The New Sudoku Players' Forum

[civiliza]

Searches on this Forum seem to suggest that other than XY Wings, Sudoku does not specifically look for ALS-XY's. (EDIT - ALS-XZ's)

My understanding of ALS-XZ's is tenuous, but I think (hope) that I can spot a 3 bivalue cell ALS intersecting with a single bivalue cell ALS as follows:

Code: Select all

+----+----+----+
|    |    |    |
|    |    | YZ |
|    |    |    |
+----+----+----+
|    |    |    |
|    | W9 | Y9 |
|    |    |    |
+----+----+----+
|    |    |    |
|    | 49 | 4Z |
|    |    |    |
+----+----+----+

Leading to the elimination of the 9 from the W9 cell.

Given that my generator uses SE Ratings, what sort of rating should I give this ALS-XZ if I were to add it to my generator? (Presumably higher than XY Wing's 4.2)

[tarek]

Hi civiliza,

I was one of the 1st to program a solver using the ALS-XY rule so I have some experience with this ...

My advice would be to research / understand the the rules (ALS-XZ & ALS-XY) before programming ... Subset counting is the ultimate way of looking at these rules because it will cover them and much more.

Your example uses only bivalue cells which can be put together to make up a larger ALS (YZ Y9 4Z) which could link up with 49 using the ALS-XZ rule ...

The issue of rating is more challenging!!! Would you go with a bigger ALS & use the ALS XZ rule or fragment it to smaller ALSs & use the ALS XY rule??? & with ALSs of variable sizes do you change the rating according to size of the ALS? (You already do that for the LSs so why not ALSs??!!)

[civiliza]

My advice would be to research / understand the the rules (ALS-XZ & ALS-XY) before programming

Fair enough, I was not aware of ALS-XZ, so I will look into that.

As to extending the methodology into larger/more generic ALS's, I am hoping to program this as a specific one off method rather like the XY Wing. Nine times out of ten when I get stuck on one of my puzzles it's because I have missed an XY Wing, so this one off will be challenging enough for my personal tastes.

My generator is very Heath Robinson with extra techniques bolted on piecemeal, it is lacking in several basics like link, set and ALS recognition, and pending a redesign from the ground up, I can only hard code specific cases.

That said, it would be interesting to get a feeling for how different ALS sizes and methodologies affect the ratings beyond this (relatively?) simple case.

EDIT - D'oh, according to Sudopedia, this is an ALS-XZ not an ALS-XY, sorry, I can't even get my terminology right on this

[civiliza]

I had originally envisaged this method as an ALS-XZ between:

(YZ, Y9, 4Z) and (49) with 4 as the X value and 9 as the Z value

However (possibly due to using only bivalue cells) it can also be seen as an ALS-XZ between:

(YZ, Y9) and (49, 4Z) with Z as the X value and 9 as the Z value (confusing I know, shouldn't have used Y and Z in the original diagram)

--

After realising this last night, I came to the conclusion that this was equivalent to the chain (94 4Z ZY Y9).

No doubt the more experienced amoungst you already realised this, but having deliberately shied away from chains up to now it is pleasing to have (re)worked this out for myself.

So now the issue boils down to what SE rating would eliminations of the digit 9 by a 4 cell chain of (9A AB BC C9) be given?

[daj95376]

So now the issue boils down to what SE rating would eliminations of the digit 9 by a 4 cell chain of (9A AB BC C9) be given

Here is a puzzle that has two, 4-cell XY-Chains after basics. Assuming that Sudoku Explainer doesn't have a separate rating for Remote Pair, input this puzzle into SE and see what rating it gives them.

Code: Select all

 +-----------------------+
 | 8 4 2 | 5 . . | . 6 . |
 | 9 1 . | . 8 . | . 7 2 |
 | 5 . . | . . . | 4 . . |
 |-------+-------+-------|
 | 6 . . | 2 . . | . . . |
 | . 7 . | . 4 . | . . . |
 | . . . | . . 5 | 2 . 4 |
 |-------+-------+-------|
 | . . 5 | . . 4 | 6 . 7 |
 | 7 2 . | . . . | . . . |
 | . 6 . | . . 2 | 8 . 9 |
 +-----------------------+

     b7  Naked  Pair                     <> 13   r8c3

 +--------------------------------------------------------------+
 |  8     4     2     |  5     19    7     |  139   6     13    |
 |  9     1     6     |  4     8     3     |  5     7     2     |
 |  5     3     7     |  19    2     6     |  4     189   18    |
 |--------------------+--------------------+--------------------|
 |  6     5     4     |  2     139   189   |  7     1389  138   |
 |  2     7     1389  |  1389  4     189   |  139   5     6     |
 | a13    89    1389  |  6     7     5     |  2     89-13 4     |
 |--------------------+--------------------+--------------------|
 | b13    89    5     |  1389  139   4     |  6     2     7     |
 |  7     2     89    |  1389  6     189   |  13    4     5     |
 |  4     6    c13    |  7     5     2     |  8    d13    9     |
 +--------------------------------------------------------------+
 # 51 eliminations remain

 RP (1=3)r6c1 - (3=1)r7c1 - (1=3)r9c3 - (3=1)r9c8  =>  r6c8<>1
 RP (3=1)r6c1 - (1=3)r7c1 - (3=1)r9c3 - (1=3)r9c8  =>  r6c8<>3


[civiliza]

Hi DAJ, thanks for responding.

Having run your example through Sudoku Explainer from your intermediate Possibility map, I see that SE can use the chain of (13) Naked Pairs to create two different Turbot Fish eliminations - one on the 1 and one on the 3.

A couple of months ago, I had familiarised myself with the dual elimination possibilities at the open ends of an even length chains of Naked Pairs, but decided at the time not to implement them in my generator.

It seems that they are a special form of the 4 cell ALS-XZ (9A AB BC C9) with A and C holding a common value and B echoing the value at the two ends (9 as I specified it).

So the most basic ALS-XZ is an XY Wing with SE Rating 4,2
While the next one up produces a shape similar to (and in certain special cases the same contents as) a Turbot Fish with an SE Rating of 6.6.

The difference being that the Turbot Fish uses a common digit throughout its chain, whereas the ALS-XZ uses a series of digits.

Even so, maybe this suggests that this ALS-XZ should also have a rating in the region of 6.6.

Thanks for nudging me in the right direction DAJ.

[daj95376]

A better nudge!

Code: Select all

 +-----------------------------------------------------------------------+
 |  3      6789   569    |  27     1      4      |  2589   89     78     |
 |  249    4789   459    |  27     89     3      |  24589  6      1      |
 |  249    4789   1      |  6      89     5      |  2489   3      478    |
 |-----------------------+-----------------------+-----------------------|
 |  46-9   469    7      |  3      2      8      |  1     a49     5      |
 |  5      49     3      |  1      7      6      |  489    2      48     |
 |  8      1      2      |  5      4      9      |  6      7      3      |
 |-----------------------+-----------------------+-----------------------|
 |  1      2      48     |  9      5      7      |  3     b48     6      |
 | d79     3      89     |  4      6      1      | c78     5      2      |
 |  467    5      46     |  8      3      2      |  47     1      9      |
 +-----------------------------------------------------------------------+
 # 53 eliminations remain

 (9=4)r4c8 - (4=8)r7c8 - (8=7)r8c7 - (7=9)r8c1  =>  r4c1<>9


BTW: I believe the most basic ALS-XZ is a Naked Pair.

Code: Select all

 +-----------------------------------+
 |  .  .  .  |  .  .  .  |  .  .  .  |
 |  . 12  .  |  .  .  .  |  . 12  .  |
 |  .  .  .  |  .  .  .  |  .  .  .  |
 |-----------+-----------+-----------|
 |  .  .  .  |  .  .  .  |  .  .  .  |
 |  .  .  .  |  .  .  .  |  .  .  .  |
 |  .  .  .  |  .  .  .  |  .  .  .  |
 |-----------+-----------+-----------|
 |  .  .  .  |  .  .  .  |  .  .  .  |
 |  .  .  .  |  .  .  .  |  .  .  .  |
 |  .  .  .  |  .  .  .  |  .  .  .  |
 +-----------------------------------+

ALS-XZ: A=r2c2, B=r2c8, X=1, Z=2  =>  r2c1345679<>2

ALS-XZ: A=r2c2, B=r2c8, X=2, Z=1  =>  r2c1345679<>1


[civiliza]

A better nudge!

(9=4)r4c8 - (4=8)r7c8 - (8=7)r8c7 - (7=9)r8c1 => r4c1<>9

Thanks, it's even got the magic 9's at the ends. Real shame that SE can't see it, but I guess that confirms that SE doesn't recognise higher ALS-XZ's.

BTW: I believe the most basic ALS-XZ is a Naked Pair.

Oops, so it is:
Naked Pair = A bivalue cell ALS-XZ with 2x1 cell ALS's (SE rating 3.0)
XY Wing = A bivalue cell ALS-XZ with 1x1 cell ALS & 1x2 cell ALS (SE rating 4.2)
? = A bivalue cell ALS-XZ with 2x2 cell ALS's (not recognised by SE, rating unknown)
? = A bivalue cell ALS-XZ with 1x1 cell ALS & 1x3 cell ALS (presumably not recognised by SE, rating unknown)

[civiliza]

A better nudge!

(9=4)r4c8 - (4=8)r7c8 - (8=7)r8c7 - (7=9)r8c1 => r4c1<>9

Naked Pair = A bivalue cell ALS-XZ with 2x1 cell ALS's (SE rating 3.0)
XY Wing = A bivalue cell ALS-XZ with 1x1 cell ALS & 1x2 cell ALS (SE rating 4.2)
? = A bivalue cell ALS-XZ with 2x2 cell ALS's (not recognised by SE, rating unknown)
? = A bivalue cell ALS-XZ with 1x1 cell ALS & 1x3 cell ALS (presumably not recognised by SE, rating unknown)

Sorry DAJ, having pathalogically avoided any techniques with a Chain in the name, I did not realise what I had partially stumbled upon.

The last time I posted here, I was considering adding a 1 cell / 3 cell ALS-XZ, with the possible addition of a 2 cell / 2 cell ALS-XZ further down the line.

Furthermore I had initially planned on "keeping it simple" by limiting myself to bivalue cells.

Anyway, while slowly working my way through a fuller 1 cell / 3 cell ALS-XZ implementation (and with 2 cell / 2 cell hovering in the back of my mind), the concept of a bivalue chain fully crystallised. On a hunch, I looked up the XY Chain method you had named earlier on JasonLion's Sudopedia stub and lo and behold, I had accidentally "rediscovered" the XY Chain as a special ALS-XZ case.

If only I had listened to your advise earlier, I could have saved myself a lot of time (although I wouldn't have had the satisfaction of (re)discovering a method).

--

Having gone on to implement XY Chains in my generator (still working on 1 cell / 3 cell ALS-XZ's), I can improve on my previous summary:

Naked Pair (2 cell XY Chain) = An ALS-XZ with 2x1 cell ALS's (SE rating 3.0)
XY Wing (3 cell XY Chain) = An ALS-XZ with 1x1 cell ALS & 1x2 cell ALS (SE rating 4.2)
4 Cell XY Chain (alternative interpretation) = An ALS-XZ with 2x2 cell ALS's (not recognised by SE, rating unknown)

4 Cell XY Chain Special Case (alternative interpretation) = A (Bivalue cell) ALS-XZ with 1x1 cell ALS & 1x3 cell ALS (single pincer) (not recognised by SE, rating unknown)
? = Mixed trivalue/bivalue cell ALS-XZ with 1x1 cell ALS & 1x3 cell ALS (presumably not recognised by SE, rating unknown)