The Empty Rectangle (ER)

Advanced methods and approaches for solving Sudoku puzzles

Postby Havard » Tue Mar 07, 2006 11:47 am

Mike Barker wrote: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)
    7) Two Strong Links (2030)
    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)
    17) Three Strong Links (441)
    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)
    31) Type 3/3B Unique Rectangles (26)
    32) Swordfish (23)
    33) Type 2/2B Unique Rectangles (22)
    34) Naked Quadruple (21)
    35) 5-node XY-ring (18)
    36) SueDeCoq (16)
    37) Grouped 6-node X-cycle (9)
    38) Four Strong Links (5)
    39) Grouped Broken Wing (5)
    40) Hidden Quadruple (1)
    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)


Hi Mike!

Wow, thanks for your speedy reply with such interesting results! Great work!:)

First of all, was this solved with the order of techniques that I proposed in the previous post? I'll just assume that and make the comments thereafter...

First of all, there is a healthy sign that your solver did not find any x-wings or Turbot Fish. That just means that your "two-strong-links" algorithm is doing its job!:) It is also a healthy sign that the number of jellyfish-squirmbags etc (finned and not) also has decreased as a result of three and four strong links. It only makes me more interested to see that one Finned Jellyfish... Can that be arranged?:D However, how is it that your solver finds 23 swordfish? If your "three-strong-links" routine is indeed before the swordfish, there should really be 0 swordfish (unless you also pick up on some of the more unusual swordfish-patterns) Can you check this?

The biggest surprise was the number of Empty Rectangles... Are you sure the two-strong-links are checked before the ER? The fact that he found more of them than locked candidates just seems a bit crazy... It is not that I am unhappy with this result (it would indeed make the ER quite useful), but it almost seems too good to be true!

The increase in number of techniques used is no big surprise to me. The two strong links can pick up on all eliminations done by a Turbot Fish, but while a Turbot Fish that have more than two strong sides can place numbers directly, the two strong links leaves them for the single (locked) candidate to pick up on. Hence you get a "easier" but longer solution since it needs more steps. I think this is really good, and I would love to see these:
29) Grouped Turbot Fish (38)
being reduced as well...:D

would it be possible for you to post them (maybe in the hierarchy thread) so they can be closely examined for easier reductions?:)

anyway, thanks for your effort! Good stuff!:D

havard
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Postby Havard » Tue Mar 07, 2006 11:54 am

ronk wrote:
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.


Hi Ron! Is this the one you had in mind?

Code: Select all
 . . . | 9 . 9 | . 9 .
 . . . | . . 9 | . . 9
 . . . | . . . | . . .
 - - - + - - - + - - -
 . . 9 | x 9 x | 9 . .
 . 9 . | . + 9 | . . .
 . | 9 | x 9 x | 9 . .
 - | - + - - - + - - -
 . | 9 | 9 . 9 | . . .
 . 9 9 | .-9 . | . . .
 . . . | . . . | . 9 9
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Postby Mike Barker » Tue Mar 07, 2006 12:23 pm

A number of things. First there are 2 types of swordfish, double implication chains (2x2x2) and higher order implication chains (3x3x2, 3x3x3,...). The strong link approach will only catch the former, not the later. Also since it is finned, basic strong links will not catch it. I hope to show that by using grouped strong links that at least the finned X-wings, grouped Turbot fish and other double implication chains can be brought in under the stong link approach.

The analysis of Ron's example shows a good example of a finned X-wing/grouped Turbot fish. By treating r46c5 and r8c5 as a grouped strong link, the strong link approach can be extended to catch this as well.

Ron my appoligies. From my narcissistic point of view I thought you were responding to my comment
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).
Not just providing a cool example of ER's. Of course I was surprized when it required a finned X-wing to solve. On the other hand, it forced me to come up with grouped strong links. Thanks.

The finned jellyfish:
..6.......2.1..79.....75...27......4....3.2....526...8.6....85.4.........1..274..
7.6..25.152.1..79.....75..227..5...46..73.215..526..78.6241.857457...12..1.5274..
Code: Select all
+------------------+------------------+---------------+
|    7 *3489     6 |  *389  489     2 |    5  48    1 |
|    5     2   348 |     1   48   368 |    7   9   36 |
|  189 *3489  1489 | *3689    7     5 |  *36  48    2 |
+------------------+------------------+---------------+
|    2     7  1389 |    89    5   189 |  369  36    4 |
|    6   489   489 |     7    3   489 |    2   1    5 |
|  *13  *349     5 |     2    6   149 |  *39   7    8 |
+------------------+------------------+---------------+
|  *39     6     2 |     4    1   *39 |    8   5    7 |
|    4     5     7 | -3689   89  3689 |    1   2   39 |
|  389     1   389 |     5    2     7 |    4  36  369 |
+------------------+------------------+---------------+


The solving order is pretty much what you asked for. One question, why ER after 2-strong links?
L_NAKED1
L_NAKED2
L_HIDDEN1
L_LOCKED1
L_NAKED3
L_HIDDEN2
L_HIDDEN3
L_NAKED4
L_HIDDEN4
L_STRONG2 'two strong links
L_EMPTY 'empty rectangle
L_XWING 'includes basic and finned
L_STRONG3 'three strong links
L_SWORDFISH 'includes basic and finned
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_ALS1xz 'same as xy-wing + xyz-wing + etc
L_ALS2xz 'xz rule (>=2 cell ALS)
L_TURBOT4 'same as locked candidates
L_TURBOT5 'includes basic, grouped, and backwing
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
L_STRONG4 'four strong links
L_ALS3xz 'xz rule (>=3 cell ALS)
L_ALS4xz 'xz rule (>=4 cell ALS)
L_TURBOT6 'includes basic, finned, and grouped
L_TURBOT7 'includes basic, grouped, and backwing
L_ALS1xy 'xy rule (A=1 cell ALS)
L_ALS2xy 'xy rule (A=2 cell ALS)
L_JELLYFISH 'includes basic and finned
L_SQUIRMBAG 'includes basic and finned
L_ALS3xy 'xy rule (A=3 cell ALS)
L_ALS4xy 'xy rule (A=4 cell ALS)
L_COLOR1 'type 1 includes basic and grouped
L_COLOR2 'type 2 includes basic and grouped
L_COLOR3 'type 3 includes basic and grouped
[/quote][/code]
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Postby ronk » Tue Mar 07, 2006 12:26 pm

Havard wrote:Is this the one you had in mind?

Code: Select all
 . . . | 9 . 9 | . 9 .
 . . . | . . 9 | . . 9
 . . . | . . . | . . .
 - - - + - - - + - - -
 . . 9 | x 9 x | 9 . .
 . 9 . | . + 9 | . . .
 . | 9 | x 9 x | 9 . .
 - | - + - - - + - - -
 . | 9 | 9 . 9 | . . .
 . 9 9 | .-9 . | . . .
 . . . | . . . | . 9 9

No, I missed that one. I was so focused on empty rectangles that I saw ...
Code: Select all
 . . . | 9 . 9 | . 9 .
 . . . | . . 9 | . . 9
 . . . | . . . | . . .
 - - - + - - - + - - -
 . . 9 | x 9 x | 9 . .
 . 9 + | . + 9 | . . .
 . . 9 | x 9 x | 9 . .
 - - - + - - - + - - -
 . . 9 | 9 . 9 | . . .
 . 9 9+| . 9-. | . . .
 . . . | . . . | . 9 9


Color me embarrassed, Ron
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Postby tarek » Tue Mar 07, 2006 2:03 pm

Mike Barker wrote:
Code: Select all
+------------------+------------------+---------------+
|    7 *3489     6 |  *389  489     2 |    5  48    1 |
|    5     2   348 |     1   48   368 |    7   9   36 |
|  189 *3489  1489 | *3689    7     5 |  *36  48    2 |
+------------------+------------------+---------------+
|    2     7  1389 |    89    5   189 |  369  36    4 |
|    6   489   489 |     7    3   489 |    2   1    5 |
|  *13  *349     5 |     2    6   149 |  *39   7    8 |
+------------------+------------------+---------------+
|  *39     6     2 |     4    1   *39 |    8   5    7 |
|    4     5     7 | -3689   89  3689 |    1   2   39 |
|  389     1   389 |     5    2     7 |    4  36  369 |
+------------------+------------------+---------------+


This is fantastic as there exists another finned Jellyfish in the same pond..... in Columns 3489 in 3s which kills also the 3 in r2c6

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Postby ravel » Tue Mar 07, 2006 3:14 pm

Just a peanut: i would recommend to put UR type 4 before 3 (ev. also type 2) , because i find, it is easier to spot. If type 3 can be used also, it can be handled as n-tupel afterwards.
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Postby Havard » Tue Mar 07, 2006 5:36 pm

Hi and thanks for that! That puzzle is amazing! It really looks like you need that finned jellyfish... never seen anything like it! I will keep pondering on it a while longer... Now how about those 38 grouped Turbot Fish... ?:D

Mike Barker wrote:The solving order is pretty much what you asked for. One question, why ER after 2-strong links?

Well, because the ER actually snatches one of the 2-strong-link patterns. The "fish" pattern that inspired the name "Turbot Fish" is covered by the ER, and hence you would get quite a few extra ER's. I wanted to see how many "really useful" ER's there were, and as it turned out, it was way more than I had expected...:)

Now for another statistic that would be very interesting: Every time a puzzle is solved, the most advanced (the one furthest down on the list) technique that was a part of solving that puzzle gets counted. So say a puzzle was solved using techniques up to (and including) the Empty Rectangle, then that technique gets +1. It would be very interesting to see how far down the list he goes for several thousand puzzles. I think this could give another perspective on the usefulness of the techniques.

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Postby aeb » Tue Mar 07, 2006 5:42 pm

tarek wrote:This is fantastic as there exists another finned Jellyfish in the same pond..... in Columns 3489 in 3s which kills also the 3 in r2c6

Yes - in this grid one can exclude 3s on (2,6), (4,7), (6,1), (8,4), (9,3).
Code: Select all
.*.|*..|...
..*|..*|..*
.*.|*..|*..
---+---+---
..*|...|**.
...|.3.|...
**.|...|*..
---+---+---
*..|..*|...
...|*.*|..*
*.*|...|.**

There are lots of different arguments. For example, look at (9,3). If there is a 3 there, then all 3s in rows 2,7,8 are covered twice by the five areas row 2, row 8, col 6, col 9, box 8. Since each of these areas contains a single 3, there can be at most 2.5 digits 3 in these three rows, a contradiction. Therefore (9,3)!3.
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Postby tarek » Tue Mar 07, 2006 7:38 pm

I think Havard that when the Finned fish doesn't follow a 2x2x2x.... formation or even harder when the fin is in a line with >2 vertices then the ER & the strong links will not catch it.

check also the other 3x2x2x2 finned jellyfish in the same formation but in 3s
Code: Select all
*--------------------------------------------------------*
| 7     3489  6    | 389   489   2    | 5     48    1    |
| 5     2    *348  |*1    -48    368  | 7     9    *36   |
| 189   3489  1489 |#3689  7     5    | 36    48    2    |
|------------------+------------------+------------------|
| 2     7    *1389 | 89    5     189  | 369  *36    4    |
| 6     489   489  | 7     3     489  | 2     1     5    |
| 13    349   5    | 2     6     149  | 39    7     8    |
|------------------+------------------+------------------|
| 39    6     2    | 4     1     39   | 8     5     7    |
| 4     5     7    |*3689  89    3689 | 1     2     39   |
| 389   1    *389  | 5     2     7    | 4    *36   *369  |
*--------------------------------------------------------*
Eliminating 3 From r2c6 (Finned Jellyfish in Columns 3489)


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Postby Mike Barker » Tue Mar 07, 2006 9:00 pm

Its taking me a little longer to get the Grouped Strong Link/Strong Cycle algorithm working, but I believe its running now. Please refer back to my earlier post regarding the Grouped Turbot Fish. The grouped strong link looks like it will take care of them. Turns out there are 5 kinds of grouped strong links that I've identifed
Code: Select all
a a a | A A A | . . b
. . . | . . . | . . b
. . . | . . . | . . b
------|-------|------
c c c | d D . | . . B
C C C | d D . | . . B
. . . | d D . | . . B
------|-------|------
e . e | . . . | . . .
. E . | . . . | . . .
. E . | . . . | . . .

The last one set me back a little. Preliminary results indicate that the grouped strong link/strong cycle (Jeff slap my hands if I'm abusing terminology too much) will handle regular/finned X-wings and double implication regular/finned swordfish, grouped Turbot Fish, 6-node continuous X-cycles, and 7-node Turbot Chains, as well as maybe a double implication jellyfish if one ever arises.

Also I found an error in the sequencing routine. Turns out that I was running the ER right after naked singles (Now I know why you shouldn't default parameter declarations.) Its still early, but with ER running after 2 strong links, I'm not seeing any ER's. I'll post the 10000 results in a couple of hours when they are done.
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Postby Havard » Tue Mar 07, 2006 9:24 pm

tarek wrote:I think Havard that when the Finned fish doesn't follow a 2x2x2x.... formation or even harder when the fin is in a line with >2 vertices then the ER & the strong links will not catch it.


Hi Tarek!

I know what you are saying, but my point is that I think that other, simpler eliminations can almost always be done insted of such complicated patterns as a finned jellyfish. In this case I can't find any other ways (yet...:)) but in most other cases I have seen, simpler reductions can be made. It is very interesting to note that after Mike Barker implemented the ER and the two and three strong links, the number of finned / Turbot / wing stuff went down a lot. Bottom line is that ER and strong links will never find the reduction in the same way a finned jellyfish will, but usually there will be a simpler reduction lurking somewhere...

In this case though, I stand corrected...:D The puzzle Mike has found is actually a really good one for showcasing the finned jellyfish! It is almost too bad that it needs a ALS-xz A=2 to solve completely after the reduction done by the finned fish, but this is after all the simplest ALS...:)

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Postby Havard » Tue Mar 07, 2006 9:29 pm

Mike Barker wrote:Also I found an error in the sequencing routine. Turns out that I was running the ER right after naked singles (Now I know why you shouldn't default parameter declarations.)Its still early, but with ER running after 2 strong links, I'm not seeing any ER's

Arrr. I knew it was too good to be true!:D

However, I would be very surprised if you could not find any!:)

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Postby Mike Barker » Tue Mar 07, 2006 11:09 pm

In answer to the question of puzzles which require an ER or a technique more advanced than a finned X-wing, grouped Turbo Fish, etc to solve. Here's one:

.8...5......64..9....97.48.27..8..4............6..18.57..23.....68...2.......9...
.8...5.....764859.6..97.48.27..8..4.8..........6..18.57..23..58.68...2.....8.9...

Code: Select all
+----------------------+------------------+--------------------+
|     49      8     49 |  *13    12     5 |   367   2367 *2367 |
|     13    123      7 |    6     4     8 |     5      9   123 |
|      6   1235    235 |    9     7    23 |     4      8   123 |
+----------------------+------------------+--------------------+
|      2      7   1359 |  *35     8    36 |  1369      4  -369 |
|      8  13459  13459 |  457  2569  2367 |  1379  12367  2367 |
|    349    349      6 |   47    29     1 |     8    237     5 |
+----------------------+------------------+--------------------+
|      7    149    149 |    2     3    46 |   169      5     8 |
|  13459      6      8 |  157    15    47 |     2     13   349 |
|   1345   1234    234 |    8   156     9 |  1367   1367  3467 |
+----------------------+------------------+--------------------+

They really aren't very common as the preceeding analysis indicated, but they do exist!
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Postby Mike Barker » Tue Mar 07, 2006 11:45 pm

Here are the results with the grouped strong link/strong cycle algorithm. It pretty much works as predicted - basically no double implication X-cycles except those identified by the GSLSC algorithm. Besides including the group algorithm, I also moved Unique Rectangles 4 before 2 as suggested and made a few other changes moving similar techniques closer to each other (I probably won't stop looking for strong links, look for xy-wings and then back to strong links). The grouping algorithm is better and simpler than I used before - there were 13 more solutions than previous (9738 out of 10000). My guess is that most result from the grouped 4 strong link algorithm.

My original concern with the strong link approach was that you still needed to do finned X-wings, grouped Turbot Fish, etc. That is no longer an issue! This algorithm handles the strong double implication X-cycle chains. For more on grouped strong nice loops go here. I'd say the next task is to expand it to triple implication chains or extending the algorithm to strong nice loops. I'm headed on vacation - I look forward to reading your progress next week.

Both swordfish were the 322 variety. The second has a finned swordfish as well:
.8.......27....1..1....3.5.9.18.5.......2..46....7........3.2..7.92......1...4.97
Code: Select all
+--------------------+-------------------+------------------+
|  3456     8   3456 |  1456  14569    2 |  3469    7    49 |
|     2     7   3456 |   456 -45689   68 |     1  368   489 |
|     1     9     46 |     7   *468    3 |  *468    5     2 |
+--------------------+-------------------+------------------+
|     9    46      1 |     8     46    5 |     7    2     3 |
|  *358    35      7 |    13      2   19 |  *589    4     6 |
|-34568     2  34568 |   346      7   69 |  -589   18  1589 |
+--------------------+-------------------+------------------+
| -4568   456   4568 |     9      3    7 |     2  168  1458 |
|     7  3456      9 |     2  -1568  168 |  3456  368   458 |
| *3568     1      2 |    56   *568    4 | *3568    9     7 |
+--------------------+-------------------+------------------+


8.7.93.......1....3....2..4.9854.1..17.....8...........5.7.............84..12..9.
8.7.93.5.....158733.58729.4.9854713.17.369.8553.2817.9.5.738...7..9543.8483126597
Code: Select all
+---------------+-----------+---------------+
|    8  14    7 |  46  9  3 |  26    5  126 |
|  269 -246  29 |  46  1  5 |   8    7    3 |
|    3 *16    5 |   8  7  2 |   9  *16    4 |
+---------------+-----------+---------------+
|   26   9    8 |   5  4  7 |   1    3   26 |
|    1   7   24 |   3  6  9 |  24    8    5 |
|    5   3  *46 |   2  8  1 |   7  *46    9 |
+---------------+-----------+---------------+
|  269   5  129 |   7  3  8 |  46 -1246  16 |
|    7 *26 *126 |   9  5  4 |   3 *126    8 |
|    4   8    3 |   1  2  6 |   5    9    7 |
+---------------+-----------+---------------+


Here are the results:
1) Naked Single (255656)
2) Hidden Single (183350)
3) Locked Candidate (13305)
4) Naked Pair (8923)
5) Two Strong Links (3758)
6) Naked Triple (2118)
7) ALS-xz rule with >=2 cells per ALS (2095)
8) Generalized WXYZ-wing (1990)
9) XY-wing (1418)
10) Generalized VWXYZ-wing (949)
11) Two Grouped Strong Links (919)
12) Hidden Pair (732)
13) ALS-xy rule with A=1 cell ALS (666)
14) ALS-xz rule with >=3 cells per ALS (547)
15) Generalized XYZ-wing (480)
16) ALS-xy rule with A=2 cell ALS (399)
17) Three Strong Links (350)
18) Three Grouped Strong Links (274)
19) 4-node XY-chain (265)
20) Type 4/4B Unique Rectangles (193)
21) Type 1 Unique Rectangles (180)
22) ALS-xy rule with A=3 cell ALS (140)
23) Hidden Triple (136)
24) ALS-xz rule with >=4 cells per ALS (87)
25) ALS-xy rule with A=4 cell ALS (58)
26) 5-node XY-chain (54)
27) Finned Swordfish (47)
28) 4-node XY-ring (42)
29) Type 3/3B Unique Rectangles (38)
30) Naked Quadruple (25)
31) Type 2/2B Unique Rectangles (24)
32) Four Grouped Strong Links (17)
33) SueDeCoq (16)
34) 5-node XY-ring (15)
35) Four Strong Links (7)
36) Empty Rectangle (5)
37) Swordfish (2)
38) Hidden Quadruple (1)
39) Jellyfish (1)
40) Finned Jellyfish (1)
41) Locked box/box = Locked box/line (0)
42) Grouped Empty Rectangle (0)
43) X-wing (0)
44) Squirmbag (0)
45) Finned X-wing (0)
46) Finned Squirmbag (0)
47) ALS-xz rule with >=1 cells per ALS = XY-wing, etc (0)
48) 4-node X-cycle = X-wing (0)
49) Turbot Fish (0)
50) 6-node X-cycle (0)
51) 7-node Turbot Chain (0)
52) Grouped 4-node X-cycle = Locked box/box (0)
53) Grouped Turbot Fish (0)
54) Grouped 6-node X-cycle (0)
55) Grouped 7-node Turbot Chain (0)
56) Broken Wing (0)
57) 7-node Broken Wing (0)
58) Grouped Broken Wing (0)
59) Grouped 7-node Broken Wing (0)
60) Simple Colouring Type 1 = X-cycle (0)
61) Simple Colouring Type 2 = X-cycle (0)
62) Simple Colouring Type 3 = X-cycle (0)
63) Grouped Simple Colouring Type 1 = X-cycles (0)
64) Grouped Simple Colouring Type 2 = X-cycles (0)
65) Grouped Simple Colouring Type 3 = X-cycles (0)

Here (I hope) is the order of techniques:
L_NAKED1
L_HIDDEN1
L_NAKED2
L_LOCKED1
L_NAKED3
L_HIDDEN2
L_HIDDEN3
L_NAKED4
L_HIDDEN4
L_TURBOT4 'same as locked candidates
L_STRONG2 'two strong links
L_EMPTY 'one strong link (empty rectangle)
L_XWING 'includes basic and finned
L_TURBOT5 'includes basic, grouped, and backwing
L_STRONG3 'three strong links
L_SWORDFISH 'includes basic and finned
L_TURBOT6 'includes basic, finned, and grouped
L_STRONG4 'four strong links
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_UNIQUE1 'type 1
L_UNIQUE4 'types 4 and 4B
L_UNIQUE2 'types 2 and 2B
L_UNIQUE3 'types 3 and 3B
L_ALS1xz 'same as xy-wing + xyz-wing + etc
L_ALS2xz 'xz rule (>=2 cell ALS)
L_ALS3xz 'xz rule (>=3 cell ALS)
L_ALS4xz 'xz rule (>=4 cell ALS)
L_TURBOT7 'includes basic, grouped, and backwing
L_JELLYFISH 'includes basic and finned
L_SQUIRMBAG 'includes basic and finned
L_ALS1xy 'xy rule (A=1 cell ALS)
L_ALS2xy 'xy rule (A=2 cell ALS)
L_ALS3xy 'xy rule (A=3 cell ALS)
L_ALS4xy 'xy rule (A=4 cell ALS)
L_COLOR1 'type 1 includes basic and grouped
L_COLOR2 'type 2 includes basic and grouped
L_COLOR3 'type 3 includes basic and grouped
Mike Barker
 
Posts: 458
Joined: 22 January 2006

Postby Havard » Wed Mar 08, 2006 2:01 pm

Hi Mike!

I must admit I am not quite following you anymore...

Can you explain how your two-grouped-strong-links algorithm works, with some examples?

thanks!

Havard
Havard
 
Posts: 377
Joined: 25 December 2005

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