## Sunday puzzle (multi steps)

Post puzzles for others to solve here.

### Re: Sunday puzzle (multi steps)

HoDoKu [from earlier than 2012] gives the following :

Forcing Net Verity => r4c1<>7
r4c4=7 r4c1<>7

r5c4=7 (r5c4<>4) (r6c5<>7) r4c4<>7 r4c4=8 (r8c4<>8 r8c4=1 r9c6<>1 r9c6=2 r9c1<>2 r9c1=8 r8c1<>8) r6c5<>8 r6c5=1 r1c5<>1 r1c5=4 r2c4<>4 r7c4=4 r7c4<>5 r8c4=5 r8c2<>5 r8c2=8 r9c1<>8 r4c1=8 r4c1<>7

r7c4=7 r7c4<>5 r8c4=5 r8c2<>5 r8c2=8 (r8c1<>8) r9c1<>8 r4c1=8 r4c1<>7

In words, for each 7 in c4, there is a chain that leads to r4c1 <>7. Two of the chains are quite short but the chain starting with r5c4 is very long. In HoDoKu, you can click on each chain in the text window to see that chain alone in the visual display.

FNV.png (144.44 KiB) Viewed 169 times
ghfick

Posts: 133
Joined: 06 April 2016

### Re: Sunday puzzle (multi steps)

yzfwsf wrote:I try to translate whip(9) into graphical representation as follows :
whip(9).PNG

7r56c2 = (7-2)r3c2 = (2-5)r2c2 = r8c2 - r8c4 = (5=7)r7c4 = r4c4|(r5c4 - ALS(7=81)r4c4r5c5 - (1=4)r1c5 - 4r2c4 =>Contradiction as No 4 in C4) => r4c1<>7

Very Nice! Like to know how you did it. Gordon Fick told me about you and your solver. I am also working on one in Python that shows all four Denis Bethier grids. It is still 'early days' on the development. You and any others are certainly welcome to my python source code. I would like for anyone to fork off my stuff or work with me on developing it - assuming our "chemistry" is good!
Big1952
RichardGoodrich

Posts: 42
Joined: 12 December 2012
Location: Tucson, AZ USA

### Re: Sunday puzzle (multi steps)

RichardGoodrich wrote:
yzfwsf wrote:I try to translate whip(9) into graphical representation as follows :
whip(9).PNG

7r56c2 = (7-2)r3c2 = (2-5)r2c2 = r8c2 - r8c4 = (5=7)r7c4 = r4c4|(r5c4 - ALS(7=81)r4c4r5c5 - (1=4)r1c5 - 4r2c4 =>Contradiction as No 4 in C4) => r4c1<>7

Very Nice! Like to know how you did it. Gordon Fick told me about you and your solver. I am also working on one in Python that shows all four Denis Bethier grids. It is still 'early days' on the development. You and any others are certainly welcome to my python source code. I would like for anyone to fork off my stuff or work with me on developing it - assuming our "chemistry" is good!

To be clear, this is not at all a representation of the intended whip. On the contrary, it reveals a total lack of understanding.
The only valid representation is the one I gave before:
denis_berthier
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Location: Paris

### Re: Sunday puzzle (multi steps)

denis_berthier wrote:
creint wrote:Image still won't load on my side (connection timeout):
https://denis-berthier.pagesperso-orange.fr/Misc/Whip9.pdf
Probably something with your site, don't know if pdfs can be loaded as in [img].

Could you try this https://denis-berthier.pagesperso-orange.fr
and also this https://denis-berthier.pagesperso-orange.fr/Misc/Whip9.png

to check what's wrong with my links?

I also try with the [img] tag

Your webserver is probably blocking some ip-ranges. Visiting the site with an proxy works.
creint

Posts: 271
Joined: 20 January 2018

### Re: Sunday puzzle (multi steps)

Recently I have been busy with some things different from playing with monsieur Berthier's whips but now I have time to answer all the questions about me and the above whip[9].
I am not anonimous, and my name is Bogdan Grigorenko. I have graduated from Moscow State University, here is my diploma in mathematics: https://disk.yandex.ru/i/1EAQYNY1BT_WaQ

This is the general graphical representation of a whip from monsieur Berthier's book: https://disk.yandex.ru/i/REIbvQateV7Z3w
This is the corresponding representation of the whip[9] discussed above (I have taken pains to show all the links): https://disk.yandex.ru/i/3XfIiUyXuoM4XQ
This is the representation of all the links on the sudoku board: https://disk.yandex.ru/i/hK-0mKODfzQwKQ

Now I have some questions to the members of the forum except for monsieur Berthier (I am not willing to speak with him until he apologizes to me).
1. Does the above whip remind you of a single continuous line?
2. Is it easy to construct such a whip by yourself without help from computer programmes?
3. Do the corresponding branches of the whip differ very much from the links of my forcing net (I give this link for the third time: https://disk.yandex.ru/i/rBJo3ttnB5sQeg )?
Bogdan
AnotherLife

Posts: 48
Joined: 07 January 2021
Location: Moscow, Russia

### Re: Sunday puzzle (multi steps)

RichardGoodrich wrote:Very Nice! Like to know how you did it. Gordon Fick told me about you and your solver. I am also working on one in Python that shows all four Denis Bethier grids. It is still 'early days' on the development. You and any others are certainly welcome to my python source code. I would like for anyone to fork off my stuff or work with me on developing it - assuming our "chemistry" is good!

Sorry for the late reply. My program uses a custom data structure of 81Bits, which I named Cellmap, so that 0/1 can be used to represent whether the number appears in a specific position, so that bit operations can be introduced. For the chain, I used a recursive data structure, and then used the BFS algorithm.
yzfwsf

Posts: 337
Joined: 16 April 2019

### Re: Sunday puzzle (multi steps)

I see that the so called 'civilized Europe' has fallen too low.
denis_berthier wrote:As you're anonymous, you can claim all the diplomas you like. No one can check.
But no, you are not ok with abstraction if you don't understand that the z- and t- candidates (and the associated links) don't belong to the whip - by definition of a whip.
And you are not ok with maths in general if you don't understand that a single line and 3 (or more) lines are not the same thing.

denis_berthier wrote:But if someone hasn't even read the definitions, the only thing he may honestly do is keep his mouth shut.

I have not got an apology so far.
Bogdan
AnotherLife

Posts: 48
Joined: 07 January 2021
Location: Moscow, Russia

### Re: Sunday puzzle (multi steps)

AnotherLife wrote:1. Does the above whip remind you of a single continuous line?
2. Is it easy to construct such a whip by yourself without help from computer programmes?
3. Do the corresponding branches of the whip differ very much from the links of my forcing net (I give this link for the third time: https://disk.yandex.ru/i/rBJo3ttnB5sQeg )?

1,3 no, in my opinion both are forcing nets.
2 no, not for me.
creint

Posts: 271
Joined: 20 January 2018

### Re: Sunday puzzle (multi steps)

creint wrote:
AnotherLife wrote:1. Does the above whip remind you of a single continuous line?
2. Is it easy to construct such a whip by yourself without help from computer programmes?
3. Do the corresponding branches of the whip differ very much from the links of my forcing net (I give this link for the third time: https://disk.yandex.ru/i/rBJo3ttnB5sQeg )?

1,3 no, in my opinion both are forcing nets.
2 no, not for me.

Bogdan
AnotherLife

Posts: 48
Joined: 07 January 2021
Location: Moscow, Russia

### Re: Sunday puzzle (multi steps)

Yeah, I wouldn't consider it a basic AIC; the point of a strong link in a basic AIC is that one end or the other must be true absent other information, and that is clearly not the case here. 7r5c2 = r3c2 is a simple enough adjustment (7r56c2 = r3c2), but "c4n7{r7 r5} - r4c4{n7..." is clearly problematic in AIC terms.

Denis, one thing that would make the whip notation clearer for me personally to read is the inclusion of the starting point (target elimination) at the beginning; something like c2n7{r6 r3} (or the c4 construction) is only meaningful in the context of 7r4c1 being assumed true (or eliminating 7r4c1 by contradiction). Obviously, we can get that information from the end of the statement, it would just be clearer to read left-to-right.

Another thing I'm unclear on is why the left-linking candidate needs to be notated at all, if t-candidates are not. Wouldn't something like c2n7{* r3}, where * is a stand-in for any and all candidate rows other than the right-linking r3, be preferable? It's the right-linking candidates that ultimately matter for the pattern to continue.
mith

Posts: 460
Joined: 14 July 2020

### Re: Sunday puzzle (multi steps)

Hi mith,
mith wrote: I wouldn't consider it a basic AIC; the point of a strong link in a basic AIC is [...]

No problem for me: it is NOT a basic AIC and I don't write it as an AIC, basic or not. Moreover, there is no strong link in my approach.
A basic AIC is what I call a bivalue-chain.

mith wrote:Denis, one thing that would make the whip notation clearer for me personally to read is the inclusion of the starting point (target elimination) at the beginning; something like c2n7{r6 r3} (or the c4 construction) is only meaningful in the context of 7r4c1 being assumed true (or eliminating 7r4c1 by contradiction). Obviously, we can get that information from the end of the statement, it would just be clearer to read left-to-right.

It wouldn't be a major change. However, when there are several eliminations, I don't think it would be clearer. And the tradition in maths is to write the conditions (here, the physical presence of a pattern) before the conclusion (here, the elimination).
Notice that, in my graphical representations, I include the target at the start.

mith wrote:Another thing I'm unclear on is why the left-linking candidate needs to be notated at all, if t-candidates are not. Wouldn't something like c2n7{* r3}, where * is a stand-in for any and all candidate rows other than the right-linking r3, be preferable?

The left-linking candidate is a testimony of the continuity condition. It is what allows to draw a single continuous line, as in the drawing I provided. In braids, the llc is what allows to find to which of z or a previous rlc it is linked. In both cases, they are essential

- If any z or t candidate disappears, the whip is UNCHANGED - by definition. Of course, this definition relies on some (obvious) reasons. Notice that this definition is not only a mathematical abstraction: in CSP-Rules, whips are coded this way and no change does happen in a whip when a z or a t candidate disappears before the whip is applied.

- If a left-linking candidate disappears, the whip is CHANGED. There is sometimes an obvious whip to replace it, differing only by this llc. If not, there may be a braid to replace it, or not.

mith wrote:It's the right-linking candidates that ultimately matter for the pattern to continue.

Absolutely. The z and t candidates play no role in continuing the sequence. They play only a momentarily role in verifying one step. That's why it's useless to notate them. They would only obscure the global continuous single-line structure.
It's right the llcs play no role either in continuing the sequence - but they play a role in displaying its continuity (be it graphically or in my notation).

I think there are two main (closely related) problems with people failing to see a whip as a single continuous line:
- they have been totally submerged by the old view of chains as chains of inferences;
- they fail to acknowledge the proper level of abstraction necessary to understand a whip.
denis_berthier
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Location: Paris

### Re: Sunday puzzle (multi steps)

denis_berthier wrote:Hi mith,
mith wrote: I wouldn't consider it a basic AIC; the point of a strong link in a basic AIC is [...]

No problem for me: it is NOT a basic AIC and I don't write it as an AIC, basic or not. Moreover, there is no strong link in my approach.
A basic AIC is what I call a bivalue-chain.

Sure, I just think there was some confusion introduced on page 1 when you were comparing them and using the AIC notation. They aren't strong links, so presenting them in that notation was misleading. (Whether this is where some of the misunderstanding came from, vs. not having a sufficient understanding of your abstraction vs. being stuck in a certain mindset vs. whatever else, I have no idea.)

mith wrote:Another thing I'm unclear on is why the left-linking candidate needs to be notated at all, if t-candidates are not. Wouldn't something like c2n7{* r3}, where * is a stand-in for any and all candidate rows other than the right-linking r3, be preferable?

The left-linking candidate is a testimony of the continuity condition. It is what allows to draw a single continuous line, as in the drawing I provided. In braids, the llc is what allows to find to which of z or a previous rlc it is linked. In both cases, they are essential

- If any z or t candidate disappears, the whip is UNCHANGED - by definition. Of course, this definition relies on some (obvious) reasons. Notice that this definition is not only a mathematical abstraction: in CSP-Rules, whips are coded this way and no change does happen in a whip when a z or a t candidate disappears before the whip is applied.

- If a left-linking candidate disappears, the whip is CHANGED. There is sometimes an obvious whip to replace it, differing only by this llc. If not, there may be a braid to replace it, or not.

mith wrote:It's the right-linking candidates that ultimately matter for the pattern to continue.

Absolutely. The z and t candidates play no role in continuing the sequence. They play only a momentarily role in verifying one step. That's why it's useless to notate them. They would only obscure the global continuous single-line structure.
It's right the llcs play no role either in continuing the sequence - but they play a role in displaying its continuity (be it graphically or in my notation).

My point (and creint's, I take it) is that it seems unnecessary to display that continuity in the form of a single llcs. If we are already abstracting, why not abstract to the ultimate point of the pattern? What matters for the pattern (whips and braids alike, as far as I understand them) is that the combination of Z and rlcs together act on all candidates for that CSP-Variable except possibly for one (the next rlc) or a related group (for g-whips). You've already established notationally that a whip is continuous (rather than a braid or some other pattern) by calling it a whip; if it is left to the reader to figure identify the t- and z- candidates to understand the validity, why not the llcs as well? The continuity is really between Z and the CSP-Variables themselves (via the other rlcs, for whips longer than whip[1]). The t-, z-, and left-linking candidates alike are all only serving to verify that continuity.

To me there is functionally no abstraction difference in c2n7{r6 r3} vs. c2n7{r56 r3} vs. c2n7{r5 r3} vs. c2n7{* r3}; all are continuous patterns (though the second has a grouped llc, while the last does not explicitly state the left-linking and t- candidates at all), and all convey the same deductive information. In some sense, the second option has an advantage to me in that losing a left-linking candidate may not necessarily change the whip at all apart from losing that candidate; if 7r6c2 is eliminated, I don't have to go looking for the r5 whip; I already have it by dropping r6 from the grouped llc.

To put it another way: In the cases where there is both a left-linking candidate and a t-candidate, losing the t-candidate leaves the whip unchanged (since we still have the llc), while losing the llc leaves us with slightly different whip using the t-candidate. To me, these could just as naturally be considered the same whip with two llcs, where losing one llc reduces it to the same whip with the remaining llc. (In cases where we only have the llc and no t-candidate, the notation would be the same as it already is.)

Anyway, it's your system. (I suppose you're welcome to call it absurd to ask these questions, or insult my intelligence for not appreciating the beauty of Notre Dame or something. That would be a shame though.)
mith

Posts: 460
Joined: 14 July 2020

### Re: Sunday puzzle (multi steps)

Hi Mith,
I've been using the AIC notation here for the first time, in order to show how a whip would be written in it. The AIC notation has been violated so many times that once more shouldn't be such a problem. I currently see undefined signs with no explanation of their meaning added to it, such as: * | ' ^ , and I probably miss a few more.

As for your proposed changes to my notation, why don't you apply them first to the AIC notation? Why don't you delete half of the candidates in it? Why don't you introduce an identical notation for rc-bivalue and rn-bivalue? ...

mith wrote:The continuity is really between Z and the CSP-Variables themselves (via the other rlcs, for whips longer than whip[1]). The t-, z-, and left-linking candidates alike are all only serving to verify that continuity.

That's total nonsense. Continuity is a structural/physical property I've defined between candidates (or g-candidates for g-continuity) - more precisely between labels (and g-labels for g-continuity); not between candidates and CSP-Variables. It is totally independent of the particular puzzle and the t and z candidates take no part in verifying it. Please read my definitions before talking about them.

As I explained in my previous post, there are ESSENTIAL differences between llcs and z or t candidates. I will not repeat them here. These differences are not arbitrary. They appear formally in the logical formula defining a whip and they appear in exactly the same form in the CSP-Rules transcription.
The explicit presence of the llcs in the pattern allows to draw the following graphical representation, clearly displaying a single continuous line:

1) Can't you see a single continuous line?
2) How do you do draw it without the llcs?
3) How do you draw it if the llcs are replaced by groups?
4) Can you explain how having a blob of 3 candidates instead of 1 would make the graphic clearer?
5) How do you make the difference between a whip and a braid without the llcs?

mith wrote:To put it another way: In the cases where there is both a left-linking candidate and a t-candidate, losing the t-candidate leaves the whip unchanged (since we still have the llc), while losing the llc leaves us with slightly different whip using the t-candidate. To me, these could just as naturally be considered the same whip with two llcs, where losing one llc reduces it to the same whip with the remaining llc. (In cases where we only have the llc and no t-candidate, the notation would be the same as it already is.)

In my definition, if the llc is changed as you say, but only when supposing the t-candidate is linked to the previous rlc, the two whips are equivalent, not identical. But the point is, if the llc disappears, what can happen in general?

It's not absurd to ask questions. It's absurd to waste time on discussions about notation.
denis_berthier
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### Re: Sunday puzzle (multi steps)

Denis, I didn't invent Eureka notation. It's a logical convention people use, and sometimes people use it in different ways, just as people speak different languages and dialects and so on. You are welcome to consider it or the way people use it flawed. I don't really care about that. (I do like your more consistent version of it as a mathematician, though I find it less clear than the cell-focused approach as a manual solver.)

I have read your definitions, and I am trying to understand why you have chosen to define and notate as you have. If your goal is for no one to ever want to understand your work, then you are doing a good job of that I suppose. For what it's worth, I find your work quite interesting - if I didn't, I wouldn't bother with having this conversation in the first place.

When I say the continuity is really between Z and the CSP-Variables, what I mean is that in order to have a complete whip, certain conditions must hold for each CSP-Variable - namely that each candidate of a CSP-Variable in the whip must be left-linking, t-, z-, or right-linking. If there is a candidate that is not one of those (what you call "pending"), then you have a partial whip (at least, if it's a candidate for Vn; presumably if the pending candidate were earlier in the pattern it would just be a shorter partial whip and stop at that point?). You have continuity as you have defined it in either case, and for the purpose of programming CSP-Rules that distinction may very well matter! But for the purpose of proving an elimination in a sudoku, continuity of a chain of left-linking and right-linking is irrelevant if there are extra candidates around which aren't t- or z- candidates. What matters is the whole structure of Z, the CSP-Variables, and candidates within those variables.

1) Can't you see a single continuous line?
2) How do you do draw it without the llcs?
3) How do you draw it if the llcs are replaced by groups?
4) Can you explain how having a blob of 3 candidates instead of 1 make the graphic clearer?
5) How do you make the difference between a whip and a braid without the llcs?

1. I never said I couldn't see a single continuous line.
2. The graphical representation of a thing (or lack thereof) in no way impacts its logical validity. (But for what it's worth, I would probably draw it similarly to your dot at the end showing there is no pending candidate for 4 in c4, except this would then link to the pending candidate; for the first link, 7r4c1 with a thin arrow to a dot in c2 with a thick arrow to 7r3c2. This would be far clearer to me as a manual solver than having one of the 7s in c2 highlighted red and another ignored. But maybe that's just me.)
3. How would you draw it for g-whip? It's exactly the same challenge. (I'd probably use a similar notation as before, with the grouped candidates connected to the dot.)
4. For understanding why it works logically, having all the candidates necessary to verify that it works logically seems clearer to me. My contention is that if verification is not the goal of the notation or graphical representation, then the llc is equally redundant. Again, YMMV.
5. I'm not sure it's a distinction that necessarily needs to be made, any more than it is necessary logically to have names for a bunch of different patterns (e.g. XY-Wing, Skyscraper) that are all types of AIC. Naming conventions are a help to the manual solver more than anything. For you, the distinctions between whips, g-whips, braids, etc. are distinctions worth making, and you have done so with the first four characters of the line when you notate it "whip". (And for someone "totally submerged by the old view of chains", maybe a whip without z-candidates vs. a whip with z-candidates would be a distinction worth making, I dunno.) What I don't understand, still, is the why - why you have defined continuity in the way you have (even noting that with braids you are relaxing continuity!), why you have made a distinction between whips and braids at all, why you have made a distinction between llcs and t-candidates, etc.
mith

Posts: 460
Joined: 14 July 2020

### Re: Sunday puzzle (multi steps)

mith wrote:When I say the continuity is really between Z and the CSP-Variables, what I mean is that in order to have a complete whip,...

Continuity is defined only in terms of links between candidates (more precisely labels). This definition applies to any binary CSP, not only to Sudoku, and it comes much before any consideration of whips or any solving method. It's pure graph-theoretical.

For the rest of your paragraph, it is based on the view of a chain as a chain of inferences, which a whip is not.

The nrc notation is a faithful graphico-symbolic representation of the logical definition of a whip, the image I gave is a (partially) faithful graphical representation of both the logical definition and the nrc notation. Both include all and only the candidates defining the whip.

Verification (of the => part of the notation) comes only on a second stage.
Verification of continuity should be trivial for the reader, given the presence of all the candidates necessary to check it. Contrary to what you say, continuity is an essential condition for the manual solver.
Verification of llc=>rlc at each stage requires some "work" from the reader, namely checking that each candidate not named in the CSP-Variable but still present in the PM is a z or t one. This may be tedious work, but it's similar to what you'll find explained out as "obvious" in any scientific book or paper.

Said otherwise:
"whip[9]: c2n7{r6 r3} - c2n2{r3 r2} - c2n5{r2 r8} - c4n5{r8 r7} - c4n7{r7 r5} - r4c4{n7 n8} - r6c5{n8 n1} - r1c5{n1 n4} - c4n4{r2 .} ==> r4c1 ≠ 7"
is the statement of a theorem about the puzzle under consideration, at a certain stage of resolution
.
By my general (obvious) whip elimination theorem, the proof of this particular form of the theorem can be reduced to checking that the conditions are satisfied for the pattern on the left; such checking doesn't require any inference; it only requires to observe the presence of some links. And, according to general mathematical practice, as this part is trivial, it is left as an "exercise for the reader".
Quick readers generally consider the author has not made errors in these trivial parts and they don't check them. Readers who want to check the details (a good practice for beginners) can easily do it.

mith wrote:1. I never said I couldn't see a single continuous line.

Good. Some people pretend they can't see it.

mith wrote:2. The graphical representation of a thing (or lack thereof) in no way impacts its logical validity.

In the present case, it does. Because, as I explained above, it is merely the logical formula written in a visual form (though some information about CSP-Variables may be lacking in the image).

mith wrote:What I don't understand, still, is the why - why you have defined continuity in the way you have (even noting that with braids you are relaxing continuity!), why you have made a distinction between whips and braids at all, why you have made a distinction between llcs and t-candidates, etc.

Beware of approximative language. I never said that I relax continuity. I have one and only one definition of continuity. What's relaxed is the continuity condition in the definition of the braid pattern.
Blurring all the definitions and all the distinctions is the ground for all the pointless discussions.

I've introduced a very limited number of universal types of chains (i.e. meaningful in any binary CSP): bivalue-chains, z-chains, t-whips, whips, braids, g-bivalue-chains, g-whips, g-braids, plus the corresponding, restricted 2D-versions of some of them. You can't compare this to the almost infinite number of pattens in the aquarium. Anyway, if you don't like some of these chains, you can forget them. Each of their definitions stands on its own.

Each type of chain has specific properties and a different range of resolution power. Why these types and not others? For the only ultimate reasons you'll ever find in science: because they form a consistent system AND they work!
denis_berthier
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