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Victor
Joined: 29 Sep 2005
Posts: 207
Location: NI
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 Posted: Fri May 09, 2008 2:15 pm Post subject: |
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Thanks folks. The half-M-wing first. Here's that thread again:
| Code: | +----------------+----------------+
| 247 34A 8 | 347B 16 29 |
| 6 5 1-37 | 37 18 29 |
| 24 9 13D | 34C 168 5 |
+----------------+----------------+ |
I do now remember reading it & thinking it was nice, but also being slightly puzzled. I've put my finger on the puzzlement. In AIC lingo: 3A = 4A - 4B = 4C - 3C = 3D. So yes, we can remove the 3 from the 137 cell. But here's the source of my puzzlement: the link from 3 to 4 in C is weak: i.e. we could have other number(s) in C. To put it another way, If A = 4, then C = 4 (by standard 'transporting' or whatever one wishes to call it). This means that D is 3 because the 3s in that row are conjugate.
So we've exploited: (a) the fact that the 34 in A are naked, (b) the weak-strong link in 4s from A to C, (c) the weak link between 3 & 4 in C, and (d) the strong link in 3 between C & D. But we haven't exploited the fact that the 3 & 4 in C are 'naked'.
That's all fine but ... I'm still not sure what the distinguishing feature of a half-M-wing is. A weak-then-strong link in a single digit is often referred to as 'transport'ing, as in ER-ing, etc. Is it just a matter of context that one sometimes talks of transporting, sometimes of half-M-winging? Or have I missed something?
[b]The M-wing stuff. Asellus wrote: | Quote: | | Quote: | Victor wrote:
This kind of stuff needs EVERY link to be conjugate (strong). |
Actually, nataraj is correct: the link can be weak. |
Quite right! Sorry, Nataraj & anyone reading that: I was guilty of casual thought, not for the first time. Here's how I see it (now that I've thought about it.)
Say you've three cells A(1,4), B(1,4), C(4,y), where the link from A to B in 1s is 'by transporting ER-style', i.e a weak followed by a strong link, but without any obvious connection between the 4s. The link from B to C in 4s is strong. Then you can say: A = 1 => B= 1, and the other way round you have B <> 1 => A <> 1. Crossing over to 4s and extending to C, you have: A<>4 => B <> 4 => C =4, and C<>4 => B = 4 => A = 4. So you can indeed eliminiate any 4s seen by both A & C.
However . . . say it's a chain 14 to 1x to 1y to 14. If the links are all strong in 1s (conjugate) then you can say that the 4s are conjugate also (3 steps of a colouring chain), and do whatever that allows you to do. If, however, you replaced the first two links by a weak-then-strong as above, you could still say eliminate any 4s seen by both ends. But they'e now linked by what some people call a 'strong-only' link, and you would need more care if you wished to extend onwards in 4s.
Coinidentally, there's one of these very chains in the puzzle posted by Norm today, competition #1034. If you look at where I've got to, consider the 58s in r7c5, r4c4, r2c4. It's effectively a weak-strong link in 8s from the first to the second (i.e if the first is 8, then so is the second), and then a strong in 5s to the third. So we can eliminate any 5 seen by both ends, i.e. in r2c5. (Note that it's an irrelevance that the last cell is also a 58 - could be 5s with anything.)
Last edited by Victor on Sat May 10, 2008 10:19 am; edited 2 times in total |
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Victor
Joined: 29 Sep 2005
Posts: 207
Location: NI
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| Posted: Sat May 10, 2008 8:34 am Post subject: |
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I realised, after I'd written, what was going on. Sorry, Keith, but I think that particular example is not the best you could have for a half-M-wing, the reason being that you can, as I explained, see the 3-elimination without using a half-M-wing.
Anyway, from my point of view, I now get it: I'm happy.
Edit. Well, sort of. Still not sure of it all - more thought required!
Last edited by Victor on Sat May 10, 2008 10:20 am; edited 1 time in total |
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keith
Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
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| Posted: Sat May 10, 2008 10:18 am Post subject: |
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| Victor wrote: | | Sorry, Keith, but I think that particular example is not the best you could have for a half-M-wing, ... |
Feel free to post additional examples in that thread 
Keith |
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Victor
Joined: 29 Sep 2005
Posts: 207
Location: NI
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| Posted: Sat May 10, 2008 11:18 am Post subject: |
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Victor wrote:
| Quote: | Sorry, Keith, but I think that particular example is not the best you could have for a half-M-wing, ...
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Sorry again, Keith! I think it probably is an OK example. It's just that I realised that you can see this sort of thing in one way that does need the second 34, at D, to be naked, whereas this puzzle doesn't need the 34s at D to be naked. But thinking again, I imagine that you can see all half-M-wings this way: i.e. I'm not sure that you ever need the second pair to be naked.
I thought this was an example:
| Code: | +----------------+
| 6 9 48 |
| 258C 1458* 145 |
| 7 3 25 |
+----------------+
| 58B 258 6 |
| 9 28 7 |
| 4 125 135 |
+---------------+
| 1 58A 9 |
| 3 7 4 |
| 28 45
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(a) What I believe to be half-M-wing thinking: A = 8 => B = 8. Therefore A <> 5 => B <> 5 => C =5. (And C <> 5 => A = 5.) So we can eliminate the 5 in *.
(b) However, you can see it more simply, as with your example. A = 8 => B = 8 => C =5 (simply because it's the only 5 left in the column). So either A is 5, or it's 8, which means that C is 5. So *can't be 5. Now THIS reasoning doesn't need B to be naked - all it needs is for it to be conjugate in 5s with C.
So unless I've got this wrong, nice though these are, they're not sort of wildly special. If you or anybody else can throw light on this (i.e. put me straight), I'd be grateful. |
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Asellus
Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
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| Posted: Sat May 10, 2008 5:05 pm Post subject: |
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Victor,
An M-Wing, half or otherwise, is just a short Alternating Implication Chain (AIC):
(x=y) - (y) = (y-x) = (x)
that involves two matching bivalue cells, the "(x=y)" and "(y-x)" part. One of these, (x=y), includes a pincer x and the other one, (y-x), does not. The one without a pincer x is a weak link and so, as you have noticed, is not required to be a bivalue.
Why the focus on bivalues? Folks scan grids frequently for bivalues... to find XY Wings, XYZ Wings, etc. In the case of matching pairs of bivalues, there are Naked Pairs, Remote Naked Pairs and Semi-Remote Naked Pairs ("W-Wings") that have been recognized for quite some time. And, the logic of those three techniques require that the two cells actually be bivalues.
Looking for other ways to exploit such bivalue pairs, folks on this board pointed out what they called the "(Half) M-Wing" pattern. But, it is different in that the logic doesn't actually require the second cell to be a bivalue. So, some identical possible eliminations will be overlooked by folks relying on the two-bivalue pattern.
However, the whole point of the "M-Wing" name is to provide a pattern that people can learn to spot while scanning for bivalues. So, using the name for the case with only one bivalue cell probably isn't quite as helpful for that purpose... even though the underlying logic is identical. |
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Victor
Joined: 29 Sep 2005
Posts: 207
Location: NI
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| Posted: Sat May 10, 2008 9:53 pm Post subject: |
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Thanks Asellus. I do get all that, tho' I'll confess to a little disappointment (not your fault of course) at the apparent dissolving of what seems a nice idea. Have a look at another 'half-M-wing' in http://www.dailysudoku.com/sudoku/forums/viewtopic.php?t=2257&highlight=half+mwing&sid=55d720b3a94ad5d0352f2bfc9788d740
| Code: | +----------------+----------------+----------------+
| 678 1 4 | 3679 5 389 | 6789 2 678 |
| 2 3 79 | 679 168 189 | 6789 4 5 |
| 678 789* 5 | 4 68 2 | 6789 3 1 |
+----------------+----------------+----------------+
| 4 29D 6 | 1 7 5 | 3 89C 28 |
| 13 27 13 | 8 9 6 | 47 5 247 |
| 79 5 8 | 2 3 4 | 1 69 67 |
+----------------+----------------+----------------+
| 1389 89A 1239 | 5 1248 7 | 2468 68B 346 |
| 5 4 123 | 36 1268 138 | 28 7 9 |
| 378 6 237 | 39 248 389 | 5 1 348 |
+----------------+----------------+----------------+ |
Ravel's reasoning amounted to this:
C <> 8 => A <> 8 (strong-weak link / inverse transporting via B). Rephrased, this is C = 9 => A =>9. [i]Now that does need both cells to be bivalued I think. [/i
Now we can just look at r4, ER style: D = 9 or C = 9 => A = 9. Either way, * <> 9. I find that reasoning seductive.
However, I do see that as an AIC, 9A = 9D, we're using a weak link within C, so that that cell doesn't have to be bivalue, and I'll take your word for it that this is always the case. Shame really! (It's not the same sort of thing as say a W-wing. I know you can always express that as a shortish AIC, but that doesn't downvalue the technique. Here, the fact that C needn't be bivalue does rather downvalue the technique. Oh well, c'est la vie.)
Having said all that, agreeing with you in fact, I would perhaps take slight issue with you in regard to some of the other ideas in Keith's stuff about M-wings etc., rather on the same lines that I would take issue with anyone who denigrated say W-wings as not being worth naming. You don't think that perhaps the merits of all-conjugate chains are undervalued by many of the cognoscenti? As with W-wings, saying that you can see these things in other ways isn't a sufficient reason to ignore them.
Many thanks. |
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storm_norm
Joined: 18 Oct 2007
Posts: 1741
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| Posted: Sun May 11, 2008 12:11 am Post subject: |
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as far as the whole remote pair patterns go, for my money, I would rank them from easiest to hardest as follows... 1 being easiest
1...classic remote pair - all cells contain the same pair of candidates.
2...W-wing - because this pattern can take the form of so many other patterns... it can also be a xy-chain or a coloring pattern, etc.
3... M-wing - since the pincer cells don't contain the original pair, this technique requires care to see the correct relationship and the correct elimination.
4... Ravel's Half M-wing... requires a very careful look to find the relationship.
what is so priceless  about all these patterns is how agreeable on the eyes it is to find a pair {x,y} and go from there. it's straight forward, unlike the search or hunt for xy-wings or xy-chains. |
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keith
Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
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| Posted: Sun May 11, 2008 1:23 am Post subject: |
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Norm,
I agree with your ranking, except you missed what I would put as No. 2:
The other remote pair, where each cell in the chain is not the same pair. You only need the end cells to be the same pair, so long as the chain is coloring on one of them.
XY=aY=bY=XY
is a remote pair so long as each = is a strong link on Y. a and b can be anything.
Keith |
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storm_norm
Joined: 18 Oct 2007
Posts: 1741
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| Posted: Sun May 11, 2008 1:26 am Post subject: |
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| keith wrote: | Norm,
I agree with your ranking, except you missed what I would put as No. 2:
The other remote pair, where each cell in the chain is not the same pair. You only need the end cells to be the same pair, so long as the chain is coloring on one of them.
XY=aY=bY=XY
is a remote pair so long as each = is a strong link on Y. a and b can be anything.
Keith |
kind of like the example in the daily puzzle?? |
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keith
Joined: 19 Sep 2005
Posts: 3355
Location: near Detroit, Michigan, USA
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| Posted: Sun May 11, 2008 3:01 am Post subject: |
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| Quote: | | kind of like the example in the daily puzzle?? |
Norm,
I don't know, I'll have to look. But, I think this one is rarely found, because people rarely look for it (not because it is truly rare).
Best wishes,
Keith |
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Asellus
Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
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| Posted: Sun May 11, 2008 5:35 am Post subject: |
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Victor,
I'm not sure what nice idea it is I am accused of dissolving. I thought I was just adding clarity to some confusion.
I also hope to make it clear that I wasn't "denigrating" anything. My point was only that which sorts of named patterns are useful, and in what ways, varies according the the preferences and knowledge level of the solver. It's fine with me if some folks find something useful that others don't.
As for me, since I am very comfortable with AICs and use them extensively, the distinction between M-Wings and Half M-Wings is not significant. But others may find that distinction useful and that's fine.
By the way, I'm not sure if I understand your point with ravel's example: The 89 bivalue labeled "C" at r4c8 does not need to be a bivalue: the elimination works even if other candidates are present in this cell. Here is the AIC:
(9=8)A-(8)B=(8-9)C=9D
The only cell that needs to be a bivalue is A. It is identical to the previous example. |
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Asellus
Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA
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| Posted: Sun May 11, 2008 5:49 am Post subject: |
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PS:
I just noticed... There IS a difference between those two examples of Half M-Wings. In the second one (ravel's), the pincer digits (the <9>s in A and D) are peers. This means that this is an AIC Loop and all the links along the chain become conjugate.
So, besides eliminating <9> from r3c2, it also eliminates <8>s from r7c157. And, IF that cell C had NOT been an 89 bivalue, it would have become one due to the AIC Loop since the 8-9 link would become conjugate.
(I'm surprised ravel didn't catch this in his original post. He's going to kick himself!) |
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Victor
Joined: 29 Sep 2005
Posts: 207
Location: NI
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| Posted: Sun May 11, 2008 10:34 pm Post subject: |
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Asellus: | Quote: | | I'm not sure what nice idea it is I am accused of dissolving. |
When I read Keith's stuff about M-wings etc., I understood it all apart from half-M-wings. (In fact at first I guessed that it must be a synonym for transporting, i.e. a weak then strong link.) So I scratched around looking for other examples, and thought I understood. I did think that C(8,9) <> 8 => A(8,9) <> 8, translated into 9s: C = 9 => A = 9 required both cells to be bivalue, and I thought "Ah, that's smart, that must be the meaning of a half-M-wing." But you are right of course, it doesn't need C to be bivalue (and silly of me not to think more carefully). So the half-M-wing idea has 'dissolved': but I'm not blaming you!
M-wings etc. Of course we all use whatever techniques we find useful & with which we are comfortable. I'm comfortable enough with AICs, but I'm even more comfortable (as are presumably Keith & others) with techniques based on all-conjugate chains. Here's an M-wing:
A = (3,5), B = (3,x), C = (3,5), D = (5,y), with conjugate links between the 3s of A & B and B & C, and the 5s of C & D. Now you remarked in a recent discussion that one doesn't need the link between A & B to be strong. (Nor need C be bivalue.) In a sense that's true: (5=3)A - 3B = (3 - 5)C = 5D. You can indeed eliminate any 5 seen by both A and D in both this chain & the M-wing. But there is a difference: the AIC with weak links in it isn't an M-wing, and shouldn't be so called. One way to put the difference is to say that with the M-wing the 5s of A & D are conjugate (one true, one false), whereas the 5s of the chain that includes the weak link from A to B are connected by a 'strong-only' link. So if you want to extend either of these chains you need more care with the second chain. (Folks do commonly talk of 'extending by colouring', and of course if you're adding an even number of conjugate steps to a chain know to be conjugate, then you can do it both directions, i.e an odd number of steps in each direction, which would be invalid with the second chain.) Even if you're xx to other numbers in a long AIC, it's just plain easier if you recgnise a conjugate chain as such, because of course you can treat that now sub-chain as weak or strong as needed.
Perhaps this doesn't make a strong point, but have a quick look:
| Code: | +--------------------------+
| 136 5 26 |
| 13689 379 268 |
| 39A 379 4 |
+--------------------------+
| 4 6 9 |
| 2 1 3 |
| 7 8 5 |
+--------------------------+
| 38 2 1 |
| 5 39B 78 |
| 69 4 67 |
+--------------------------+ |
You'll note that A =3|9 <=> B = 3|9, without any thought. So they are 'complementary', and you can step on from B in first 3, then 9 in that box and thus kill the 3s & 9 in r12c1.
Now I know that you can see these as AICs, easily enough if you're good at it, but I think it's much easier for anyone who's familiar with conjugate chains to see it as I've explained. Moral, if any: a familiarity with conjugate chains is worth acquiring. |
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