A new endgame beastie.
(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!
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| Precision, please http://www.lifein19x19.com/viewtopic.php?f=15&t=17450 |
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| Author: | Bill Spight [ Thu Apr 30, 2020 1:07 pm ] |
| Post subject: | Precision, please |
A new endgame beastie. (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!
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| Author: | Knotwilg [ Thu Apr 30, 2020 2:24 pm ] |
| Post subject: | Re: Precision, please |
First calculation: 1) White has on average 2817/512 = 5.501953125 2) First play gains 545/512 = 1.064453125 |
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| Author: | Bill Spight [ Thu Apr 30, 2020 6:04 pm ] |
| Post subject: | Re: Precision, please |
Knotwilg wrote: First calculation: 1) White has on average 2817/512 = 5.501953125 2) First play gains 545/512 = 1.064453125 Thanks. Edit: Correction. |
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| Author: | Bill Spight [ Fri May 01, 2020 1:53 am ] |
| Post subject: | Re: Precision, please |
Failure 1 |
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| Author: | dany [ Fri May 01, 2020 4:10 am ] |
| Post subject: | Re: Precision, please |
| Author: | Knotwilg [ Fri May 01, 2020 4:12 am ] |
| Post subject: | Re: Precision, please |
Second attempt: White has 821/256 on average; a play gains 245/256 on average |
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| Author: | Bill Spight [ Fri May 01, 2020 6:41 am ] |
| Post subject: | Re: Precision, please |
Knotwilg wrote: Second attempt: White has 821/256 on average; a play gains 245/256 on average Edit: |
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| Author: | Bill Spight [ Fri May 01, 2020 8:46 am ] |
| Post subject: | Re: Precision, please |
Difference game for White's first play. |
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| Author: | dany [ Fri May 01, 2020 2:39 pm ] |
| Post subject: | Re: Precision, please |
1) White has on average 1307/256 = 5.10546875 |
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| Author: | Bill Spight [ Sat May 02, 2020 12:53 am ] |
| Post subject: | Re: Precision, please |
I am afraid my hint gave the impression that the answers have a large denominator. That may not be the case.
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| Author: | Knotwilg [ Sat May 02, 2020 1:33 am ] |
| Post subject: | Re: Precision, please |
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? |
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| Author: | Bill Spight [ Sat May 02, 2020 2:02 am ] |
| Post subject: | Re: Precision, please |
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.  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.  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. |
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| Author: | Bill Spight [ Sat May 02, 2020 2:48 am ] |
| Post subject: | Re: Precision, please |
Difference game for Black's first play. |
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| Author: | Bill Spight [ Sat May 02, 2020 3:48 am ] |
| Post subject: | Re: Precision, please |
Another difference game. Warning, this one gives a big hint. |
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| Author: | dany [ Sat May 02, 2020 6:51 am ] |
| Post subject: | Re: Precision, please |
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 1) White has on average 83/16 = 5.1875 |
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| Author: | Bill Spight [ Sun May 03, 2020 2:21 am ] |
| Post subject: | Re: Precision, please |
Well, the cat's out of the bag, so here we 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. OK, the first moves are obvious. OK,  yields 5 pts. for WhiteIf Black plays  instead,  yields 4 pts. for White.If  makes a false eye,  yields 3 pts. for White. You get the picture.IOW, after  the resulting position is like this one.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 .  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. The sequence, ,  , 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 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.
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| Author: | Bill Spight [ Sun May 03, 2020 8:39 am ] |
| Post subject: | Re: Precision, please |
Black's first move. Here's one candidate move for Black. 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 - is actually sente. But - does look like a reverse. After all, White could have made 5 pts. simply by playing at instead of .  can take away the possible eye and 5th pt. makes an eye and 4 pts. At this point we may realize that this is like a simple closed corridor. Then we know that gains ¾ pt. and that the position after is worth 3¼ pts. for White. Also that gains ⅞ pt. and that the position after is worth 4⅛ pt. for White. If we don't, we can continue and figure it out.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.As above we can figure this to be worth 3¼ pts. for White. Then after the territorial value for White is 4½ pts. and 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.) is pretty obviously a loser. At first glance, after it looks like White has 5 9/16 pts. on average. 5 1/16 + ½, but the evaluation is a little tricky.After White has miai to make an eye at D-18 or G-18, so the result is worth 5 pts. for White.After the result is worth 6½ pts. for White. That means that the result after is worth 5¾ pts. for White on average, with the next play gaining ¾ pt.After White has 5 pts. plus, and if we compare this with our first candidate we can stop here. But plus what?Black to play can of course save  . But - captures and then  makes ko, for 5⅔ pts. for White. Is - sente, or does - simply reverse?If White plays at 7 White gets 7 pts., which means that the position after is worth 6⅓ pts. for White and gains ⅔ pt. Since White gets only 5 pts. if Black saves , that means that - gains ⅔ pt. and the position after  is worth 5⅔ pts., on average, for White. It also means that - is ambiguous between sente and gote. (See https://senseis.xmp.net/?AmbiguousPosition ). OC, for evaluation it does not matter which way we consider it.
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| Author: | Bill Spight [ Mon May 04, 2020 12:50 am ] |
| Post subject: | Re: Precision, please |
White's first move. Here is one candidate move for White and a continuation. The result is 6 pts. for White. If Black plays again after we get 4⅛ pts. for White. Now or later we should realize that the position after is like a closed corridor with an average territorial value of 5 1/16 pts. for White.After this , yields 6½ pts. for White, on average. The other Black reply is better.White's second candidate move Black plainly has no better reply than , because a is equivalent. After 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.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 is worth 5 3/16 for White, with each play gaining 1 1/16 pt.If plays elsewhere, obviously reverses, for a result of 7 pts. for White. The result after is 5 3/16. If is gote, the territorial value after is 6 3/32, with each play gaining 29/32 pts. on average. But that is less than the gain after , so is not gote, but sente. And that means that the territorial value after is the same as after , , 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 is actually ambiguous between sente and gote. That's because, even though it raises the local temperature to 1 1/16, after 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 .
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| Author: | Bill Spight [ Mon May 04, 2020 1:29 am ] |
| Post subject: | Re: Precision, please |
Putting it all together results in 4⅛ pts. for White, on average, with each play gaining ⅞ pts. on average.The sequence, , , 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 , , reverses because is equivalent to . You can do that in your head, which may be easier than figuring out that is sente, as in the previous note. The point is that if threatens to do at least as well, if 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. |
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| Author: | Bill Spight [ Mon May 04, 2020 3:04 pm ] |
| Post subject: | Re: Precision, please |
In the FWIW department, here is another difference game. This one shows that the position after is equivalent to the position after , , . Which means that we may consider the sequence , ,  to be a reverse. Well, which sequence is the reverse? It can't be both!  I was surprised, too. Live and learn. True, it can't be both, but it can be one or the other. Either way produces the same simplified game tree and the same values. In actual play, it's Black who gets to make the decision. Does she respond to now, or wait? IMO, Black should usually wait, in case she wants to use as a ko threat.
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