Precision, please

Bill Spight · #1

A new endgame beastie.

Click Here To Show Diagram Code: [go]$$ New beastie $$ ------------------------------------- $$ , X O O O O O O O . O O . O . O X X , $$ , X O . . . . O . O O . O O O . O X , $$ , X O . . . O O X O . X O X O O O X , $$ , X X X X X X X X X X X X X X X X X , $$ , , , , , , , , , , , , , , , , , , ,[/go]

(By convention, the Black stones are alive.)

1) How much territory, on average, does White have?

2) How much does a play gain, on average?

Please be precise.

Enjoy!

Knotwilg · #2

First calculation:

1) White has on average 2817/512 = 5.501953125
2) First play gains 545/512 = 1.064453125

Bill Spight · #3

Knotwilg wrote:

First calculation:

1) White has on average 2817/512 = 5.501953125
2) First play gains 545/512 = 1.064453125

Thanks.

Edit: Correction.

Bill Spight · #4

Failure 1

dany · #5

Click Here To Show Diagram Code: [go]$$c $$ ------------------------------------- $$ |, X O O O O O O O . O O . O . O X X ,| $$ |, X O . . . . O . O O . O O O . O X ,| $$ |, X O 1 . . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

Knotwilg · #6

Second attempt: White has 821/256 on average; a play gains 245/256 on average

Bill Spight · #7

Knotwilg wrote:

Second attempt: White has 821/256 on average; a play gains 245/256 on average

Edit:

Bill Spight · #8

Difference game for White's first play.

dany · #9

1) White has on average 1307/256 = 5.10546875

Bill Spight · **#10**

I am afraid my hint gave the impression that the answers have a large denominator. That may not be the case.

Knotwilg · **#11**

Bill Spight wrote:

I am afraid my hint gave the impression that the answers have a large denominator. That may not be the case.

You have not commented on that aspect before. Me too I get a 256 denominator. It's not the hint that set me up for it, but the problem composition, which suggests the two false eyes are unsettled points, depending on follow-up play.

If it were only about the big eye where the action is, then why the complex problem composition with two real eyes at the tail?

Bill Spight · **#12**

Knotwilg wrote:

Bill Spight wrote:

I am afraid my hint gave the impression that the answers have a large denominator. That may not be the case.

You have not commented on that aspect before. Me too I get a 256 denominator.

So did I, and that's when I posted the problem. But I was mistaken. :oops:

Quote:

It's not the hint that set me up for it, but the problem composition, which suggests the two false eyes are unsettled points, depending on follow-up play.

If it were only about the big eye where the action is, then why the complex problem composition with two real eyes at the tail?

I was inspired by a similar position I ran across in a pro endgame. I thought that the false eyes (Edit: technically, half eyes) made for a cool problem. :cool:

This position is slightly simpler than the game position. The half eyes are still significant.

BTW, the game is in the GoGoD collection. 2018-07-19u.

Bill Spight · **#13**

Difference game for Black's first play.

Bill Spight · **#14**

Another difference game.

Warning, this one gives a big hint.

dany · **#15**

Click Here To Show Diagram Code: [go]$$c $$ ------------------------------------- $$ |, X O O O O O O O . O O . O . O X X ,| $$ |, X O . . . . O . O O . O O O . O X ,| $$ |, X O a . b O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

In such positions, often the best play is asymmetric.
So I think: a - the best for black, b - the best for White. But I was wrong - a the best for the both.

Bill Spight wrote:

Warning, this one gives a big hint.
...
Now we have a simpler problem than the original

Click Here To Show Diagram Code: [go]$$c $$ ------------------------------------- $$ |, X O O O O O O O . O O . O . O X X ,| $$ |, X O . . . . O . O O . O O O . O X ,| $$ |, X O O X . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

1) White has on average 83/16 = 5.1875

Bill Spight · **#16**

Well, the cat's out of the bag, so here we go.

Click Here To Show Diagram Code: [go]$$c Evaluate this $$ ------------------------------------- $$ |, X O O O O O O O . O O . O . O X X ,| $$ |, X O . . . . O . O O . O O O . O X ,| $$ |, X O . . . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

We checked, for some reason, and found that the next position has the same territorial value and each move gains the same, on average.

Click Here To Show Diagram Code: [go]$$c Evaluate that $$ ------------------------------------- $$ |, X O O O O O O O . O O . O . O X X ,| $$ |, X O . . . . O . O O . O O O . O X ,| $$ |, X O W B . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

OK, the first moves are obvious.

Click Here To Show Diagram Code: [go]$$c Black first, White reply $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O . 1 2 C O . O O . O O O C O X ,| $$ |, X O W B . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

OK,

yields 5 pts. for White

Click Here To Show Diagram Code: [go]$$c $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O . 1 3 . O 4 O O . O O O C O X ,| $$ |, X O W B . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

If Black plays :b3:

instead,

yields 4 pts. for White.

Click Here To Show Diagram Code: [go]$$c $$ ------------------------------------- $$ |, X O O O O O O O . O O C O C O X X ,| $$ |, X O . 1 3 . O 5 O O 6 O O O C O X ,| $$ |, X O W B . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

If

makes a false eye, :w6:

yields 3 pts. for White. You get the picture.

IOW, after the resulting position is like this one.

Click Here To Show Diagram Code: [go]$$c $$ ------------------------------- $$ |, X O O O O O O O O C O X X , $$ |, X O . 1 , , , , O O C O X , $$ |, X O W B . O O O O O O O X , $$ |, X X X X X X X X X X X X X , $$ |, , , , , , , , , , , , , , ,[/go]

The succession of half eyes is like a simple closed corridor.

This is one of the first endgame positions we learn about. This corridor is worth, on average, 2⅛ pts. for White. Adding that to the 2 marked, safe pts, gives us 4⅛ pts. for White after :b1:

.

Click Here To Show Diagram Code: [go]$$Wc White first $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O C 1 . . O . O O . O O O C O X ,| $$ |, X O W B . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

makes an eye and yields 5 pts. for White plus a short corridor, which has 1¼ pts. for White, on average. The result is 6¼ pts. for White.

This problem is a gote, then, with an original territorial value of 5 3/16 pts. for White, on average. And each player gains 1 1/16 pts. by playing first.

Applying all this to the original problem yields these plays.

Click Here To Show Diagram Code: [go]$$Bc Black first $$ ------------------------------------- $$ |, X O O O O O O O , O O , O C O X X ,| $$ |, X O . . , , O . O O . O O O C O X ,| $$ |, X O 1 . . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

Click Here To Show Diagram Code: [go]$$Wc White first $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O C 3 , , O . O O . O O O C O X ,| $$ |, X O 1 2 , O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

The sequence, :w1:

,

, and

, form a unit of play called a reverse or reversal, ( https://senseis.xmp.net/?Reversible ). Many such sequences, such as the hane-and-connect on the first line, are obvious; this one is not. The fact that :w1:

and

make an eye shows how this problem differs from a simple corridor.

----

Now, I used difference games to find the first move for each player and the reverse, but not every go player knows about them. Besides, it is possible to figure the problem out without using them. I'm off to bed. More on that later.

Bill Spight · **#17**

Black's first move.

Click Here To Show Diagram Code: [go]$$Bc Black first $$ ------------------------------------- $$ |, X O O O O O O O , O O , O C O X X ,| $$ |, X O . . , , O . O O . O O O C O X ,| $$ |, X O 1 . . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

Here's one candidate move for Black.

Click Here To Show Diagram Code: [go]$$Bc Black first $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O . 3 4 C O . O O . O O O C O X ,| $$ |, X O 1 . 2 O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

And one continuation. This sequence is almost surely correct, and tells us that the original territorial value for White is at most 5 pts. We hardly expect that :b1:

-

is actually sente. But :w2:

-

does look like a reverse. After all, White could have made 5 pts. simply by playing at :w4:

instead of :w2:

.

Click Here To Show Diagram Code: [go]$$Bc Black follow-up, White reply $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O . . 3 , O 4 O O . O O O C O X ,| $$ |, X O 1 . . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

can take away the possible eye and 5th pt. :w4:

makes an eye and 4 pts. At this point we may realize that this is like a simple closed corridor. Then we know that :w4:

gains ¾ pt. and that the position after :b3:

is worth 3¼ pts. for White. Also that :b3:

gains ⅞ pt. and that the position after :b1:

is worth 4⅛ pt. for White.

If we don't, we can continue and figure it out.

Click Here To Show Diagram Code: [go]$$Bc Black first, second candidate $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O . . 2 C O . O O . O O O C O X ,| $$ |, X O . . 1 O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

After

,

makes 5½ pts. for White, on average. If we have already tried the first candidate, we can stop right here, we know this is not as good. But let's continue with Black's follow-up.

Click Here To Show Diagram Code: [go]$$Bc Black first, Black follow-up $$ ------------------------------------- $$ |, X O O O O O O O , O O , O C O X X ,| $$ |, X O . . 3 , O . O O . O O O C O X ,| $$ |, X O . . 1 O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

As above we can figure this to be worth 3¼ pts. for White. Then after :b1:

the territorial value for White is 4½ pts. and :w2:

and

each gain 1¼ pts. At first blush that seems like it's too good for White. And, OC, when we compare this with the fact that after our first candidate, White has only 4⅛ pts. and only threatens to make 5 pts., we can eliminate this option.

(3d candidate discussion edited for correctness.)

Click Here To Show Diagram Code: [go]$$Bc Black third candidate, White reply $$ ------------------------------------- $$ |, X O O O O O O O , O O , O C O X X ,| $$ |, X O . 2 , , O . O O . O O O C O X ,| $$ |, X O . 1 . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

is pretty obviously a loser.

At first glance, after :w2:

it looks like White has 5 9/16 pts. on average. 5 1/16 + ½, but the evaluation is a little tricky.

Click Here To Show Diagram Code: [go]$$Bc Black third candidate, continuation $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O . 2 , , O . O O . O O O C O X ,| $$ |, X O . 1 3 O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

After

White has miai to make an eye at D-18 or G-18, so the result is worth 5 pts. for White.

Click Here To Show Diagram Code: [go]$$Bc Black third candidate, B3 elsewhere, W4 instead $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O . 2 C C O . O O . O O O C O X ,| $$ |, X O . 1 4 O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

After

the result is worth 6½ pts. for White. That means that the result after :w2:

is worth 5¾ pts. for White on average, with the next play gaining ¾ pt.

Click Here To Show Diagram Code: [go]$$Bc Black fourth candidate, White reply $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O . 1 2 C O . O O . O O O C O X ,| $$ |, X O . . . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

After

White has 5 pts. plus, and if we compare this with our first candidate we can stop here.

But plus what?

Click Here To Show Diagram Code: [go]$$Bc Plus What? White to play. $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O 6 B W C O . O O . O O O C O X ,| $$ |, X O 5 4 7 O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

Black to play can of course save :bc:

. But

-

captures

and then

makes ko, for 5⅔ pts. for White. Is :w4:

-

sente, or does :w4:

-

simply reverse?

If White plays at 7 White gets 7 pts., which means that the position after :w6:

is worth 6⅓ pts. for White and :b7:

gains ⅔ pt. Since White gets only 5 pts. if Black saves :bc:

, that means that :w4:

-

gains ⅔ pt. and the position after :wc:

is worth 5⅔ pts., on average, for White. It also means that :w4:

-

is ambiguous between sente and gote. (See https://senseis.xmp.net/?AmbiguousPosition ). OC, for evaluation it does not matter which way we consider it.

Bill Spight · **#18**

White's first move.

Click Here To Show Diagram Code: [go]$$Wc White first $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O 4 5 C C O . O O . O O O C O X ,| $$ |, X O 2 3 1 O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

Here is one candidate move for White and a continuation. The result is 6 pts. for White.

Click Here To Show Diagram Code: [go]$$Wc White first, B2, B4 $$ ------------------------------------- $$ |, X O O O O O O O , O O , O C O X X ,| $$ |, X O . 4 , , O . O O . O O O C O X ,| $$ |, X O 2 . 1 O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

If Black plays again after :b2:

we get 4⅛ pts. for White. Now or later we should realize that the position after :b2:

is like a closed corridor with an average territorial value of 5 1/16 pts. for White.

Click Here To Show Diagram Code: [go]$$Wc White first, different reply $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O . 3 C C O . O O . O O O C O X ,| $$ |, X O . 2 1 O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

After this :b2:

,

yields 6½ pts. for White, on average. The other Black reply is better.

White's second candidate move

Click Here To Show Diagram Code: [go]$$Wc White's second candidate with continuation $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O C 3 , , O . O O . O O O C O X ,| $$ |, X O 1 2 a O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

Black plainly has no better reply than :b2:

, because a is equivalent. After :w3:

the result is 6¼ pts. for White, on average. This is better than after the other candidate move. If we had already done the analysis after Black plays first, we could figure out that this sequence is a reverse, but let's continue.

Click Here To Show Diagram Code: [go]$$Wc White first, B2, B4 $$ ------------------------------------- $$ |, X O O O O O O O , O O , O C O X X ,| $$ |, X O . 4 , , O . O O . O O O C O X ,| $$ |, X O 1 2 . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

As above, we can figure out that this position is like a closed corridor with 4⅛ pts. for White, on average. That means that the result after :b2:

is worth 5 3/16 for White, with each play gaining 1 1/16 pt.

Click Here To Show Diagram Code: [go]$$Wc White W1, W3 $$ ------------------------------------- $$ |, X O O O O O O O C O O C O C O X X ,| $$ |, X O C 5 C C O . O O . O O O C O X ,| $$ |, X O 1 4 3 O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

If

plays elsewhere, :w3:

obviously reverses, for a result of 7 pts. for White. The result after :b2:

is 5 3/16. If :b2:

is gote, the territorial value after :w1:

is 6 3/32, with each play gaining 29/32 pts. on average. But that is less than the gain after :b2:

, so

is not gote, but sente. And that means that the territorial value after :w1:

is the same as after :w1:

,

,

above, or 6¼ pts. Technically, this is not quite enough to show that that sequence is a reverse, but computationally the result is the same.

Edit: Oh, I see that :b2:

is actually ambiguous between sente and gote. That's because, even though it raises the local temperature to 1 1/16, after :w3:

the local temperature has only come back down to ¾ (i.e., the next play gains ¾), but that's the same as White's play to a White score of 7 pts. gains, so it could be played as part of a gote sequence. However, that makes no difference to the calculation of the value of positions. Besides, it is still part of a reverse that ends with :w3:

.

Bill Spight · **#19**

Putting it all together

Click Here To Show Diagram Code: [go]$$c Evaluate this $$ ------------------------------------- $$ |, X O O O O O O O . O O . O . O X X ,| $$ |, X O . . . . O . O O . O O O . O X ,| $$ |, X O . . . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

Click Here To Show Diagram Code: [go]$$c Black first $$ ------------------------------------- $$ |, X O O O O O O O . O O . O . O X X ,| $$ |, X O . . . . O . O O . O O O . O X ,| $$ |, X O 1 . . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

results in 4⅛ pts. for White, on average, with each play gaining ⅞ pts. on average.

Click Here To Show Diagram Code: [go]$$Wc White first $$ ------------------------------------- $$ |, X O O O O O O O . O O . O . O X X ,| $$ |, X O . 3 . . O . O O . O O O . O X ,| $$ |, X O 1 2 . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

The sequence, :w1:

,

,

results in 6¼ pts. for White, on average, with each play gaining ¾ pts. on average.

So the original position has a territorial value of 5 3/16 pts. for White on average, with each play gainig 1 1/16 pts. on average.

If you are of a mind to, you can show that :w1:

,

,

reverses because

Click Here To Show Diagram Code: [go]$$Wc W3 elsewhere $$ ------------------------------------- $$ |, X O O O O O O O . O O . O . O X X ,| $$ |, X O . 4 . . O . O O . O O O . O X ,| $$ |, X O 1 2 . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

is equivalent to

Click Here To Show Diagram Code: [go]$$Bc Black first $$ ------------------------------------- $$ |, X O O O O O O O . O O . O . O X X ,| $$ |, X O . . . . O . O O . O O O . O X ,| $$ |, X O 1 . . O O X O . X O X O O O X ,| $$ |, X X X X X X X X X X X X X X X X X ,| $$ |, , , , , , , , , , , , , , , , , , ,|[/go]

.

You can do that in your head, which may be easier than figuring out that :b2:

is sente, as in the previous note.

The point is that if :b2:

threatens to do at least as well, if :w3:

plays elsewhere, as playing first from the original position, White loses nothing by continuing and preventing Black from playing that threat.

----

1/16 pt. precision rarely matters in a real game, but if you keep miai and reversal in mind, and know your corridors, at least approximately, you can figure out over the board that a play here gains a little more than 1 pt. Recognizing reversal helps a lot, not only for computation, but for reading and play, as well. Often, reversal is obvious, often it is not.