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 Post subject: Is 7 the magic number in L&D? #1 Posted: Sun Mar 31, 2024 6:17 am
 Oza

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I am one of the world's anti-number people. To give an example of what that means for mindset, I was learning a dance some time ago in which two couples approach each other and one has to give way to the other. Couples are numbered 1 to 5. I was in couple 2 and the teacher was in couple 1. It was unclear at that early stage who gave way to whom. Any impasse ensured. The teacher, a scientist and statistician, said in a slightly irritated "as any fule kno" voice, "2s give way to 1s." My reaction (showing I was no fule, incidentally!) was, "Cows eat grass. So what?"

But that does not mean I totally ignore umbers. I was editing a book of L&D problems, adding the captions to the diagrams. This involved pitting in the number of moves in each sequence, e.g. "Diagram 1 (1 ~ 9)". After a while, my subconscious (yes, it really works) told me that the number 7 was coming yup rather a lot. That got me thinking. It also got me counting.

I took out a copy of the Guanzi Pu. Incidentally, this does not have 1473 problems as SL and other places says. And that, as any fule kno, tells us that the editors of SL and the like don't actually read these books. There are about that number of positions, but a VERY large number of these are variations - some problems have at least five variations. In fact, what makes the Guanzi Pu so famous and important is precisely that it was the first book to give extensive variations.

Anyway, I decided to count the number of moves given in each line for the first 100 or so positions. The following was the result, with the number of moves followed by the number of positions giving that number of moves.

1: 0
2: 0
3: 10
4: 1
5: 20
6: 4
7: 28
8: 10
9: 13
10: 9
11: 5
12: 1
13: 6
14: 2
15: 1

7 is the winner! Because most L&D problems (and all in this case) involve the first side killing the other, we can expect odd numbers to predominate - even numbers usually come up in variations. I looked at 105 positions (68 problems), so there were a fair number of variations, but even there odd numbers dominate. Since that is relevant to a point I want to make later, I will therefore recast the table as follows:

1-2: 0
3-4: 11
5-6: 24
7-8: 38
9-10: 22
11-12: 6
13-14: 3
15-16: 1

Or, to put it another for my purpose, let us view it as pairs of moves. I think that can be justified for tactical situations, on sente and other grounds) just as happens in chess (a two-move problem is really three moves there!).

1 pair (1-2:) 0
2 pairs (3-4): 11
3 pairs (1-6): 24
4 pairs (1-8): 38
5 pairs (1-10): 22
6 pairs (1-12: 6
7 pairs (1-14: 3
8 pairs (1-16): 1

At the risk of showing my ignorance, this seems to be a sort of Bell curve. Whatever you call it, it gives some interesting points to speculate on.

First, let it be said that the moves shown go up to the point where the editor thinks the full solution is then instantly obvious by inspection. For example, nakade move inside a five-point void would be the last move shown. FWIW that editorial assumption matched my own perception at least close to 100%. A lower kyu player may have to stop and think for a while, but I think it is reasonable to expect any amateur to easily get to the status of a competent L&D Inspector.

Second, there is also an editorial hand at work in that the positions chosen limit the number of moves somewhat - but do note that the Guanzi Pu problems are still considered to be hard on the whole.

Now what this suggests to me is that L&D is really quite easy. We only have to use the frontal short-term memory portion of our brain. Again, I may be showing my ignorance, but as I understand it, our short-term memory is very, very fast but can only handle about 6 or 7 bits of information at once. But those bits of information can be put into chunks, and if you can get those chunks down to 6 or 7 max, your STM can often still handle them. We see this process at work with telephone numbers. 9497859209 is very hard to remember, but if we split it up as 9-497 859 209 it is already a lot easier to handle and memorise. But if we spot the pattern (something humans are very good at) that there is a 9 in each segment, memorisation gets another boost.

What I suspect L&D devisers are doing is, first, assuming solutions can be found within a very limited number of moves (7 being the magic number there). Second, they are assuming that these moves can be split up into recognisable chunks of paired action-reaction moves. Furthermore, this tends to reduce the number of chunks to the lower end spectrum (2 to 5 pairs), easing the work of the STM considerably.

But chunking can go even further. Pairs of moves can be grouped together - for example a connect-and-die (oi-otoshi) sequence will comprise two, three or even four pairs of moves. My research indicates that at least 50 chunk-type patterns can be found. This is far more than the usual named varieties, but you choose your own flavours and add your own names, as I have done. For example, one of the commonest themes in L&D problems is a throw-in kind of move I call an elbow lock. I have never seen it named or even discussed anywhere else. But it's so common that I instinctively look for it - and very often find it - in L&D positions I look at.

I think you need to get these namable chunks into the surface areas of your longer-term memory, so that the STM can look them up instantly, but 50 is no big deal as a total (and even only about 12 will cover about 90% of problems) , and keeping these chunks on the surface of LTM is just a better of going to the L&D gym regularly.

You can also follow the ancients and give your LTM some help in keeping L&D shapes readily available on the front shelf. Many famous problems have names. These help either as clues or as aids to memorisation, either of themes or even of whose positions.

In short, L&D can become easy with the right approach (which, regrettably does include some work, of course). There is no need to get bogged down in tedious and often impossible "if he goes there, I go there" calculations. It becomes easy to understand how pros can make lightning-fast calculations - but understanding that means you can do it, too.

It's an awful lot easier to catch rabbits by setting a trap rather than chasing them down rabbit-holes.

 This post by John Fairbairn was liked by 2 people: gowan, xela
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 Post subject: Re: Is 7 the magic number in L&D? #2 Posted: Sun Mar 31, 2024 9:03 am
 Oza

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To give some substance to my claim that L&D can be made easy as described above, and because it's Easter, I thought I'd add an Easter egg of a problem. White to kill Black. It's better to go easy on the chocolate and do problems like this to get your dose of serotonin.

`[go]\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . O O . . . . . . . . . . . . . . . . |\$\$ | . . . O . . . . . . . . . . . . . . . |\$\$ | . X X . . . . . . . . . . . . . . . . |\$\$ | X . X . O . . . . . . . . . . . . . . |\$\$ | X O X , O . . . . , . . . . . , . . . |\$\$ | . O X X . O . . . . . . . . . . . . . |\$\$ | . . O X . O . . . . . . . . . . . . . |\$\$ | . . O . . . . . . . . . . . . . . . . |\$\$ ----------------------------------------|[/go]`

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 Post subject: Re: Is 7 the magic number in L&D? #3 Posted: Sun Mar 31, 2024 9:59 am
 Oza

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Six is the magic number (Easter Hare to the rescue)

`[go]\$\$W Rabbity six\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . O O . . . . . . . . . . . . . . . . |\$\$ | . . . O . . . . . . . . . . . . . . . |\$\$ | . X X . . . . . . . . . . . . . . . . |\$\$ | X . X . O . . . . . . . . . . . . . . |\$\$ | X O X , O . . . . , . . . . . , . . . |\$\$ | 4 O X X . O . . . . . . . . . . . . . |\$\$ | 1 . O X . O . . . . . . . . . . . . . |\$\$ | . 3 O 2 . . . . . . . . . . . . . . . |\$\$ ----------------------------------------|[/go]`

W1 at W3 also works
Black dies in gote. Later ...

`[go]\$\$B Rabbity six\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . O O . . . . . . . . . . . . . . . . |\$\$ | . . . O . . . . . . . . . . . . . . . |\$\$ | . X X . . . . . . . . . . . . . . . . |\$\$ | X 1 X . O . . . . . . . . . . . . . . |\$\$ | X O X , O . . . . , . . . . . , . . . |\$\$ | X W X X . O . . . . . . . . . . . . . |\$\$ | O 3 O X . O . . . . . . . . . . . . . |\$\$ | . O O X . . . . . . . . . . . . . . . |\$\$ ----------------------------------------|[/go]`

W2 pass. W4 at WW.

`[go]\$\$B Rabbity six\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . O O . . . . . . . . . . . . . . . . |\$\$ | . . . O . . . . . . . . . . . . . . . |\$\$ | . X X . . . . . . . . . . . . . . . . |\$\$ | X X X . O . . . . . . . . . . . . . . |\$\$ | X . X , O . . . . , . . . . . , . . . |\$\$ | X O X X . O . . . . . . . . . . . . . |\$\$ | O C O X . O . . . . . . . . . . . . . |\$\$ | . O O X . . . . . . . . . . . . . . . |\$\$ ----------------------------------------|[/go]`

If Black goes on to capture the rabitty six, White throws in at the vital point.

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 Post subject: Re: Is 7 the magic number in L&D? #4 Posted: Sun Mar 31, 2024 11:17 am
 Oza

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Except your first var is wrong as black 2 would be at 3

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 Post subject: Re: Is 7 the magic number in L&D? #5 Posted: Mon Apr 01, 2024 1:48 am
 Oza

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Ah of course. Strange things happen ...

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 Post subject: Re: Is 7 the magic number in L&D? #6 Posted: Mon Apr 01, 2024 2:30 am
 Lives in gote

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My attempt:
`[go]\$\$W\$\$ | . . . , . . . . . , . . . . . , . . . |\$\$ | . . . . . . . . . . . . . . . . . . . |\$\$ | . O O . . . . . . . . . . . . . . . . |\$\$ | . . . O . . . . . . . . . . . . . . . |\$\$ | . X X . . . . . . . . . . . . . . . . |\$\$ | X . X . O . . . . . . . . . . . . . . |\$\$ | X O X , O . . . . , . . . . . , . . . |\$\$ | . O X X . O . . . . . . . . . . . . . |\$\$ | b . O X . O . . . . . . . . . . . . . |\$\$ | . 1 O a . . . . . . . . . . . . . . . |\$\$ ----------------------------------------|[/go]`

If black is kind enough to reply at 'a', then white can play 'b' for Knotwilg's rabbity variation. Or if black 'b' then white 'a' connects out.

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 Post subject: Re: Is 7 the magic number in L&D? #7 Posted: Mon Apr 01, 2024 2:34 am
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Oh, and John, I'm impressed to see you compiling a table of statistics! I think I owe you a poem some time.

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 Post subject: Re: Is 7 the magic number in L&D? #8 Posted: Mon Apr 01, 2024 6:26 am
 Gosei

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John Fairbairn wrote:
I am one of the world's anti-number people. To give an example of what that means for mindset, I was learning a dance some time ago in which two couples approach each other and one has to give way to the other. Couples are numbered 1 to 5. I was in couple 2 and the teacher was in couple 1. It was unclear at that early stage who gave way to whom. Any impasse ensured. The teacher, a scientist and statistician, said in a slightly irritated "as any fule kno" voice, "2s give way to 1s." My reaction (showing I was no fule, incidentally!) was, "Cows eat grass. So what?"

But that does not mean I totally ignore umbers. I was editing a book of L&D problems, adding the captions to the diagrams. This involved pitting in the number of moves in each sequence, e.g. "Diagram 1 (1 ~ 9)". After a while, my subconscious (yes, it really works) told me that the number 7 was coming yup rather a lot. That got me thinking. It also got me counting.

I took out a copy of the Guanzi Pu. Incidentally, this does not have 1473 problems as SL and other places says. And that, as any fule kno, tells us that the editors of SL and the like don't actually read these books. There are about that number of positions, but a VERY large number of these are variations - some problems have at least five variations. In fact, what makes the Guanzi Pu so famous and important is precisely that it was the first book to give extensive variations.

Anyway, I decided to count the number of moves given in each line for the first 100 or so positions. The following was the result, with the number of moves followed by the number of positions giving that number of moves.

1: 0
2: 0
3: 10
4: 1
5: 20
6: 4
7: 28
8: 10
9: 13
10: 9
11: 5
12: 1
13: 6
14: 2
15: 1

7 is the winner! Because most L&D problems (and all in this case) involve the first side killing the other, we can expect odd numbers to predominate - even numbers usually come up in variations. I looked at 105 positions (68 problems), so there were a fair number of variations, but even there odd numbers dominate. Since that is relevant to a point I want to make later, I will therefore recast the table as follows:

1-2: 0
3-4: 11
5-6: 24
7-8: 38
9-10: 22
11-12: 6
13-14: 3
15-16: 1

Or, to put it another for my purpose, let us view it as pairs of moves. I think that can be justified for tactical situations, on sente and other grounds) just as happens in chess (a two-move problem is really three moves there!).

1 pair (1-2:) 0
2 pairs (3-4): 11
3 pairs (1-6): 24
4 pairs (1-8): 38
5 pairs (1-10): 22
6 pairs (1-12: 6
7 pairs (1-14: 3
8 pairs (1-16): 1

At the risk of showing my ignorance, this seems to be a sort of Bell curve. Whatever you call it, it gives some interesting points to speculate on.

First, let it be said that the moves shown go up to the point where the editor thinks the full solution is then instantly obvious by inspection. For example, nakade move inside a five-point void would be the last move shown. FWIW that editorial assumption matched my own perception at least close to 100%. A lower kyu player may have to stop and think for a while, but I think it is reasonable to expect any amateur to easily get to the status of a competent L&D Inspector.

Second, there is also an editorial hand at work in that the positions chosen limit the number of moves somewhat - but do note that the Guanzi Pu problems are still considered to be hard on the whole.

Now what this suggests to me is that L&D is really quite easy. We only have to use the frontal short-term memory portion of our brain. Again, I may be showing my ignorance, but as I understand it, our short-term memory is very, very fast but can only handle about 6 or 7 bits of information at once. But those bits of information can be put into chunks, and if you can get those chunks down to 6 or 7 max, your STM can often still handle them. We see this process at work with telephone numbers. 9497859209 is very hard to remember, but if we split it up as 9-497 859 209 it is already a lot easier to handle and memorise. But if we spot the pattern (something humans are very good at) that there is a 9 in each segment, memorisation gets another boost.

What I suspect L&D devisers are doing is, first, assuming solutions can be found within a very limited number of moves (7 being the magic number there). Second, they are assuming that these moves can be split up into recognisable chunks of paired action-reaction moves. Furthermore, this tends to reduce the number of chunks to the lower end spectrum (2 to 5 pairs), easing the work of the STM considerably.

But chunking can go even further. Pairs of moves can be grouped together - for example a connect-and-die (oi-otoshi) sequence will comprise two, three or even four pairs of moves. My research indicates that at least 50 chunk-type patterns can be found. This is far more than the usual named varieties, but you choose your own flavours and add your own names, as I have done. For example, one of the commonest themes in L&D problems is a throw-in kind of move I call an elbow lock. I have never seen it named or even discussed anywhere else. But it's so common that I instinctively look for it - and very often find it - in L&D positions I look at.

I think you need to get these namable chunks into the surface areas of your longer-term memory, so that the STM can look them up instantly, but 50 is no big deal as a total (and even only about 12 will cover about 90% of problems) , and keeping these chunks on the surface of LTM is just a better of going to the L&D gym regularly.

You can also follow the ancients and give your LTM some help in keeping L&D shapes readily available on the front shelf. Many famous problems have names. These help either as clues or as aids to memorisation, either of themes or even of whose positions.

In short, L&D can become easy with the right approach (which, regrettably does include some work, of course). There is no need to get bogged down in tedious and often impossible "if he goes there, I go there" calculations. It becomes easy to understand how pros can make lightning-fast calculations - but understanding that means you can do it, too.

It's an awful lot easier to catch rabbits by setting a trap rather than chasing them down rabbit-holes.

I wonder whether these number patterns would be consistent between different sources of problems. For example what would happen if the problems in Xuanxuan Qijing were analysed, or Gokyo Shumyo, or Shikatsu Myoki, or even a modern composer such as Cho U.

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 Post subject: Re: Is 7 the magic number in L&D? #9 Posted: Mon Apr 01, 2024 7:45 am
 Oza

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Quote:
I wonder whether these number patterns would be consistent between different sources of problems. For example what would happen if the problems in Xuanxuan Qijing were analysed, or Gokyo Shumyo, or Shikatsu Myoki, or even a modern composer such as Cho U.

Having already published e.g. the complete Xuanxuan Qijing and Shikatsu Myoki and having prepared other old works for publication (being proofread as I write), and bearing in mind that most of Genbi's works are taken from the classics (that's what Gokyo refers to, after all), I am confident that the various numbers would more or less hold up there. The corpus I have in mind would be maybe over 2,000 distinct problems - most books crib extensively from earlier ones, and so I am disregarding duplicates even when superficial changes in configuration have been made.

I am less sure about about the modern masters (Hashimoto, Shioiri, Maeda, Cho U, Go, etc) in one respect. I think the numbers themselves would hold up (e.g. length of solutions), but the modern masters often seem concerned, as far as I can see, with trying to introduce novel themes which few of us have in our memory databanks. That makes the problem harder and entertaining, of course, but they also seem less rewarding in a way. Because these new themes are seemingly rare, they may have little relevance to our actual play even if we learn them. But precisely because they are rare, they are not so easy to remember. But, that said, I still think the moderns' problems overlap in themes and length with the old masters' problems to the degree of something like 95%+.

To make this more concrete, here are two problems which are not entirely similar but were similar enough for me to bring them together in my mind (both visually and thematically). One is by Hashimoto and one by A. Non from some 500 years earlier.

As you derive or look at the solutions (these are built into the sgf), ask yourself questions like:

1. which one is modern and which one is ancient?
2. Is either one novel?
3. Is one aesthetically better then the other?
4. Is one more practical use than the other?
5. What themes are there?

Let us all know your thoughts.

In this one, it is Black to play.

In this next one it is White to play. Do not fall into the common trap of thinking that White to play means it is an ancient problem. In any case, I have flipped a coin.

One is called "Workings of an immortal" and the other (Hashimoto's) is Koromogaseki. There are strong clues in there as to theme and level of difficulty, but I'll leave that for another time.

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 Post subject: Re: Is 7 the magic number in L&D? #10 Posted: Tue Apr 02, 2024 6:16 am
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John Fairbairn wrote:
What I suspect L&D devisers are doing is, first, assuming solutions can be found within a very limited number of moves (7 being the magic number there). Second, they are assuming that these moves can be split up into recognisable chunks of paired action-reaction moves. Furthermore, this tends to reduce the number of chunks to the lower end spectrum (2 to 5 pairs), easing the work of the STM considerably.

I will make another observation that may offer another speculation. 7 is roughly the number of internal spaces available in most L&D problems. Your Easter problem had 6, the more recent examples in this thread have 7-9.

I watched a Michael Redmond video last week where he put forth the rule of thumb that a group that has 8 internal spaces available is quite likely to be able to make a living shape. Not a perfect rule, but a useful heuristic for judging how weak a position is. Flipping that around, it would seem that 7 spaces may be just close enough to living that nontrivial problems can be created.

We might plausibly expect the number of moves in a solution to be bounded by the amount of internal space available. Not all internal spaces will need to be filled, but there may be additional moves like a hane that are required to collapse the eye space.

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 Post subject: Re: Is 7 the magic number in L&D? #11 Posted: Tue Apr 02, 2024 8:03 am
 Oza

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Quote:
I watched a Michael Redmond video last week where he put forth the rule of thumb that a group that has 8 internal spaces available is quite likely to be able to make a living shape.

Interesting idea. I looked quickly at a few positions but got the impression that in devised, as opposed to naturally occurring, L&D contexts, it doesn't seem to work all that well. Devised problems are meant to defy the normal rules, of course. One big complication, though, is that internal spaces may contain enemy stones. Do they count as spaces in Redmond's heuristic? Also, am I right in assuming he meant "eight spaces plus sente?"

Your post also reminds me of "five alive", a Korea notion that is really for use in fights rather than L&D. If a string has five liberties it becomes very hard to capture. It seems to work quite well in L&D, though, at least turning it around a bit. If you get a string with at least five liberties, there is a good chance the group it belongs to will live. Conversely, if you are looking for a kill and you move gives the enemy a string of five+ liberties, you are probably going to fail on that line. If you want to look at it, there are some points to observe. One i recall is that you should count the liberties after mentally making all forcing moves.

Here's an example (a boundary play problem) where at first glance suggests Black is easily safe (lots of space, a solid eye) but an almost as fast inspection using liberty counts tells a different story (White to play). Answer embedded in file. This is "Happy magpie perched on a branch." I have, this morning, happily been watching fat magpies enjoying worm spaghetti in my back garden.

There is a vaguely similar idea used in an early LD progam by a guy whose name I have forgotten (Thomas somebody, German/Austrian but based in Canada???). There (as I recall) the solution is defined as being found once you capture the biggest enemy string in the position. Does save a huge amount of work!

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 Post subject: Re: Is 7 the magic number in L&D? #12 Posted: Tue Apr 02, 2024 9:01 am
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John Fairbairn wrote:
Interesting idea. I looked quickly at a few positions but got the impression that in devised, as opposed to naturally occurring, L&D contexts, it doesn't seem to work all that well. Devised problems are meant to defy the normal rules, of course. One big complication, though, is that internal spaces may contain enemy stones. Do they count as spaces in Redmond's heuristic? Also, am I right in assuming he meant "eight spaces plus sente?"

Yes, I think it was 8 spaces plus sente. The context I saw it had empty internal spaces, and it wasn't plastered around the outside with stones. It's certainly not a firm rule, but I have noticed it works well in evaluating whether/when to get nervous as a group starts getting enemy stones nearby.

John Fairbairn wrote:
There is a vaguely similar idea used in an early LD progam by a guy whose name I have forgotten (Thomas somebody, German/Austrian but based in Canada???). There (as I recall) the solution is defined as being found once you capture the biggest enemy string in the position. Does save a huge amount of work!

You must certainly be thinking of Thomas Wolfe, who created the GoTools tsume-go solver. He teaches at a university just outside Toronto.

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 Post subject: Re: Is 7 the magic number in L&D? #13 Posted: Thu Apr 04, 2024 10:08 am
 Judan

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John Fairbairn wrote:
In short, L&D can become easy with the right approach (which, regrettably does include some work, of course). There is no need to get bogged down in tedious and often impossible "if he goes there, I go there" calculations.

As I have proved to you before, as a general statement this is wrong.

- L-D problems composed of chunks may be accessible to simplified solutions using chunks. (As you have noticed, many classics belong to this type.)

- L-D problems that cannot be decomposed for simplification into chunks during the early plies require detailed tactical reading then.

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 Post subject: Re: Is 7 the magic number in L&D? #14 Posted: Fri Apr 05, 2024 2:21 am
 Oza

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I think it's a composer's bias towards what is interesting for a targeted audience.

`[go]\$\$W Problem - experienced\$\$ ------------------\$\$ | . . . a . . . . .\$\$ | b . X X O . . . .\$\$ | . . X O . O . . .\$\$ | . X X O . . . . .\$\$ | . O O . O . . . .\$\$ | . . . . . . . . .\$\$ | . O . . . . . . .\$\$ | . . . . . . . . .\$\$ | . . . . . . . . .[/go]`

For an experienced player, the above problem is trivial. The most elementary of the killing moves is White's hane at A. With Black the elementary way to live is B. The number of moves one needs to show to an audience like ours is 0-1. One might include it into a collection for this audience, if the question is made more interesting: which other living/killing moves are there, and why are A and B preferred?

`[go]\$\$W Solution - beginners\$\$ ------------------\$\$ | . . 2 1 . . . . .\$\$ | . 5 X X O . . . .\$\$ | 4 . X O . O . . .\$\$ | 3 X X O . . . . .\$\$ | . O O . O . . . .\$\$ | . . . . . . . . .\$\$ | . O . . . . . . .\$\$ | . . . . . . . . .\$\$ | . . . . . . . . .[/go]`

For a beginner, the depth of the solution shown is more likely to be 5, the point where they recognize the bulky five's vital point. It's also instructive for the beginner that the hane's don't need to be protected and this is a hint as to why White may need a strong surrounding group.

`[go]\$\$W Variant - beginners\$\$ ------------------\$\$ | . 3 4 1 . . . . .\$\$ | 2 7 X X O . . . .\$\$ | 6 . X O . O . . .\$\$ | 5 X X O . . . . .\$\$ | . O O . O . . . .\$\$ | . . . . . . . . .\$\$ | . O . . . . . . .\$\$ | . . . . . . . . .\$\$ | . . . . . . . . .[/go]`

In such a beginner's book on life and death, this variation will be shown and it has depth 7, the point where a beginner recognizes this "nakade".

`[go]\$\$W Variant 2 - beginners\$\$ ------------------\$\$ | . 6 5 1 . . . . .\$\$ | 7 2 X X O . . . .\$\$ | 3 . X O . O . . .\$\$ | 4 X X O . . . . .\$\$ | . O O . O . . . .\$\$ | . . . . . . . . .\$\$ | . O . . . . . . .\$\$ | . . . . . . . . .\$\$ | . . . . . . . . .[/go]`

Another variation, another 7 moves. Probably a beginner's book will include a note that capturing those two stones is futile due to the throw-in. This raises the mental depth to 9.

On a final note, the bias is 7 (or 5) rather than 6 (or 4) because a success diagram for the player to start will have an odd number. Success is shown by the glorious completion of the challenge by the challenger, rarely by the failure of the defender.

`[go]\$\$B Black to live\$\$ ------------------\$\$ | . 2 . . . . . . .\$\$ | 1 3 X X O . . . .\$\$ | . . X O . O . . .\$\$ | . X X O . . . . .\$\$ | . O O . O . . . .\$\$ | . . . . . . . . .\$\$ | . O . . . . . . .\$\$ | . . . . . . . . .\$\$ | . . . . . . . . .[/go]`

Here the challenge is Black to live, which can be made apparent to a beginner in 3 moves

`[go]\$\$B Black to live\$\$ ------------------\$\$ | . 6 . 5 . . . . .\$\$ | 4 1 X X O . . . .\$\$ | 2 . X W a O . . .\$\$ | 3 X X O . . . . .\$\$ | . O O . O . . . .\$\$ | . . . . . . . . .\$\$ | . O . . . . . . .\$\$ | . . . . . . . . .\$\$ | . . . . . . . . .[/go]`

Failure diagrams usually end in an even number. In this case one might appeal to a more experienced audience to foresee how this situation will evolve. Incidentally, if e.g. the marked stone is at A instead, works because Black can suffocate the white stones inside, thanks to the extra outside liberty. This is an awareness I have but definitely not something I instantly see when looking at this problem.

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