## Help with next steps

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

### Help with next steps

Hi all, first time posting as I need some help with solving techniques. I am working on Expert puzzles and can get so far using everything up to X-/XY-Wing and Swordfish but have hit a brick wall at a similar point on a number of similar puzzles, leaving me the only option to guess. I obviously don't want to guess, so I'm looking for a hint on a technique to uncover the next clue on this puzzle, see the attached image.

(as you can see I've been at this one for 3 hours...)
Attachments
JosephMcDermott 2018-08-30 at 19.04.28.jpg (57.33 KiB) Viewed 420 times
facehuk

Posts: 3
Joined: 30 August 2018

### Re: Help with next steps

Update: I've since eliminated 4 pencil mark from centre of top right (Bh) cell
facehuk

Posts: 3
Joined: 30 August 2018

### Re: Help with next steps

where you are, the next logic step is a Unique Rectangle r18c56
r8c5=1

If you don't like it, you have to look for a forcing chain on digit "1" (kite)
champagne
2017 Supporter

Posts: 7163
Joined: 02 August 2007
Location: France Brittany

### Re: Help with next steps

Many thanks for the reply Champagne, I had to look up what you meant by "r18c56" so I'm learning things already!

I will try to figure the Unique Rectangle using your hint (and answer) and will also look up "forcing chain", exactly the assistance I was hoping for.

Thanks again!
facehuk

Posts: 3
Joined: 30 August 2018

### Re: Help with next steps

Note, that forcing chains are somewhat out of fashion.
For the second move i would recommend Strong Links for Beginners, which covers skyscrapers and kites.
eleven

Posts: 2461
Joined: 10 February 2008

### Re: Help with next steps

facehuk wrote:Many thanks for the reply Champagne, I had to look up what you meant by "r18c56" so I'm learning things already!

Yeah, the notations can be a bit cryptic at first, but soon they become a second nature

I will try to figure the Unique Rectangle using your hint (and answer) and will also look up "forcing chain", exactly the assistance I was hoping for.

Champagne already gave you the correct hints, but I'll give you a couple of sources that might help understanding them better.

The Unique Rectangle (UR) you can use here is the simplest kind, UR Type 1. More on that (and other kinds) here:

http://sudopedia.enjoysudoku.com/Uniqueness_Test.html
http://hodoku.sourceforge.net/en/tech_ur.php#u1
http://www.sudokuwiki.org/Unique_Rectangles

In this puzzle it looks like this:

Code: Select all
`.-----------------.------------------.----------------.| 34   2     9    | 1 *(58)    *(58) | 7    6    34   || 34   7     8    | 9  6        2    | 134  13   5    || 56   56    1    | 3  7        4    | 9    2    8    |:-----------------+------------------+----------------:| 167  146   3    | 8  2        17   | 5    147  9    || 2    15    57   | 6  4        9    | 8    137  13   || 9    8     47   | 5  3        17   | 6    147  2    |:-----------------+------------------+----------------:| 15   9     46   | 7  15       3    | 2    8    46   || 157  1345  4567 | 2 *(58)#1  *(58) | 134  9    1346 || 8    13    2    | 4  9        6    | 13   5    7    |'-----------------'------------------'----------------'`

If candidate (1) were removed from r8c5, the four marked (58)s would form a deadly rectangle and we'd have two solutions. Since valid puzzles can't have two solutions, we can safely place 1r8c5. (But if you don't know for sure that your puzzle provider guarantees single solutions, then you can't use that method -- however, I'd find a different puzzle provider in that case.) Unique Rectangles are often found and one of the easiest non-basic patterns to spot I think (actually easier than some basic ones like triples and quads), so they're worth learning.

The other option champagne mentioned, 2-String Kite, is another relatively easy pattern to look for (as is its buddy, Skyscraper). Here's some links about that:

http://sudopedia.enjoysudoku.com/2-String_Kite.html
http://hodoku.sourceforge.net/en/tech_sdp.php#t2sk

In this puzzle it looks like this:

Code: Select all
`.-----------------.-------------.----------------.| 34   2     9    | 1  58   58  | 7    6    34   || 34   7     8    | 9  6    2   | 134  13   5    || 56   56    1    | 3  7    4   | 9    2    8    |:-----------------+-------------+----------------:| 167  146   3    | 8  2    17  | 5    147  9    || 2   x5-1   57   | 6  4    9   | 8    137 *(1)3 || 9    8     47   | 5  3    17  | 6    147  2    |:-----------------+-------------+----------------:| 15   9     46   | 7  15   3   | 2    8    46   || 157  1345  4567 | 2  158  58  | 134  9   *1346 || 8   *(1)3  2    | 4  9    6   |*13   5    7    |'-----------------'-------------'----------------'`

The two candidate (1)s in r5c9 and r9c2 are connected by a short chain which ensures that at least one of those cells must have a (1) as its true value. Thus there can't be a 1 in any cell that sees both ends. The victim cell in this case is r5c2, and candidate 1 can be eliminated from that. The logic in Eureka notation:

(1)r5c9 = r8c9 - r9c7 = (1)r9c2 => -1 r5c2

Some background information about these and other kinds of chains:

2-String Kite is a specific kind of Turbot Fish (not really a fish, though):

http://sudopedia.enjoysudoku.com/Turbot_Fish.html
http://hodoku.sourceforge.net/en/tech_sdp.php#tf

Turbot Fishes are special kinds of X-Chains:

http://sudopedia.enjoysudoku.com/X-Chain.html
http://hodoku.sourceforge.net/en/tech_chains.php#x

And X-Chains are special kinds of AICs (Alternating Inference Chains):

http://sudopedia.enjoysudoku.com/Alternating_Inference_Chain.html
http://hodoku.sourceforge.net/en/tech_chains.php

PS. All of these are effectively forcing chains, but usually that term is reserved for more complex types of chains (that are called Krakens on this forum) which can't be easily converted into AICs, so I don't think using that term ("forcing chain") here is necessarily helpful. (Edit: eleven seems to have mentioned that also above while I was typing.)
-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: 2550
Joined: 22 May 2017

### Re: Help with next steps

eleven wrote:Note, that forcing chains are somewhat out of fashion.
For the second move i would recommend Strong Links for Beginners, which covers skyscrapers and kites.

That link has very good stuff about those. However -- and this gets off topic -- I noticed something else that bothered me:

Havard wrote:First of all: What is a strong link?
A strong link (or a conjugate pair as it is also called) is when you only have two candidates of one number in any row, column or box!

That is quite a limited definition of a strong link. It leaves out even group links, as well as internal ALS links, not to even mention all derived strong links (most of which are strong-only and not conjugate). I thought you'd probably have an issue with that as well, because you're so good at using those derived strong links, most recently here (which I loved, by the way).
-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: 2550
Joined: 22 May 2017

### Re: Help with next steps

SpAce wrote:
Havard wrote:First of all: What is a strong link?
A strong link (or a conjugate pair as it is also called) is when you only have two candidates of one number in any row, column or box!

That is quite a limited definition of a strong link. It leaves out even group links, as well as internal ALS links, not to even mention all derived strong links (most of which are strong-only and not conjugate). I thought you'd probably have an issue with that as well, because you're so good at using those derived strong links, most recently here (which I loved, by the way).

Quit right.

I am not expert in this field, but after the kite seen here I was wondering what would be the simplest "XY" corresponding pattern.
The trivial answer is a naked pair.

But assume we have this

a12 12
- -
12 b12 ab

a UR with 2 different digits in diagonal and a pair of the 2 digits seeing one of the 2 digits

a|b in the UR is a strong inference, one of them (or both) must be true
This works as a kite.

any cell seeing "a12" and "ab" can not contain the digit "a".
champagne
2017 Supporter

Posts: 7163
Joined: 02 August 2007
Location: France Brittany

### Re: Help with next steps

SpAce wrote:
Havard wrote:First of all: What is a strong link?
A strong link (or a conjugate pair as it is also called) is when you only have two candidates of one number in any row, column or box!

That is quite a limited definition of a strong link.

Yes, at that time (Feb 2006) this was the definition of a strong link (on this forum). No one would have said, that there is a strong link between the candidates ab of a cell (maybe on the Eureka forum, i don't know).
Now every (a or b) relation is called a strong link, and as it seems, strong inference. So we have 2 names for the same, but no different for trivial 1-digit and monster AIC links.
[Added:] For some time there was a distinction between strong links and inferences: for a strong link exactly one of the candidates had to be true (a xor b), for an inference at least one (a or b). I am not sure, if this is common sense anymore.
[Edit: corrected 'or' relation, thx to SpAce]
Last edited by eleven on Sun Sep 02, 2018 7:17 pm, edited 1 time in total.
eleven

Posts: 2461
Joined: 10 February 2008

### Re: Help with next steps

eleven wrote:Now every (!a or b) relation is called a strong link, and as it seems, strong inference. So we have 2 names for the same, but no different for trivial 1-digit and monster AIC links.
[Added:] For some time there was a distinction between strong links and inferences: for a strong link exactly one of the candidates had to be true (a xor b), for an inference at least one (!a or b). I am not sure, if this is common sense anymore.

I always thought that AIC was a short for Alternate Inference Chain.

AIC is a very old definition (more then 10 years ??), If this is right, one could say as well that any such link is called "strong inference", a strong link is then somehow a subset of the strong inference class.

What remains important is the necessary conditions to be part of an AIC
champagne
2017 Supporter

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Joined: 02 August 2007
Location: France Brittany

### Re: Help with next steps

Yes, the last years was a big shift to use AIC's for everything, where only strong inferences are of interest (same for nice loops).
So no wonder, that Andrew Stuart writes in his sudokuWiki 'Strong links are "links with strong inference"'.

However it is not the same. A trivial example is a remote pair, where you need 2 AIC's to describe, that exactly one of the cells must be a, the other b.
eleven

Posts: 2461
Joined: 10 February 2008

### Re: Help with next steps

To me it seems that there are three kinds of links:

1. strong-weak (XOR) (aka conjugate)
2. strong-only (OR)
3. weak-only (NAND)

The strong-weak links (normally called just strong links) can be used in chains for either kind of inference, but the other two work only one way. Would that be a correct way to see the difference between a strong link and a strong inference, and the reason for the distinction, as most so called strong links (i.e. strong-weak links) can be used for weak inferences as well?
Last edited by SpAce on Sun Sep 02, 2018 8:35 pm, edited 1 time in total.
-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: 2550
Joined: 22 May 2017

### Re: Help with next steps

champagne wrote:a12 12
- -
12 b12 ab

a UR with 2 different digits in diagonal and a pair of the 2 digits seeing one of the 2 digits

a|b in the UR is a strong inference, one of them (or both) must be true
This works as a kite.

any cell seeing "a12" and "ab" can not contain the digit "a".

A nice example! As a chain (with imagined cell positions and an inlined derived strong link):

(a)r1c1 =[UR(12)r16c13]= (b)r6c3 - (b=a)r6c5 => -a r1c5

A UR-Wing, perhaps? Not really, as at least one definition of a wing has three strong links and this has only two, so it's even simpler than a wing -- maybe more like UR-XZ (x=b, z=a).
-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: 2550
Joined: 22 May 2017

### Re: Help with next steps

SpAce wrote:A UR-Wing, perhaps? Not really, as at least one definition of a wing has three strong links and this has only two, so it's even simpler than a wing -- maybe more like UR-XZ (x=b, z=a).

It has probably been described somewhere in the past,
If I had to name it, I would say a XY_UR kite, the reason why I thought of it. It can be possible to have a similar pattern with an ALS replacing the UR, but this one surely exists.
champagne
2017 Supporter

Posts: 7163
Joined: 02 August 2007
Location: France Brittany

### Re: Help with next steps

champagne wrote:If I had to name it, I would say a XY_UR kite, the reason why I thought of it. It can be possible to have a similar pattern with an ALS replacing the UR, but this one surely exists.

I wasn't seriously suggesting any names myself, and you can naturally name it however you want (if it doesn't already have one) However, I can't honestly see the Kite resemblance (but I'm glad you did, if it made you mention this!). Even if I did, I wouldn't like that name because it's so clearly tied to a known single-digit pattern. I think it would just cause confusion in a totally different context.

There are a few fundamental differences to Kites, besides the obvious ones (UR, double-digit). As a chain, a Kite always has four nodes and two strong links, while this can be written with just two nodes and one derived strong link (if two nodes are combined into an ALS). Furthermore, a Kite always has at most a single elimination, while this can have variants with more eliminations and shapes that don't resemble a Kite at all. Thus I think it behaves more like a wing or an ALS-XZ, but is even simpler (but that's just my opinion).

Unless I've misunderstood something fundamental, I can think of at least three different shapes using the same logic. It wouldn't make sense to have different names for all of them -- and even less to call them all Kites, as only the first one has any resemblance to the real thing:

Code: Select all
`Shape 1:.----------------------.-----------.---------.|a12(#5)*  12*         |    -5     |         ||                      |           |         ||                      |           |         |:----------------------+-----------+---------:|                      |           |         ||                      |           |         || 12*     b12#7*       |   b(5)7   |         |:----------------------+-----------+---------:|                      |           |         ||                      |           |         ||                      |           |         |'----------------------'-----------'---------'(5)r1c1 == (75)r6c25 => -5 r1c5Shape 2:.------------------------.----------.---------.|a12(#5)*  12*     -5    |          |         ||                        |          |         ||                        |          |         |:------------------------+----------+---------:| -5                     |          |         || -5                     |          |         || 12*     b12#7*   b(5)7 |          |         |:------------------------+----------+---------:|                        |          |         ||                        |          |         ||                        |          |         |'------------------------'----------'---------' (5)r1c1 == (75)r6c23 => -5 r1c3, r45c1 Shape 3:.----------------------.----------.---------.|a12(#5)*  12*         |          |         ||          -5          |          |         ||          -5          |          |         |:----------------------+----------+---------:|                      |          |         ||                      |          |         || 12*     b12#7*       |          |         |:----------------------+----------+---------:|-5                    |          |         ||-5                    |          |         ||-5        b(5)7       |          |         |'----------------------'----------'---------'(5)r1c1 == (75)r69c2 => -5 r23c2, r789c1`
-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: 2550
Joined: 22 May 2017

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