Life In 19x19

Bill Spight wrote:
OK. We can write G this way, using more slashes.

G = {21 ||| 18 | 4 || 0 | -14}

I'll draw the thermograph. Be back soon.

I'm back. Here is the thermograph.

Gérard TAILLE wrote:OK Bill, now take the complete game (see the game I proposed but here with a simplier upper side)
{21 ||| 18 | 4 || 0 | -14} + {20|0} + {0|-6} + {4|0}
What is the best sequence for this game if it is white to play?

RobertJasiek wrote:Yes, the two games / ensembles of games are quite different indeed. Your diagram position contained a separate large simple gote endgame and a separate want-be double sente region. Separate games are combined by forming their sum.

G = {21 ||| 18 | 4 || 0 | -14} can be studied for different purposes but, with another branch of the tree, do not fall into the scope of my non-existence proofs. Anyway, we can also study values of that larger game tree.

So let me calculate the count of the right part {18 | 4 || 0 | -14}, of which I have already determined that it is a local gote:

C = (Cb + Cw)/2 = (11 + (-7)) / 2 = 2.

Now, we can proceed studying G = {21 ||| 18 | 4 || 0 | -14}.

It might be White's long gote, else White's local sente, else a simple gote.


We already know

21 count after move 1 by Black of Black's alternating sequence
2 count after move 1 by White of White's alternating sequence
11 count after move 2 by Black of White's alternating sequence
4 count after move 3 by White of White's alternating sequence

Gw2 = 11 - 2 = 9 gain of move 2 by Black of White's alternating sequence
Gw3 = 11 - 4 = 7 gain of move 3 by White of White's alternating sequence


First, we make the hypothesis of White's long gote.

M(G) = (21 - 4) / 2 = 8 1/2 tentative gote move value
C(G) = (21 + 4) / 2 = 12 1/2 tentative gote count
Gb1 = 21 - 12 1/2 = 8 1/2 tentative gain of move 1 by Black of Black's alternating sequence
Gw1 = 12 1/2 - 2 = 10 1/2 tentative gain of move 1 by White of White's alternating sequence

M(G) <= Gb1, Gw1, Gw2, Gw3 <=> 8 1/2 <= 8 1/2, 10 1/2, 9, 7 is partially violated so we do not have White's long gote.


Second, we make the hypothesis of White's local sente.

M(G) = 21 - 11 = 10 tentative sente move value
C(G) = 11 tentative sente count
Gb1 = 21 - 11 = 10 tentative gain of move 1 by Black of Black's alternating sequence
Gw1 = 11 - 2 = 9 tentative gain of move 1 by White of White's alternating sequence

M(G) <= Gb1, Gw1, Gw2 <=> 10 <= 10, 9, 9 is partially violated so we do not have White's local sente.


Third, we make the hypothesis of a simple gote.

M(G) = (21 - 2) / 2 = 9 1/2 tentative gote move value
C(G) = (21 + 2) / 2 = 11 1/2 tentative gote count
Gb1 = 21 - 11 1/2 = 9 1/2 tentative gain of move 1 by Black of Black's alternating sequence
Gw1 = 11 1/2 - 2 = 9 1/2 tentative gain of move 1 by White of White's alternating sequence

M(G) <= Gb1, Gw1 <=> 9 1/2 <= 9 1/2, 9 1/2 is (of course) fulfilled so G = {21 ||| 18 | 4 || 0 | -14} is a simple gote with M(G) = 9 1/2 as the move value and C(G) = 11 1/2 as the count.

Seems to agree to Bill's thermograph:)

Gérard TAILLE wrote:

Bill Spight wrote:
OK. We can write G this way, using more slashes.

G = {21 ||| 18 | 4 || 0 | -14}

I'll draw the thermograph. Be back soon.

I'm back. Here is the thermograph.

Surely you know where I encountered difficulties Bill

Starting from G = {21 ||| 18 | 4 || 0 | -14}
if use tax = 7 then the game becomes
G = {14 ||| 11 | 11 || 7 | 7}

Now what will happen with tax = 8?
I see that the games {11 | 11} and {7 | 7} are blocked and I do not know how to go to tax = 8.
Can you help me Bill?

Gérard TAILLE wrote: in the game {21 ||| 18 | 4 || 0 | -14} + {20|0}
where will you play first? In the simple 9½ gote G={21 ||| 18 | 4 || 0 | -14} or in the simple 10 gote {20|0} ?

Bill Spight wrote:

Gérard TAILLE wrote:OK Bill, now take the complete game (see the game I proposed but here with a simplier upper side)
{21 ||| 18 | 4 || 0 | -14} + {20|0} + {0|-6} + {4|0}
What is the best sequence for this game if it is white to play?

Let's call the games, G, H, M, and N. I'll us a minus sign to indicate a White moves, a plus sign to indicate a Black move. We start with a mean territorial value of 11½ + 10 - 3 + 2 = 20½.

First, let White play the ever popular hotstrat strategy, taking the average gains in order.

1) - G + H - G + G - M + N = 20½ - 11½ + 10 - 9 + 7 - 3 + 2 = 16

After Black replies in H, the rest is provable correct. Can White do better? Let's see.

2) - H + G - M + N = 20½ - 10 + 11½ - 3 + 2 = 21

That's worse. Can Black do better than 1)?

3 - H + G - H + G - M + N = 20½ - 11½ + 9 - 10 + 7 - 3 + 2 = 14

That's 2 points worse than 1). Hotstrat wins.

Bill Spight wrote:

Gérard TAILLE wrote:

Bill Spight wrote:
OK. We can write G this way, using more slashes.

G = {21 ||| 18 | 4 || 0 | -14}

I'll draw the thermograph. Be back soon.

I'm back. Here is the thermograph.

Surely you know where I encountered difficulties Bill

Starting from G = {21 ||| 18 | 4 || 0 | -14}
if use tax = 7 then the game becomes
G = {14 ||| 11 | 11 || 7 | 7}

Now what will happen with tax = 8?
I see that the games {11 | 11} and {7 | 7} are blocked and I do not know how to go to tax = 8.
Can you help me Bill?

Sure. Replace {11|11} and {7|7} with their mean (mast) values. That corresponds with using the mast instead of the lower walls. That yields.

G = {14 || 11 | 7}

When t = 8 we have

G = {13 || 11 | 9}

When t = 9 we have

G = {12 || 11 | 11} -> {12 | 11}

When t = 9½ we have

G = {11½ | 11½}

Gérard TAILLE wrote:

Bill Spight wrote:

Gérard TAILLE wrote:OK Bill, now take the complete game (see the game I proposed but here with a simplier upper side)
{21 ||| 18 | 4 || 0 | -14} + {20|0} + {0|-6} + {4|0}
What is the best sequence for this game if it is white to play?

Let's call the games, G, H, M, and N. I'll us a minus sign to indicate a White moves, a plus sign to indicate a Black move. We start with a mean territorial value of 11½ + 10 - 3 + 2 = 20½.

First, let White play the ever popular hotstrat strategy, taking the average gains in order.

1) - G + H - G + G - M + N = 20½ - 11½ + 10 - 9 + 7 - 3 + 2 = 16

After Black replies in H, the rest is provable correct. Can White do better? Let's see.

2) - H + G - M + N = 20½ - 10 + 11½ - 3 + 2 = 21

That's worse. Can Black do better than 1)?

3 - H + G - H + G - M + N = 20½ - 11½ + 9 - 10 + 7 - 3 + 2 = 14

That's 2 points worse than 1). Hotstrat wins.

Yes Bill here is my point. In this example you have to begin by this 9½ gote instead of the simple 10 gote.
This is very strange. Isn't it?

It seems to me it is due to the potential double sente hidden in the game G.

RobertJasiek wrote:Yes, the two games / ensembles of games are quite different indeed. Your diagram position contained a separate large simple gote endgame and a separate want-be double sente region. Separate games are combined by forming their sum.

G = {21 ||| 18 | 4 || 0 | -14} can be studied for different purposes but, with another branch of the tree, do not fall into the scope of my non-existence proofs. Anyway, we can also study values of that larger game tree.

Bill Spight wrote: The double sente aspect has nothing to do with it. For instance, let's replace {18|4||0|-14} with {18|4||-7}. Remember, H = {20|0}, M = {0|-6}, and N = {4|0}.

First White plays the hotstrat strategy of playing the largest play (on average) first.

1) - H + G - M + N = 20½ - 10 + 9½ - 3 + 2 = 19

Can White do better? Let's see.

2) - G + H - G + M - N = 20½ - 9½ + 10 - 9 + 3 - 2 = 13

That's 6 points better, not just 1 point better.

Gérard TAILLE wrote:a double sente (with our definition)

RobertJasiek wrote:

Gérard TAILLE wrote:a double sente (with our definition)

Global double sente or what definition do you refer to?

Gérard TAILLE wrote:

Bill Spight wrote: The double sente aspect has nothing to do with it. For instance, let's replace {18|4||0|-14} with {18|4||-7}. Remember, H = {20|0}, M = {0|-6}, and N = {4|0}.

First White plays the hotstrat strategy of playing the largest play (on average) first.

1) - H + G - M + N = 20½ - 10 + 9½ - 3 + 2 = 19

Can White do better? Let's see.

2) - G + H - G + M - N = 20½ - 9½ + 10 - 9 + 3 - 2 = 13

That's 6 points better, not just 1 point better.

Very good Bill I agree.