## Degenerate XWings

Post the puzzle or solving technique that's causing you trouble and someone will help

### Degenerate XWings

.2456179..914372....68294.1958.4......29785......5.9...85.14..916.2958.4..9.86...

6FebPost.png (28.38 KiB) Viewed 507 times

Here, r68c3 & r8c8 are all (3,7). Or should that be (3-7) or (3=7)? Anyway, those three cells are all limited to candidates 3 and 7
and they form three corners of an XWing. What can we infer from that in Box6?
Yogi
2017 Supporter

Posts: 178
Joined: 05 December 2015
Location: New Zealand

### Re: Degenerate XWings

Not much. The reason is that r6c3 and r8c8 have the same parity ie they are both 3 or both 7. If they are both 7 you can't say anything directly about Box 6. Instead there is a Sashimi finned Jellyfish in 3's that resolves their position as follows.

Code: Select all
`*-----------------------------------------------*| *38    2     4  | 5  6 1  | 7    9      *38   || 58     9     1  | 4  3 7  | 2    568     568  ||*357   *37    6  | 8  2 9  | 4   *35      1    ||-----------------+---------+-------------------|| 9      5     8  | 16 4 23 | 136  12367   2367 ||*346   *134   2  | 9  7 8  | 5   *1346   *36   || 3467   1347  37 | 16 5 23 | 9    123468  2368 ||-----------------+---------+-------------------|| 27-3   8     5  | 37 1 4  | 36   2367    9    || 1      6    F37 | 2  9 5  | 8   *37      4    || 247-3  47-3  9  | 37 8 6  | 13   12357   2357 |*-----------------------------------------------*`

It doesn't solve the whole puzzle but you resolve your cell values of interest.

Code: Select all
`*-----------------------------------------*| 38  2    4 | 5  6 1  | 7    9      38   || 58  9    1 | 4  3 7  | 2    568    568  || 357 37   6 | 8  2 9  | 4    35     1    ||------------+---------+------------------|| 9   5    8 | 16 4 23 | 136  1236   7    || 346 134  2 | 9  7 8  | 5    1346   36   || 346 134 #7 | 16 5 23 | 9    123468 2368 ||------------+---------+------------------|| 27  8    5 | 37 1 4  | 36   236    9    || 1   6   #3 | 2  9 5  | 8   #7      4    || 247 47   9 | 37 8 6  | 13   1235   235  |*-----------------------------------------*`

Leren
Leren

Posts: 3928
Joined: 03 June 2012

### Re: Degenerate XWings

Leren is right that the (37)-pattern itself doesn't produce anything useful (not even UR potential here, because the corners are in four boxes). Then again, catching a Finned Sashimi Jellyfish doesn't seem like a very realistic option for most manual solvers either (certainly not for me). There's a much simpler X-Chain of 3s that does the same trick, though:

Code: Select all
`.------------------.------------.---------------------.| 38    2      4   | 5   6   1  |   7    9       38   || 58    9      1   | 4   3   7  |   2    568     568  || 357   37     6   | 8   2   9  |   4    35      1    |:------------------+------------+---------------------:| 9     5      8   | 16  4  b23 |  c136  12367   2367 || 346   134    2   | 9   7   8  |   5    1346    36   || 3467  1347  a37  | 16  5  b23 |   9    123468  2368 |:------------------+------------+---------------------:| 237   8      5   | 37  1   4  | c(3)6  2367    9    || 1     6    a(3)7 | 2   9   5  |   8    7-3     4    || 2347  347    9   | 37  8   6  | c(3)1  12357   2357 |'------------------'------------'---------------------'`

Extended Grouped Skyscraper:

(3)r[8=6]c3 - r[6=4)c6 - (3)r[4=79]c7 => -3 r8c8

Standard Eureka: Show
(3)r8c3 = r6c3 - r6c6 = r4c6 - r4c7 = (3)r79c7 => -3 r8c8

Incidentally that pattern is also a Finned Sashimi Swordfish (with the same elimination), but I only noticed that fact afterwards (apparently having more aptitude for being a chain-ganger than a fisherman):

Finned Sashimi Swordfish: (3)c367\r468 f:r79c7 => -3 r8c8

Either that or the X-Chain should be easier to see than the Jellyfish variant. (Unfortunately it's not a very effective move here no matter which way you spot it).
-SpAce-: Show
Code: Select all
`   *             |    |               |    |    *        *        |=()=|    /  _  \    |=()=|               *            *    |    |   |-=( )=-|   |    |      *     *                     \  ¯  /                   *    `

"If one is to understand the great mystery, one must study all its aspects, not just the dogmatic narrow view of the Jedi."

SpAce

Posts: 2559
Joined: 22 May 2017

### Re: Degenerate XWings

I think I’m more into chaining than Jellying also, although 2Fish seem routine to me. That’s why I’m examining these variations of XWings.
I notice that nominating 3r6c3 quickly leads to 3r58c8, proving 7r6c3, with more singles flowing on, although the puzzle remains difficult.
This does seem to be related to SpAce’s suggestion, although it joins the chain at a different point. An ordinary Skyscraper is easy enough to describe as two parallel CPs in the same candidate with their ends occupying four boxes arranged rectangularly, with one end cell of each of the CPs being in the same row or column, but their other ends NOT being in the same row or column. Maybe you can help me see this pattern here as an ‘Extended Skyscraper’.
Yogi
2017 Supporter

Posts: 178
Joined: 05 December 2015
Location: New Zealand

### Re: Degenerate XWings

Yogi wrote:An ordinary Skyscraper is easy enough to describe as two parallel CPs in the same candidate with their ends occupying four boxes arranged rectangularly, with one end cell of each of the CPs being in the same row or column, but their other ends NOT being in the same row or column. Maybe you can help me see this pattern here as an ‘Extended Skyscraper’.

An Extended Grouped Skyscraper. It has thus two separate complications when compared to a regular Skyscraper: one is the grouping (one end-point is a group instead of a single candidate, in this case 3r79c7) and the other is the extension (the chain has an extra linking pair, in this case 3r46c6). Otherwise it works just like a Skyscraper (or any other chain) by proving that at least one of the end-points must be true, which causes the eliminations. (Note: using the "Extended" keyword here is probably non-standard or at least non-common).

1. Skyscraper:

Code: Select all
`.------------------.------------.---------------------.|                  |            |                     ||                  |            |                     ||                  |            |                     |:------------------+------------+---------------------:|                  |            |                     ||                  |            |                     ||               3  |            |  3                  |:------------------+------------+---------------------:|                  |            |                     ||              (3) |            |          x       x  ||  x      x        |            | (3)                 |'------------------'------------'---------------------'`

2. Grouped Skyscraper

Code: Select all
`.------------------.------------.---------------------.|                  |            |                     ||                  |            |                     ||                  |            |                     |:------------------+------------+---------------------:|                  |            |                     ||                  |            |                     ||               3  |            |  3                  |:------------------+------------+---------------------:|                  |            | (3)                 ||              (3) |            |          x       x  ||                  |            | (3)                 |'------------------'------------'---------------------'`

3. Extended Skyscraper

Code: Select all
`.------------------.------------.---------------------.|                  |            |                     ||                  |            |                     ||                  |            |                     |:------------------+------------+---------------------:|                  |          3 |  3                  ||                  |            |                     ||               3  |          3 |                     |:------------------+------------+---------------------:|                  |            |                     ||              (3) |            |          x       x  ||  x      x        |            | (3)                 |'------------------'------------'---------------------'`

4. Extended Grouped Skyscraper

Code: Select all
`.------------------.------------.---------------------.|                  |            |                     ||                  |            |                     ||                  |            |                     |:------------------+------------+---------------------:|                  |          3 |  3                  ||                  |            |                     ||               3  |          3 |                     |:------------------+------------+---------------------:|                  |            | (3)                 ||              (3) |            |          x       x  ||                  |            | (3)                 |'------------------'------------'---------------------'`

Legend: Show
x: possible elimination (of a candidate 3)
(3): chain end point
(note: all marked candidate 3s are the only ones in their columns)

Did that help?

SpAce

Posts: 2559
Joined: 22 May 2017

### Re: Degenerate XWings

This is a one-stepper, which i don't want to formulate in AIC, but i hope, Yogi would accept it.
Code: Select all
` *--------------------------------------------------------------* |  38     2      4    |  5    6  1    |  7     9       b38     | |  58     9      1    |  4    3  7    |  2    b568      568    | |  357    37     6    |  8    2  9    |  4    b35       1      | |---------------------+---------------+------------------------| |  9      5      8    |  16   4  23   |  136   12367    2367   | |  346    134    2    |  9    7  8    |  5     1346     36     | |  3467   1347   37   |  16   5  23   |  9     123468   2368   | |---------------------+---------------+------------------------| |  237    8      5    |  37   1  4    |  36   c2367     9      | |  1      6      37   |  2    9  5    |  8     37       4      | |  2347   347    9    |  37   8  6    |  13   a12357    35-27  | *--------------------------------------------------------------*`

-5r9c9 => 5r9c9 => 386 r3c8,r1c9,r2c8 => 27r78c8
==> r9c9<>27
eleven

Posts: 2461
Joined: 10 February 2008

### Re: Degenerate XWings

eleven wrote:This is a one-stepper, which i don't want to formulate in AIC, but i hope, Yogi would accept it.

-5r9c9 => 5r9c8 => 386 r3c8,r1c9,r2c8 => 27r78c8
==> r9c9<>27

Nice solution! I'd write it like this, though it's not necessarily standard Eureka:

Code: Select all
`.----------------.-----------.------------------------.| 38    2     4  | 5   6  1  | 7     9       b38      || 58    9     1  | 4   3  7  | 2    b568     b568     || 357   37    6  | 8   2  9  | 4    b35       1       |:----------------+-----------+------------------------:| 9     5     8  | 16  4  23 | 136   12367    2367    || 346   134   2  | 9   7  8  | 5     1346     36      || 3467  1347  37 | 16  5  23 | 9     123468   2368    |:----------------+-----------+------------------------:| 237   8     5  | 37  1  4  | 36  c(27)36    9       || 1     6     37 | 2   9  5  | 8   c(7)3      4       || 2347  347   9  | 37  8  6  | 13    12357   a3(5)-27 |'----------------'-----------'------------------------'`

(5)r9c9 = (5,6,8,3)b3p6538 - (6|3=27)r78c8 => -27 r9c9; stte

SpAce

Posts: 2559
Joined: 22 May 2017

### Degenerate XWings

Thanx to SpAce for that explanation. I see it works just like a Sashimi or Finned XWing with the Either or Neither reasoning in Box 9:

1) If Either of the two fins r79c7 are 3, then -3r8c8 => 3r8c3
||
2) If Neither of r79c7 are 3, then 3r4c7 => => 3r8c3

The 23Locked Pair r46c6 connects r4c7 to r6c3 beautifully to keep the forcing chain going in that direction.

However, it is true that there is still another step required to finally solve the puzzle.
Yogi
2017 Supporter

Posts: 178
Joined: 05 December 2015
Location: New Zealand

### Re: Degenerate XWings

Yogi wrote:Thanx to SpAce for that explanation. I see it works just like a Sashimi or Finned XWing with the Either or Neither reasoning in Box 9:

1) If Either of the two fins r79c7 are 3, then -3r8c8 => 3r8c3
||
2) If Neither of r79c7 are 3, then 3r4c7 => => 3r8c3

Exactly.

The 23Locked Pair r46c6 connects r4c7 to r6c3 beautifully to keep the forcing chain going in that direction.

That's one way to see it, but it's more complicated because then it's no longer a single-digit technique. You don't actually need the locked pair for anything because you have a conjugate pair of 3s in that column, which is enough for the linking action. It would work just as well if there were many other digits in those two cells -- it's just a coincidence that it's also a locked pair.

SpAce

Posts: 2559
Joined: 22 May 2017

### Degenerate XWings

Let’s resurrect that discussion again as I’ve found another puzzle which seems to clearly illustrate this idea that a skyscraper can work even if there is apparently no direct CP connection between all the boxes: 5.7394.81.9386157..182759..35978.1..74291..5818645..978751.946...15478.99.46.8715

ESky.png (14.16 KiB) Viewed 295 times

Here, the Box Analysis rules quickly identify that only candidate 2 could provide a single digit elimination in boxes 1379, but there is no direct connection of a 2CP between boxes 7&9. However, we do seem to have a skyscraper with a kink: there is a zig-zag connection from r9c2 to r7c9 via the locked pair in box8, which allows the skyscraper elimination of 2 from any cell that can see both r1c7 and r7c9 => 2r2c1 stte.
Yogi
2017 Supporter

Posts: 178
Joined: 05 December 2015
Location: New Zealand

### Re: Degenerate XWings

Code: Select all
`*-----------------------------------*| 5   26  7 | 3 9  4  | 26  8   1   ||a24  9   3 | 8 6  1  | 5   7  b24  || 46  1   8 | 2 7  5  | 9   34  346 ||-----------+---------+-------------|| 3   5   9 | 7 8  26 | 1   24  246 || 7   4   2 | 9 1  36 | 36  5   8   || 1   8   6 | 4 5  23 | 23  9   7   ||-----------+---------+-------------|| 8   7   5 | 1 23 9  | 4   6  c23  || 6-2 236 1 | 5 4  7  | 8  d23  9   || 9   23  4 | 6 23 8  | 7   1   5   |*-----------------------------------*`

(2) r2c1 = r2c9 - r7c9 = (2) r8c8 => - 2 r8c1; stte

Leren
Leren

Posts: 3928
Joined: 03 June 2012

### Re: Degenerate XWings

I always have to refer everybody to the UFG with regards to terms because of Sashimi

The fish is either Finned or not Finned (omitting Not finned is common practice)
The fish is Sashimi/degenerate if the fish structure in the absence of fins degenerates into smaller Fish, or not Sashimi/Degenrate (again omitting Not sashimi/Degenerate is common practice) . The use of Sashimi to describe Finned Sashimi is unfortunate but it has gone mainstream now with no way of stopping it
So in theory you could have a sashimi fish that is not finned
Why would you do that in the presence of easier techniques is left to the solver but using the logic is fine by me in that situation too

tarek

tarek

Posts: 3745
Joined: 05 January 2006

### Re: Degenerate XWings

Hi Yogi,

Yogi wrote:Here, the Box Analysis rules quickly identify that only candidate 2 could provide a single digit elimination in boxes 1379, but there is no direct connection of a 2CP between boxes 7&9. However, we do seem to have a skyscraper with a kink: there is a zig-zag connection from r9c2 to r7c9 via the locked pair in box8, which allows the skyscraper elimination of 2 from any cell that can see both r1c7 and r7c9 => 2r2c1 stte.

The elimination is valid, and so is your logic, but calling it any kind of Skyscraper is confusing. In both normal and extended Skyscrapers all the links (except the elimination links) are along rows and columns. Here you're using a weak link in box 7 which disqualifies it from being a Skyscraper type.

Furthermore, that logic is unnecessarily complicated (even if we forget about the irrelevant locked pair, already mentioned here):

Code: Select all
`.----------------.------------.---------------.|   5     26   7 | 3   9   4  | 26  8     1   || a[2]4   9    3 | 8   6   1  | 5   7     4-2 ||   46    1    8 | 2   7   5  | 9   34    346 |:----------------+------------+---------------:|   3     5    9 | 7   8   26 | 1   24    246 ||   7     4    2 | 9   1   36 | 36  5     8   ||   1     8    6 | 4   5   23 | 23  9     7   |:----------------+------------+---------------:|   8     7    5 | 1  e23  9  | 4   6   f(2)3 ||  b26    236  1 | 5   4   7  | 8   23    9   ||   9    c23   4 | 6  d23  8  | 7   1     5   |'----------------'------------'---------------'`

(2)r2c1 = r8c1 - r9c2 = r9c5 - r7c5 = (2)r7c9 => -2 r2c9

As a fish: Show
Finned Mutant Swordfish: (2)r79c1\r2c5b7 f:r7c9 => -2 r2c9

Obi-Fish: (2)R79C1\r2c59b7 => -2 r2c9

That's an X-Chain of length 6 (or a 3-fish). As Leren already demonstrated, there's a shorter and simpler X-Chain to an equivalent elimination -- and a similar one to yours too (which Leren did not demonstrate). It's length 4 so it's a Turbot Fish type, just like a Skyscraper or a Kite (which also means it's a 2-fish).

However, it's using a box-based strong link, which means it's neither of those two most familiar Turbot Fish subtypes. It's the third kind: a Turbot Crane. In fact, it's a dual Turbot Crane, because you can get both -2 r8c1 and -2 r2c9 using the same logic and cells. (Perhaps a more familiar POV is a dual ER, but to me ER is a grouped type, so I don't like calling it that any more than just Turbot Fish, which is the parent type of Skyscrapers, Kites, and Cranes.)

Code: Select all
`.--------------.-----------.---------------.|  5    26   7 | 3  9   4  | 26   8    1   || *24   9    3 | 8  6   1  | 5    7   *4-2 ||  46   1    8 | 2  7   5  | 9    34   346 |:--------------+-----------+---------------:|  3    5    9 | 7  8   26 | 1    24   246 ||  7    4    2 | 9  1   36 | 36   5    8   ||  1    8    6 | 4  5   23 | 23   9    7   |:--------------+-----------+---------------:|  8    7    5 | 1  23  9  | 4    6   *23  || *6-2  236  1 | 5  4   7  | 8   *23   9   ||  9    23   4 | 6  23  8  | 7    1    5   |'--------------'-----------'---------------'`

Turbot Crane 1: (2)r2c1 = r8c1 - r8c8 = (2)r7c9 => -2 r2c9
Turbot Crane 2: (2)r2c1 = r2c9 - r7c9 = (2)r8c8 => -2 r8c1

As fishes: Show
As UFG::Finned Franken X-Wings:

(2)c1b9\r28 f:r7c9 => -2 r2c9
(2)r2b9\c19 f:r8c8 => -2 r8c1

As Obi-Fishes:

(2)C1B9\r28c9 => -2 r2c9
(2)R2B9\r8c19 => -2 r8c1

Or as a Dual Turbot Crane:

(2)r2c1 = r2c9&r8c1 - b9p35 = (2)r7c9&r8c8 => -2 r2c9,r8c1

...or perhaps more simply:

(2)r2c1 = r2c9&r8c1 - b9p35 = contradiction => +2 r2c1

Lesson learned: You should remember to consider box-based conjugate pairs as well.

SpAce

Posts: 2559
Joined: 22 May 2017

### Re: Degenerate XWings

tarek wrote:I always have to refer everybody to the UFG with regards to terms because of Sashimi

I don't know if you're noticed, but I've changed my uses of "Sashimi" to comply with your definition! In other words, I now call the normal cases "Finned Sashimi" instead of just "Sashimi". You managed to convince me of the logic in some earlier discussion.

Then again, I mostly prefer the Obi-Fish style nowadays because it's shorter and simpler than the UFG style and similar to Allan Barker's set logic. With that style exact UFG fish names rarely make sense (because they can often map into multiple UFG fishes, depending on which is considered the fin sector).

SpAce

Posts: 2559
Joined: 22 May 2017

### Degenerate XWings

This discussion has wandered somewhat from the original idea, but I should acknowledge Leren's very simple solution to the second puzzle I posted in this thread, which clarifies that a 3-link chain can do the job just as well, even if it doesn't conform to the structure of a skyscraper. The principle still holds good that if it's an odd-numbered AIC then one of its ends must be true and any cell which can see both ends of it cannot be the candidate of the chain.
Last edited by Yogi on Tue Oct 08, 2019 7:05 pm, edited 1 time in total.
Yogi
2017 Supporter

Posts: 178
Joined: 05 December 2015
Location: New Zealand

Next