Gérard TAILLE wrote:
Bill Spight wrote:
OK. Let's try again.
- Click Here To Show Diagram Code
[go]$$Bc Black first
$$ -----------------
$$ | X 1 . . . W X X 2 |
$$ | X W W W W W X O 3 |
$$ | X X X X X O X 5 O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]

,

elsewhere
Result: 25 - 2t
- Click Here To Show Diagram Code
[go]$$Wc White first
$$ -----------------
$$ | X 3 . 5 . W X X 1 |
$$ | X W W W W W X O 2 |
$$ | X X X X X O X 4 O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]

elsewhere
Result: 4 + t
Assuming that these are the sequences of play when
t is small enough, we find that
t by solving the equation,
25 - 2t = 4 + t
t = 21/3 = 7
And we can verify that when Black plays first, White throws in to make the ko when t ≤ 13¾ , and when White plays first and Black takes the ko, White plays the ko threat when t ≤ 11. So these are the normal sequences of play when t ≤ 7.

The mast value of the position is 11.
If I haven't goofed, that is.

I agree that black will play first if the temperature has droped to 7.
But I do not understand why the mast value of the position is 11 instead of 14⅜.
The mast value is equal to 4 + 7 or 25 - 2*7.

14⅜ is the temperature below which White threatens to win the ko.
Gérard TAILLE wrote:
If for example the temperature is equal to 13 then white should immediatly provoque the ko without waiting for temperature droping to 11 shouldn't he?
If White throws in at temperature 13, then:
1) if White lets Black win the ko the result is 26 - 13 = 13, which is worse for White than 11;
2) if White plays the threat then we reach this position with Black to play.
- Click Here To Show Diagram Code
[go]$$Wc White first
$$ -----------------
$$ | X W . . . W X X . |
$$ | X W W W W W X O B |
$$ | X X X X X O X . O |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
One

prisoner
If

plays elsewhere White can take and win the ko for a result of -18 + 3t (from the original position), which is equal to 39 - 18 = 21. That's even worse for White than above, so Black will play elsewhere and so will White.
What is the mast value of this position? From here White can take and win the ko for a result of -18 + 2t.
If instead Black wins the ko we have this position.
- Click Here To Show Diagram Code
[go]$$Wc White first
$$ -----------------
$$ | X W . . . W X X . |
$$ | X W W W W W X . B |
$$ | X X X X X O X B . |
$$ | . . . . X O X X X |
$$ | X X X X X X O O O |
$$ | . . . . . . O . . |
$$ | . . . . . . O O . |
$$ | . . . . . . . O . |
$$ | . . . . . . . O . |
$$ -----------------[/go]
Three

prisoners
If White lives the result is 4. If Black kills the result is 26. The mean value is 15 and at or below temperature 11 White will live.
To find the temperature for this position at or below which the players will play locally, let's first assume that White lives. Then we solve for
-18 + 2t = 4
t = 22/2 = 11
Fine.

We are done. The mast value of this position is 4.
Assuming that at temperature 13 White makes the ko and then plays the threat, the result will be 4 + 13 = 17 (from the original position), which is worse for White than 11, so White does best to wait until t = 7.
Gérard TAILLE wrote:
BTW how will you eventually draw the thermograph Bill? It looks quite unusual doesn't it?
Actually, it looks like a simple ko thermograph. Above temperature 7 the mast rises at a territory value of 11. Below that the right wall is 4 + t and the left wall is 25 - 2t. I'll add the thermograph below, probably tomorrow.

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Everything with love. Stay safe.