinatorial games form a group, so for every game G there exists a game, -G. In go, you just switch the colors of the stones.

What do you mean Bill?
My question was about the equation G + G = 0, not the equation G - G = 0.

[Gérard TAILLE]

For the equation
G + G + G = 0
does it exist another solution than G = 0 ?

I don't know, but I don't think so. G has to be fuzzy, and it cannot be impartial

What does mean impartial Bill?