## The Empty Rectangle (ER)

Advanced methods and approaches for solving Sudoku puzzles

### The Empty Rectangle (ER)

Hi.

First of all, this is no new technique. It falls under the catagory of grouped strong links. However, I think this one has it's worth in that it is much easier to spot and to deal with than grouped x-cycles.

Some of this have already been posted in the "skyscraper & 2-string kite" thread, but I thought it deserved a thread on it's own, besides I have thought of a few improvements

Here goes:

definition: An Empty Rectangle (ER) is when four cells in a box does not contain a specific candidate, and these four cells form a rectangle.

so that with this candidate-distribution: (theoretical example)
Code: Select all
`. 1 . | . . . | . . .1 1 1 | . . . | . . .. 1 . | . . . | . . .------+-------+------1 . 1 | . 1 . | . . .. . . | 1 . . | . . .. . 1 | 1 . . | . . .------+-------+------1 . . | . . 1 | . . .. 1 . | . . 1 | . . .. 1 . | 1 1 1 | . . .`

you would have these empty rectangles:
Code: Select all
`X 1 X | . . . | . . .1 1 1 | . . . | . . .X 1 X | . . . | . . .------+-------+------1 . 1 | . 1 . | . . .X X . | 1 X X | . . .X X 1 | 1 X X | . . .------+-------+------1 . . | X X 1 | . . .X 1 X | X X 1 | . . .X 1 X | 1 1 1 | . . .`

Now with every Empty Rectangle, you will get two Empty Rectangle Lines! (ERL). These are the two straight lines you can draw inside the box with the ER without touching the ER itself:

Example:
Code: Select all
`X | X | . . . | . . .--|-------ERL | . . .X | X | . . . | . . .--|---+-------+------. | . | . . . | . . .. | . | . . . | . . ..ERL. | . . . | . . .------+-------+------. . . | . . . | . . .. . . | . . . | . . .. . . | . . . | . . .`

Now luckily we don't have to concider these ERL's at all! Our only concern is the point where they intersect, and this we call the Empty Rectangle Intersection. (ERI)

Here are all the ERI's marked with a "+" in the above example (without the candidates)
Code: Select all
`X . X | . . . | . . .. + . | . . . | . . .X . X | . . . | . . .------+-------+------. . + | + . . | . . .X X . | . X X | . . .X X . | . X X | . . .------+-------+------. + . | X X . | . . .X . X | X X . | . . .X . X | . . + | . . .`

And now for the use of this thing:

If you have a strong link and an ERI that shares a line (row or column) then you can eliminate a candidate at the point where the other end of the strong link and the ERI intersect.

pics says more than words, so here are some examples:
Code: Select all
`. . . | . . . | X . X . . . | . . . | X . X . a . | . . . | . + . --|------------------ . | . | . . . | . . . . | . | . . . | . . . . | . | . . . | . . . --|------------------ . b . | . . . | . * . . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | X X . . a . | . . . | . . + . | . | . . . | X X . --|------------------ . | . | . . . | . . . . | . | . . . | . . . . | . | . . . | . . . --|------------------ . b . | . . . | . . * . . . | . . . | . . . . . . | . . . | . . . . a . | . . . | + . . . | . | . . . | . X X . | . | . . . | . X X --|------------------ . | . | . . . | . . . . | . | . . . | . . . . | . | . . . | . . . --|------------------ . b . | . . . | * . . . . . | . . . | . . . . . . | . . . | . . . a-b mark the strong linksX shows the Empty Rectangle (ER)+ marks the Empty Rectangle Intersection (ERI)* shows the candidate that can be eliminated`

*It does not matter what the distribution of the candidates in the box are! We are only looking where there are NO candidates!
*It is VERY easy to spot!
*Can eliminate candidates that escapes the easier methods, and would require extensive knowledge of grouped x-cycles to find otherwise.
*Absorbs one of the three basic patterns in the Turbot Fish. (The one that is not Skyscraper or 2-string kite)

hope you like it!

havard
Havard

Posts: 378
Joined: 25 December 2005

### Re: The Empty Rectangle (ER)

Havard wrote:
ronk wrote:
Havard wrote:x marks the ER (Empty Rectangle
+ marks the ERI (Empty Rectangle Intersection)

What is the significance of the ERI? IOW please explain the connection between the ERI and the elimination.

If you have a strong link and an ERI that shares a line (row or column) then you can eliminate a candidate at the point where the other end of the strong link and the ERI intersect.

That's a statement of conclusion. It doesn't explain how to arrive at that conclusion.

And I fail to see how the ERI ... one cell on the grid ... can intersect with the strong links in your illustrations.

Ron
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

### Re: The Empty Rectangle (ER)

ronk wrote:
Havard wrote:
ronk wrote:
Havard wrote:x marks the ER (Empty Rectangle
+ marks the ERI (Empty Rectangle Intersection)

What is the significance of the ERI? IOW please explain the connection between the ERI and the elimination.

If you have a strong link and an ERI that shares a line (row or column) then you can eliminate a candidate at the point where the other end of the strong link and the ERI intersect.

That's a statement of conclusion. It doesn't explain how to arrive at that conclusion.
And I fail to see how the ERI ... one cell on the grid ... can intersect with the strong links in your illustrations.
Ron

ok, I'll prove it!

The easiest way to explain how the ER works, is to concider the options of "a" in the strong link being "true" (a number), and hence "b" is false, and the opposite. (b true, a false)

Now with b true and a false the elimination of * is obvious.

If a is true on the other hand, we have to prove that this will always lead to the elimination of *.

Now the placement of the ERI (Empty Rectangle Intersection, "+" in the illustations) is aligned both with "a" and the *, and it is also important to remember that all candidates that are distributed in the ER box, are either aligned with "a" or with *.

Now, if a is true, then you get the following scenarios (from the examples above)
Code: Select all
`. . . | . . . | X c X . . . | . . . | X c X . a . | . . . | X X X --|------------------ . | . | . . . | . . . . | . | . . . | . . . . | . | . . . | . . . --|------------------ . b . | . . . | . * . . . . | . . . | . . . . . . | . . . | . . . . . . | . . . | X X c . a . | . . . | X X X. | . | . . . | X X c --|------------------ . | . | . . . | . . . . | . | . . . | . . . . | . | . . . | . . . --|------------------ . b . | . . . | . . * . . . | . . . | . . . . . . | . . . | . . . . a . | . . . | X X X . | . | . . . | c X X . | . | . . . | c X X --|------------------ . | . | . . . | . . . . | . | . . . | . . . . | . | . . . | . . . --|------------------ . b . | . . . | * . . . . . | . . . | . . . . . . | . . . | . . . c - possible place for a candidate`

As you can see, there are only two possible places that are NOT empty if "a" is true (marked with "c", and if there are candidates in both those places, then we have a strong link in that box that will eliminate *, and if there is a candidate in only one of the two places, then we have a basic elimination of *.

hence we have proved that * will be eliminated if a is true, and if a is false.

havard
Havard

Posts: 378
Joined: 25 December 2005

Well it works but its pretty rare. I've not thrown my solved library at it yet but against my unsolvable library (appox 2500) it helped solve two. These two needed an Empty Rectangle strat:

Code: Select all
`. . .|5 6 4|. . . . 4 .|. . .|. 1 . 3 . .|. 9 .|. . 6 -----+-----+-----7 . .|9 4 .|. . 1 . . 4|. 5 .|2 . . 9 . .|. 1 2|. . 5 -----+-----+-----8 . .|. 7 .|. . 9 . 6 .|. . .|. 7 . . . .|6 2 8|. . . . . 7|. 8 .|5 . . 4 2 .|. 6 .|. . 1 8 . .|. . .|. 4 6 -----+-----+-----. . .|3 . 2|. . . . 7 .|. . .|. 1 . . . .|9 . 4|. . . -----+-----+-----1 9 .|. . .|. . 4 6 . .|. 2 .|. 7 . . . 3|. 4 .|2 . . `

In the first Sudoku the solving gets to a point where we need and ER on number 5. These are the distribution of 5s (candidates only). The ER is outlined using Havard's symbols. Removing the 5 at R8C7 leave an 8 and it solves.

Code: Select all
`. . . | . . . | . . .   . . . | . . . | . . . 5 . . | . . . | 5 . .   a---------------b . . . . . | . . . | 5 5 .   . . . | . . . | . . . ------+-------+------   ------+-------+------ . 5 5 | . . . | . . .   . . . | . . . | . . . . . . | . . . | . . .   . . . | . . . | . . . . . . | . . . | . . .   . . . | . . . | . . . ------+-------+------   ------+-------+------ . . . | . . . | . . .   . X X | . . . | . . . 5 . 5 | . . . | 5 . .   + . c | . . . | * . . 5 . . | . . . | 5 5 .   c X X | . . . | . . . `

In the second Sudoku the number under the spotlight is 9:

Code: Select all
`9 . . | . . . | . . 9   . . . | . . . | . . . . . 9 | . . . | 9 . .   . . b-----------a . . . . . | . . . | . . .   . . . | . . . | . . . ------+-------+------   ------+-------+------ 9 . 9 | . . . | 9 . .   . . . | . . . | c X X . . 9 | . . . | 9 . 9   . . * | . . . | + . c . . . | . . . | . . .   . . . | . . . | . X X ------+-------+------   ------+-------+------ . . . | . . . | . . .   . . . | . . . | . . . . . . | . . . | . . .   . . . | . . . | . . . . . . | . . . | . . .   . . . | . . . | . . .`

Removing the 9 at R5C3 leaves a 4 and it solves. One point to note 'a', 'b' and '*' must all be in different boxes for this to work.

I really don't think this is an easy method to spot manually, IMHO. But it adds to the armoury. (I've noticed a strong diminishing returns in these advanced strats, chipping flakes off the vast rock of the unknown, ho hum)
AndrewStuart

Posts: 21
Joined: 28 December 2005

AndrewStuart,

I've actually found empty rectangle reductions to be quite ubiquitous and to be very useful. Take for instance Menneske Very Hard #5340665.

Code: Select all
`. 4 . | . . 9 | . . 7. . . | 2 . . | . 9 1. . 8 | . 7 . | . 3 6------+-------+------. . 2 | . . 7 | 6 . .7 . . | . . . | . . 9. 6 . | 1 . . | . 8 .------+-------+------6 8 . | . 3 . | 1 . .2 5 . | 7 . . | . . 33 . . | . . 4 | . . .`

After standard moves, one gets to the position:

Code: Select all
`1 4 . | . . 9 | . . 75 . . | 2 . . | . 9 19 2 8 | . 7 1 | . 3 6------+-------+------8 . 2 | . . 7 | 6 1 .7 1 5 | . . . | 3 . 94 6 . | 1 . . | 7 8 .------+-------+------6 8 . | . 3 . | 1 . .2 5 . | 7 . . | 9 . 33 . . | . . 4 | . . 8`

Now an empty rectangle (or "hinge" as it was originally called by Rod Hagglund in http://www.sudoku.org.uk/discus/messages/29/414.html?1135572720) in box 1 forces (6,3)!3 (or if you like, (2,6)!3) and the puzzle solves easily from here. Without this move, Sudoku Susser uses a simple forcing chain and 2 comprehensive forcing chains to solve the puzzle. Maybe I would find the simple forcing chain, but the comprehensive ones, yikes. If you are not able with practice to find the empty rectangle reductions, look at the post of Rod Hagglund above and search for his hinge pattern (as I do) which is equivalent to finding the empty rectangles.
re'born

Posts: 551
Joined: 31 May 2007

rep'nA wrote:Now an empty rectangle (or "hinge" as it was originally called by Rod Hagglund in http://www.sudoku.org.uk/discus/messages/29/414.html?1135572720) in box 1 forces (6,3)!3 (or if you like, (2,6)!3) and the puzzle solves easily from here. Without this move, Sudoku Susser uses a simple forcing chain and 2 comprehensive forcing chains to solve the puzzle

Code: Select all
` 1     4     36    | 3568  568   9     | 258   25    7 5     37    367   | 2     468   368   | 48    9     1 9     2     8     | 45    7     1     | 45    3     6-------------------+-------------------+------------------ 8     39    2     | 3459  459   7     | 6     1     45 7     1     5     | 468   2468  268   | 3     24    9 4     6     39    | 1     259   235   | 7     8     25-------------------+-------------------+------------------ 6     8     479   | 59    3     25    | 1     2457  24 2     5     14    | 7     168   68    | 9     46    3 3     79    179   | 569   12569 4     | 25    2567  8`

... Angus Johnson's Simple Sudoku's (SS) next step is simple coloring (single digit) for 3s:
Code: Select all
` .  .  B  | G  .  .  | .  .  .   .  B  3  | .  .  B  | .  .  .   .  .  .  | .  .  .  | .  .  .  ----------+----------+---------- .  G  .  | B  .  .  | .  .  .   .  .  .  | .  .  .  | .  .  .   .  .  B  | .  .  G  | .  .  .  ----------+----------+---------- .  .  .  | .  .  .  | .  .  .   .  .  .  | .  .  .  | .  .  .   .  .  .  | .  .  .  | .  .  .  `

Blue (B) and green (G) are conjugate colors in SS. The above coloring places two Bs in box 1 which is impossible. Therefore, 3 can be eliminated from ALL cells colored B ... which "breaks" the puzzle. The Empty Rectangle (Hinge) to which you apparently refer, can be seen in box 1.

Ron
Last edited by ronk on Thu Feb 23, 2006 2:57 pm, edited 1 time in total.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

Ron,

I go back and forth about which is easier to look for, simple coloring or the hinge (on paper, the hinge seems easier to spot but on the computer it is a bit of a toss-up). The advantage of the hinge is that I don't need strong links everywhere. Thus, I can capture many of the simple coloring patterns and more by looking for the hinge first. On the other hand, looking for the hinge, I will miss many simple (or multiple) coloring patterns.

Thanks for pointing out this alternative method. I often overlook coloring when using the Susser.

P.S. Does anybody know why coloring is not singled out as a heuristic in Sudoku Susser?
re'born

Posts: 551
Joined: 31 May 2007

All of the puzzles so far proposed can be solved with basic techniques (finned X-wing, XY-wings, etc). Even Rod Hagglund's puzzle is just a grouped Turbot Fish (we know now). I'm still trying to find a puzzle which needs ER assuming basic techniques (I'd include a grouped Turbot Fish as basic since it requires the same structure as an ER). That ER is rare makes sense.
Code: Select all
`. . . | X . X. . . | X . X. a . | # + #--|-----------. a . | . * .. . . | . . .`

If the "+" doesn't contain the candidate value then the ER is a grouped Turbot Fish and "*" is "deleted". If one of the "#" doesn't contain the candidate, then box/line will "delete" "*". If "+" and "*" are conjugate pairs, then finned X-wing will "delete" "#" and box/line take care of "*". So as far as I've made it in order for an ER to further advance a puzzle you need at least 10 cells (6 with and 4 without the candidate):
Code: Select all
`. . . | X . X. . . | X . X. a . | a + .--|-----------. a . | . * .. . . | . a .`

where the "a's" can move to equivalent positions and "+" and "*" contain "a" as well. Given these conditions and a puzzle with no other simple reductions elsewhere in the board, then an ER could help (unless I've missed further limitations). I just haven't found any. I haven't tried it yet, but the same logic can apply to a 7-node (almost) Grouped Turbot Chain (almost because the "+" disallows the grouping).

Please don't get me wrong, I did not set out to put down ER, simple techniques are what I do best and this qualifies. Its just that I'm not sure how valuable it really is. At this point, maybe its greatest value is emphasizing finding the conjuate pair as a precurser to finding a Turbot fish or X-wing.
Mike Barker

Posts: 458
Joined: 22 January 2006

Mike Barker wrote:All of the puzzles so far proposed can be solved with basic techniques (finned X-wing, XY-wings, etc). Even Rod Hagglund's puzzle is just a grouped Turbot Fish (we know now). I'm still trying to find a puzzle which needs ER assuming basic techniques (I'd include a grouped Turbot Fish as basic since it requires the same structure as an ER). That ER is rare makes sense.

I think the ER is for solving puzzles manually, and believe that Havard would argue that the ER is easier to spot than either a finned x-wing or a grouped turbot fish. I agree with that POV.

Ron
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

Mike Barker wrote:All of the puzzles so far proposed can be solved with basic techniques (finned X-wing, XY-wings, etc). Even Rod Hagglund's puzzle is just a grouped Turbot Fish (we know now). I'm still trying to find a puzzle which needs ER assuming basic techniques (I'd include a grouped Turbot Fish as basic since it requires the same structure as an ER). That ER is rare makes sense. .

Hi Mike!

The value of the ER is not primarily as a "new" technique to solve more puzzles with, it is to easier be able to solve puzzles that normally would require much more complicated techniques as ronk pointed out. The fact that it takes care of one of the three Turbot Fish Patterns, and can do eliminations that you would need finned X-wing or grouped Turbot to do, is something that I concider a strenght and not a weakness with the pattern.

Say, I would be very interested to see your solver doing your 25000 puzzles with this configuration:
L_NAKED1
L_NAKED2
L_HIDDEN1
L_LOCKED1
L_LOCKED2
L_NAKED3
L_HIDDEN2
L_HIDDEN3
L_NAKED4
L_HIDDEN4

L_TWOSTRONGLINKS (as described here. In other words including x-wing and Turbot Fish)
L_EMPTYRECTANGLE
L_FINNEDXWING
L_THREESTRONGLINKS (including swordfish, and all the patterns I pointed out here.)
L_FINNEDSWORDFISH

L_XY
L_XYZ 'generalized (allows 2-3 cell pilots)
L_WXYZ 'generalized (allows 2-4 cell pilots)
L_VWXYZ 'generalized (allows 2-5 cell pilots)
L_XYRING4
L_XYCHAIN4
L_XYRING5
L_XYCHAIN5
L_SUEDECOQ
L_ALS1 'xz only, same as xy-wing, xyz-wing, etc
L_ALS2 'xz only
L_TURBOT4 'same as locked candidates
L_TURBOT5 'includes basic and grouped
L_ALS3 'xz only
L_ALS4 'xz only
L_JELLYFISH 'includes basic and finned
L_SQUIRMBAG 'includes basic and finned
L_TURBOT6 'includes basic, finned, and grouped
L_TURBOT7 'includes basic and grouped
L_COLOR1 'type 1 includes basic and some grouped
L_COLOR2 'type 2 includes basic and some grouped
L_COLOR3 'type 3 includes basic and some grouped
L_UNIQUE1 'type 1
L_UNIQUE2 'includes types 2 and 2B
L_UNIQUE3 'includes types 3 and 3B
L_UNIQUE4 'includes types 4 and 4B

that would give a good indication at least of how useful the ER can be to manual solvers. (and a lot of other interesting data more relating to the "Method hierachy"-thread, which I hope you have not abandoned... )

Havard
Havard

Posts: 378
Joined: 25 December 2005

I haven't given up on the technique hierarchy. I'd implemented the empty rectangle and when I didn't see any exclusions went on to some more exotic techniques (I've implemented ALS xy-rule based on the success of the xz-rule with good results) and was looking at nice loops. Its nice to get back to something simple. I went ahead and implemented 2, 3 and added 4 strong link algorithms in addition to the empty rectangle that I already had. Not unexpectedly with these algorithms operating prior to NxN Fish and Turbot Fish there are a significant number of eliminations. It took about 4% more steps to complete all 10000 puzzles and that got me thinking. The strong links as described only deal with eliminations common to the start and end cells. If the start and end cells share the same house then eliminations are possible not only there but between the end and start nodes of each conjugate pair (its a continuous X-cycle) - a strong cycle. I've implemented this and will post the results later. I guess my conclusion would be the same as Ron so correctly pointed out: These techniques are great for catching many eliminations and their simplicity makes them very useful especially for beginners. The point I tried to make previously is that once you've identified one of these patterns identifying a finned X-wing, a grouped Turbot fish, etc is easy and should also be performed. (That and expressing surprise that I didn't actually solve any more puzzles - sorry about that).

Here are the results (number of times a technique is used based only on successfully solved puzzles (9725 out of 10000)
1) Naked Single (322048)
2) Hidden Single (92521)
3) Naked Pair (42327)
4) Empty Rectangle (13916)
5) Locked Candidate (12185)
6) ALS-xz rule with >=2 cells per ALS (2378)
8) Generalized WXYZ-wing (1968)
9) Naked Triple (1964)
10) XY-wing (1403)
11) Generalized VWXYZ-wing (962)
12) ALS-xy rule with A=1 cell ALS (685)
13) Hidden Pair (649)
14) ALS-xz rule with >=3 cells per ALS (571)
15) Generalized XYZ-wing (494)
16) Finned X-wing (468)
18) ALS-xy rule with A=2 cell ALS (413)
19) 4-node XY-chain (253)
20) Finned Swordfish (174)
21) ALS-xy rule with A=3 cell ALS (135)
22) Hidden Triple (128)
23) ALS-xz rule with >=4 cells per ALS (91)
24) Type 1 Unique Rectangles (79)
25) Type 4/4B Unique Rectangles (66)
26) ALS-xy rule with A=4 cell ALS (64)
27) 5-node XY-chain (58)
28) 4-node XY-ring (45)
29) Grouped Turbot Fish (38)
30) Grouped 7-node Turbot Chain (36)
31) Type 3/3B Unique Rectangles (26)
32) Swordfish (23)
33) Type 2/2B Unique Rectangles (22)
35) 5-node XY-ring (18)
36) SueDeCoq (16)
37) Grouped 6-node X-cycle (9)
39) Grouped Broken Wing (5)
41) Finned Jellyfish (1)
42) Grouped 7-node Broken Wing (1)
43) Unused (0)
44) X-wing (0)
45) Jellyfish (0)
46) Squirmbag (0)
47) Finned Squirmbag (0)
48) ALS-xz rule with >=1 cells per ALS = XY-wing, etc (0)
49) 4-node X-cycle = X-wing (0)
50) Turbot Fish (0)
51) 6-node X-cycle (0)
52) 7-node Turbot Chain (0)
53) Grouped 4-node X-cycle = Locked box/box (0)
54) Broken Wing (0)
55) 7-node Broken Wing (0)
56) Simple Colouring Type 1 = X-cycle (0)
57) Simple Colouring Type 2 = X-cycle (0)
58) Simple Colouring Type 3 = X-cycle (0)
59) Grouped Simple Colouring Type 1 = X-cycles (0)
60) Grouped Simple Colouring Type 2 = X-cycles (0)
61) Grouped Simple Colouring Type 3 = X-cycles (0)
Mike Barker

Posts: 458
Joined: 22 January 2006

After applying basic techniques to this puzzle ...
Code: Select all
` ..8|.2.|7.1 .34|6..|... 9..|...|... ---+---+--- ...|..5|... ..2|.1.|8.. ...|4..|... ---+---+--- ...|...|..5 ...|..7|42. 7.1|.8.|3..`

... we get this 9s candidate grid ...
Code: Select all
` . . . | 9 . 9 | . 9 . . . . | . . 9 | . . 9 . . . | . . . | . . . - - - + - - - + - - - . . 9 | . 9 . | 9 . . . 9 . | . . 9 | . . . . . 9 | . 9 . | 9 . . - - - + - - - + - - - . . 9 | 9 . 9 | . . . . 9 9 | . 9 . | . . . . . . | . . . | . 9 9 `

... with elimination r8c5<>9.

The chain is a grouped x-cycle of length 7 including three strong links due to three Empty Rectangles. I leave identification of the ERs as an exercise.

Ron

P.S. This elimination first noted by aeb here.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

I'm not sure where exactly you stopped, at one point there is an Empty Rectangle on 3's
Code: Select all
`+------------------+---------------+------------------+|    56   56     8 |  39    2  349 |    7   349     1 ||    12    3     4 |   6    7   19 |    5     8    29 ||     9   12     7 |   8    5  134 |   26   346  2346 |+------------------+---------------+------------------+|   148  178  *369 |   2 *369    5 |   69    14    47 ||  3456  569     2 |   7    1  369 |    8  3456   346 ||  1356   17  3569 |   4  369    8 |  269  1356  2367 |+------------------+---------------+------------------+|    28   28   369 |  39    4  369 |    1     7     5 ||   356  569 *3569 |   1 -369    7 |    4     2     8 ||     7    4     1 |   5    8    2 |    3    69    69 |+------------------+---------------+------------------+`

but also a finned X-wing with 9's:
Code: Select all
`+------------------+---------------+------------------+|    56   56     8 |  39    2  349 |    7   349     1 ||    12    3     4 |   6    7   19 |    5     8    29 ||     9   12     7 |   8    5  134 |   26   346  2346 |+------------------+---------------+------------------+|   148  178   369 |   2 *369    5 |   69    14    47 ||  3456 *569     2 |   7    1 -369 |    8  3456   346 ||  1356   17  3569 |   4 *369    8 |  269  1356  2367 |+------------------+---------------+------------------+|    28   28   369 |  39    4  369 |    1     7     5 ||   356 *569  3569 |   1 *369    7 |    4     2     8 ||     7    4     1 |   5    8    2 |    3    69    69 |+------------------+---------------+------------------+`

Using the X-wing, later there is a "skyscraper" with 3's
Code: Select all
`+---------------+---------------+-----------------+|   56   56   8 |  39    2  349 |    7  349     1 ||   12    3   4 |   6    7   19 |    5    8    29 ||    9   12   7 |   8    5  134 |   26  346  2346 |+---------------+---------------+-----------------+|  148  178 *36 |   2 *369    5 |   69   14    47 || -346    9   2 |   7    1   36 |    8    5   346 || -136   17   5 |   4  369    8 |  269  136  2367 |+---------------+---------------+-----------------+|   28   28 -36 |  39    4  369 |    1    7     5 || *356   56   9 |   1  *36    7 |    4    2     8 ||    7    4   1 |   5    8    2 |    3   69    69 |+---------------+---------------+-----------------+`

or another finned X-wing with 3's
Code: Select all
`+---------------+---------------+-----------------+|   56   56   8 |  39    2  349 |    7  349     1 ||   12    3   4 |   6    7   19 |    5    8    29 ||    9   12   7 |   8    5  134 |   26  346  2346 |+---------------+---------------+-----------------+|  148  178 *36 |   2 *369    5 |   69   14    47 ||  346    9   2 |   7    1   36 |    8    5   346 ||  136   17   5 |   4  369    8 |  269  136  2367 |+---------------+---------------+-----------------+|   28   28 *36 | *39    4 *369 |    1    7     5 ||  356   56   9 |   1  -36    7 |    4    2     8 ||    7    4   1 |   5    8    2 |    3   69    69 |+---------------+---------------+-----------------+`

No need for anything fancier. I admit that my solver is definitely a work in progress, but based on the results I've observed the puzzle cannot be solved with empty rectangles/strong links and basic techniques alone. On the other hand it can be solved with a finned X-wing, grouped Turbot fish, or something equally sophisticated. It appears that this is a good example to show that empty rectangles/strong links are great for speeding through many puzzles, but be prepared to use the focus these techniques provide to apply slightly bigger guns (finned X-wing, etc) to solve some problems.

Another approach is to go as far as possible with ER/strong links and throw in an XY-wing besides. With this I get to a finned X-wing with 9's which will again crack the puzzle, but the finned X-wing is still required.
Code: Select all
`+------------------+---------------+------------------+|    56   56     8 |  39    2  349 |    7   349     1 ||    12    3     4 |   6    7   19 |    5     8    29 ||     9   12     7 |   8    5  134 |   26   346   234 |+------------------+---------------+------------------+|   148  178   369 |   2 *369    5 |   69    14    47 ||  3456 *569     2 |   7    1  -69 |    8  3456   346 ||  1356   17  3569 |   4 *369    8 |  269  1356  2367 |+------------------+---------------+------------------+|    28   28    69 |  39    4  369 |    1     7     5 ||   356 *569  3569 |   1  *69    7 |    4     2     8 ||     7    4     1 |   5    8    2 |    3    69    69 |+------------------+---------------+------------------+`

I guess there are lots of approaches, but I still come to the same conclusion. Maybe the solution is to identify grouped strong links, in which case at least the above problem could be solved. I'll look into it.
Last edited by Mike Barker on Tue Mar 07, 2006 1:28 am, edited 1 time in total.
Mike Barker

Posts: 458
Joined: 22 January 2006

By-the-by I did implement the strong cycles in addition to the strong links and it did reduce the number of 2-strong link eliminations, but didn't significantly decrease the total number of eliminations. That must be the result of some of the changes to the order precedence and not the addition of the ER/strong links. Still I'll happily take a 10% reduction in the number of strong link eliminations I need to do!
Mike Barker

Posts: 458
Joined: 22 January 2006

Mike Barker wrote:I'm not sure where exactly you stopped

I stopped at the 9s candidate grid I posted ... to illustrate a (relatively) unusual rectangle of 3 ERs ... in line with the thread topic.

I included the starting grid as a courtesy, because I appreciate it when others do the same. However, to be frank, it's irritating when people then go off-topic to show there are "other" and/or "better" ways to advance the puzzle. Isn't that usually the case?

Ron
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