The Effortless Extremes thread

Everything about Sudoku that doesn't fit in one of the other sections

Postby ronk » Sat Jul 01, 2006 11:36 am

Mike Barker wrote:63 ALS xz-rule (A=2 cells)
......9.........6742....3...5...4..3...1.2....87.9......2.81.....64...5.......274

FWIW there is also a smaller ALS xz-rule (five cell total) that solves the puzzle.
Code: Select all
 67    67    1358  | 35    1345  358   | 9     1248  1258
 13589 139   13589 | 2     1345  3589  | 1458  6     7
 4     2     1589  | 5679  156   56789 | 3     18    158
-------------------+-------------------+------------------
-1269  5    #19    | 8    #67    4     |#17    129   3
*369  *369   4     | 1    -3567  2     | 578   89    568
 1236  8     7     | 356   9     356   | 145   124   1256
-------------------+-------------------+------------------
 57    4     2     | 57    8     1     | 6     3     9
 379   379   6     | 4     2     379   | 18    5     18
 13589 139   13589 | 3569  356   3569  | 2     7     4

x=9, z=6, r4c1<>6 and r5c5<>6
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Postby Mike Barker » Sat Jul 01, 2006 5:39 pm

Ron, thanks. My solver looks to keep the "A" set as small as possible, but doesn't look for the smallest combination of "A" and "B" sets. Yours is the better solution. Here are some more puzzles. Many are included because they only require singles in addition to the one "extreme" technique. Others are to try to fill out the set of puzzles. I've added what I've called "advanced colours" to my solver. As I've implemented it, they are nice loops with only strong links. I'm not sure if this is exactly "advanced colours" so I'll change the name if there is an issue. Anyway I've included a few of those as well. In addition there are a couple of wings. They don't appear to be affected by the new class of eliminations that RW has come up with, which are ALS with hidden sets. I'm anxious to see what he does with these! As far as the Advanced BUG-lite puzzle. It actually requires two so is not valid for this data set.
Code: Select all
14 Empty Rectangle
4....3.8.5.....6.9.1...7...6..3..5.8..26..........1....765.........8..9.......175 >column

16 XYZ-wing
..4....87..38.2..6.6...5....5..9.1.....3.........1.8.2..5....6....1267....1...9..

21 UR+1 (Type 1)
......7..6..8...5...8.91..2...6...4..54......82.3..........8..9.3...7.6.5.9.3....

24 UR+2X (Type 3/3B)
..7....4..2..67...9..4....1...3.1.....4.2.67.........8......35...5..9....3.6.4.2.
.8...5.7..4..3.9...1.....2..2..9....7...48..563..5.8.....3.2.........1.........34

25 UR+2rd
67.5.8.9.............3...2.5.......9....4.2...4...1.8...147...3.2...31..49....5..
.........24..5193.9..4..........3.7..8..7.421....1...5.5..94....28...6..36...2...
.....8.1...51..7.....92........6.....73...94....8..6.742...5....3..4.16...1....5.

29 5-node XY-chain
.29....15........315...7..4...1....84......9....5.....6..31..4..8....73.5...6.8..

31 6-node XY-chain
...95...2872..6.....34.....4......6..........231.4..5...8.2....9..7......5...86..

19 WXYZ-wing
.....1.54.9..3.2..374.....1....2..8.5.946....2.8.1.......9......4.6...1...6.....5

34 UVWYXZ-wing
3.....1..7.5....2.......4.6..8.....193..........51.34.4.32.9.....21...6.....47... >
..8....17.......9.164....5.....32..8....7.4....34.1.2.8....5...5.....236.96..3... >

35 Advanced Colouring3
1...6.....8.5.7....6..8.3.....9..23.2...14..........967.6.....4.......1..9.2.6..7 (6-node XY-chain)
.4.9...2........612....45....63.28....7..1..25.........2..5...395..48.........18. (6-node XY-chain)
.....52.8.87.21..3..1.....6.4.1.2...8...965......5....5..........9.7.........3.42 >
5.....67.2..9..1.........9..1.4..8.......7.6946.2...1..2...3....5..14......8....5 >
2...4...8...2...9...56.....3.....7.4..7.2...........5..2.4..6..........7.569..31. >

43 ALS xz-rule
...4.......5....3.....3.8.1..9..8..5.5...3..28.6..14..24......8....7.....1.26.... >
..8....9519.....7..5.6.....2.37....1....2...4.....3..9.3....1....647.....8..5.2.. >
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Postby ronk » Sat Jul 01, 2006 8:35 pm

Mike Barker wrote:I've added what I've called "advanced colours" to my solver. As I've implemented it, they are nice loops with only strong links. I'm not sure if this is exactly "advanced colours" so I'll change the name if there is an issue. Anyway I've included a few of those as well.

That's definitely one of the techniques within advanced coloring. For your advanced coloring puzzles, I was able to find nice loops of the following form ... r6c2-1-r5c2=1=r5c7=3=r5c5=7-r6c5=9=r6c2, implies r6c2<>1 ... in all except this one ...
2...4...8...2...9...56.....3.....7.4..7.2...........5..2.4..6..........7.569..31. >

... so I need a hint. The nice loop above can be found in:
29 5-node XY-chain
.29....15........315...7..4...1....84......9....5.....6..31..4..8....73.5...6.8..
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Postby Mike Barker » Sat Jul 01, 2006 9:54 pm

I'm not quite sure how much of a hint you want. Here's what I think the link is:
Code: Select all
Advanced Colouring: r6c7=2=r6c3=4=r2c3-4-r3c2=4=r3c7~4~r6c7 => r3c7<>2
+----------------------+----------------------+------------------+
|     2   13679    139 |  1357      4   13579 |    15  367     8 |
|   168  134678  *1348 |     2  13578   13578 |   145    9  1356 |
|   189 *134789      5 |     6  13789   13789 |  -124  237    13 |
+----------------------+----------------------+------------------+
|     3    1689   1289 |   158    159    1569 |     7   26     4 |
|     5   14689      7 |   138      2   13469 |   189   36  1369 |
|  1689   14689 *12489 |  1378   1379  134679 | *1289    5  1369 |
+----------------------+----------------------+------------------+
|     7       2    139 |     4    135     135 |     6    8    59 |
|   189    1389   1389 |   135      6       2 |    59    4     7 |
|     4       5      6 |     9     78      78 |     3    1     2 |
+----------------------+----------------------+------------------+
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Postby ronk » Sun Jul 02, 2006 3:10 pm

Mike Barker wrote:Here's what I think the link is:
Advanced Colouring: r6c7=2=r6c3=4=r2c3-4-r3c2=4=r3c7~4~r6c7 => r3c7<>2

That's certainly valid, but I'm surprised your solver comes up with that since here ...
you wrote:I modified my nice loop algorithm to implement a form of advanced colors - nice loops with only strong links.

By my count, this nice loop has two weak inferences -- one of which is from an actual weak link -- wihich is easier to see when written ...

r3c7-2-r6c7=2=r6c3=4=r2c3-4-r3c2=4=r3c7 => r3c7<>2

I also note this was found by L_ADVANCED3 ... meaning 3 "links". Now there happens to be both 3 links in the nice loop (excluding the links to the discontinuity) and 3 conjugate chains (one in 2s and two in 4s). Which, if either, actually applies for your "3"?

Mike wrote:I'm not quite sure how much of a hint you want.
Usually, only the exclusion is sufficient.

[edit: removed errant BBcode]
Last edited by ronk on Sun Jul 02, 2006 7:51 pm, edited 1 time in total.
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Postby Ocean » Sun Jul 02, 2006 6:33 pm

Here are three puzzles where eliminations can be done by xy-rings. The two first puzzles each have a short and simple ring - very easy to find because there are so few bivalue cells. The third contains two functionally equal (overlapping) 6-rings, which both give the same 3 eliminations.

Code: Select all
# XY-ring
 *-----------*
 |...|...|...|
 |..1|.2.|3..|
 |.3.|1.4|.5.|
 |---+---+---|
 |..6|...|2..|
 |.4.|...|.7.|
 |..8|...|9..|
 |---+---+---|
 |.1.|3.7|.4.|
 |..2|.9.|8..|
 |...|...|...|
 *-----------*

# XY-ring
 *-----------*
 |...|...|...|
 |..1|...|2..|
 |.2.|3.4|.5.|
 |---+---+---|
 |..2|.6.|7..|
 |...|4.8|...|
 |..6|.1.|9..|
 |---+---+---|
 |.8.|2.5|.4.|
 |..7|...|1..|
 |...|...|...|
 *-----------*

# XY-ring
 *-----------*
 |...|..1|2..|
 |..3|.42|...|
 |5..|...|.6.|
 |---+---+---|
 |72.|...|...|
 |.6.|...|.5.|
 |...|...|.84|
 |---+---+---|
 |.1.|...|..8|
 |...|75.|9..|
 |..6|4..|...|
 *-----------*
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Postby Mike Barker » Sun Jul 02, 2006 9:03 pm

The way I was counting there are three strong links in
    r6c7=2=r6c3=4=r2c3-4-r3c2=4=r3c7~4~r6c7 => r3c7<>2
    r6c7=2=r6c3
    r6c3=4=r2c3
    r3c2=4=r3c7
The first two connect directly. The last two are disjoint and the connection is shown with a weak link. Since this does not involve a bivalue or ALS, I considered this to be part of connecting the strong links so valid for coloring. In the same manner I would consider an X-wing (which is a special form of advanced coloring) to be composed of 2 disjoint strong links. This may be confusing and I can modify my description if it is an issue. I guess I find the POV useful since it doesn't matter whether the link is strong or weak inference, it just connects two strong links with the same label, period.

As far as the elimination the discontinuity is as shown. As you know, the start and end nodes share a house, but cannot be linked together. This can allow "4" to be eliminated from r6c7 (if one existed) and "2" from r3c7. For this reason I prefer the way I wrote the loop, but both ways are valid.
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Postby ronk » Mon Jul 03, 2006 12:27 am

Mike Barker wrote:As far as the elimination the discontinuity is as shown. As you know, the start and end nodes share a house, but cannot be linked together. This can allow "4" to be eliminated from r6c7 (if one existed) and "2" from r3c7. For this reason I prefer the way I wrote the loop, but both ways are valid.

I arrived at my "advanced coloring" perspective via Myth Jellies's POM vulnerable pairs after having read numerous posts by Lummox JR on the Programmer's Forum. You might find the MJ thread -- only 3 pages -- an interesting read.

In brief, the technique finds pure bilocation chains with a discontinuity. One chain segment has an even number of links and all other segments have an odd number of conjugate (strong) links. A segment has conjugate links for one digit only, and digits may re-occur in non-adjacent segments. Then the colored digit at each end of the even-length segment (and all similarly colored digits in that segment) may be excluded. Of course, since it's a conjugate chain, excluding just one will exclude the others consequentially.

Mike Barker wrote:The last two [edit: strong links] are disjoint and the connection is shown with a weak link. Since this does not involve a bivalue or ALS, I considered this to be part of connecting the strong links so valid for coloring.

That's definitely less restrictive for the odd-length segments and I think I can incorporate that into my solver with little difficulty.

Mike Barker wrote:In the same manner I would consider an X-wing (which is a special form of advanced coloring) to be composed of 2 disjoint strong links. This may be confusing and I can modify my description if it is an issue.

I only take issue with considering an x-wing as any form of advanced coloring. It is "multi-coloring" in the sense of multiple chains -- only one link in each of two chains, in this case -- of a single digit. "Advanced coloring" and "super-coloring" always involve chains of multiple digits AFAIK.
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Postby Mike Barker » Mon Jul 03, 2006 4:17 am

Thanks Ron, I'll look into the post. In the mean time here are some interesting BUGs and a few other techniques. Again I've included techniques already submitted, but which here only require singles. Let me know if this is of interest.
Code: Select all
12 Franken Swordfish
.......8..4.3.......6.75..97.....6....8....7..9.1....5....6.9.12.7.......39.2.4..  row
..7..219...5.....8...9..2...3.6...2......7..9....25...6..23....9..4..31.5.....6.. ~row

14 BUG+2x (Type 2/2B)
1...6....6..8....5....52....9....4.886.....9......3.1.7.2.4..6....5.83...5.......

14 BUG+2X (Type 3/3B)
.4.36.1........7...8.19...22.5............3.....2....661.....9895..4....7..6..2..

15 UR+1 (Type 1)
...1.49..7..9..42....7..6...6........2...5...8...3...4.8.....7....5.....231.6...5

16 UR+2x (Type 2)
.9...4...4..38........5.1.7..4...9.......9..3...2.3.64.8.........671.8.2..2....3.

20 UR+2kx
1239........1...5...7........83..64...9..8..5.1..65.3.8...9...3....21.....5...8.4

22 Strong Link BUG Eliminations
....6..3..849...........8.1...2.8.....5.....4..6..3..71...8..7.2...4...5.7...14.. >
.148.29...........8.26......78..4..9.2.....4....5.1...56.......7...9..16.8..3...7 ~
.9...1..5.4....37..1.2....96.....5...5..2.9.7...3.....8....2.......597684.9.6.... >

26 4-node XY-ring
8.4.52.1.7.2.4.......1...49.3.7.....5......82.....1.3...76..........51.6....7.... ~

32 5-node XY-ring
.2...16....79.8.......5.4.........37381.......65...8.1..6872....4......8..9....2. >

33 VWXYZ-wing
...1......3..5..1241.23..5......2..316......9.2.37...8..9.....5..8......24....8..
.....56.2.3..98.17..9........1..3...2..4...8....8......1.5.....3...42.5...6...9.1 ~

44 7-node XY-chain
....12...8...7........953.7..4...78..5....1......5...9.1.....3..2..8.9...73.4..6. ~
.34...6..8.5.......2.5.79.....865...7..2..5.1........2.596..2....6....4.3........ >
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Postby Mike Barker » Tue Jul 04, 2006 4:25 am

In trying to generalize RW's elimination in the following puzzle
Code: Select all
.4..9...76.85....2..5....8....2..7..41..36........1.3.2..3...1.9...4.........9.2.
+---------------------+-------------------+--------------------+
|    13      4      2 |   168      9   38 |   1356   56      7 |
|     6    379      8 |     5     17  347 |   1349   49      2 |
|   137    379      5 |  1467      2  347 |  13469    8   3469 |
+---------------------+-------------------+--------------------+
|  *358  *3568    369 |     2    #58 -458 |      7  469      1 |
|     4      1     79 |   789      3    6 |      2   59    589 |
|  *578      2    679 |  4789    578    1 |   4689    3   4689 |
+---------------------+-------------------+--------------------+
|     2   5678    467 |     3   5678  578 |  45689    1  45689 |
|     9   3568    136 |    18      4    2 |   3568    7   3568 |
|  3578  35678  13467 |   178  15678    9 |  34568    2  34568 |
+---------------------+-------------------+--------------------+

I was thinking that this approach might be a precursor to using almost hidden subsets in a manner similar to ALS xz-rule, etc. So far that generalization has eluded me. What seems to be happening here is a case of "almost locked candidates". Consider a box with 2 candidates, "a" and "b", existing only in a box-line plus one cell and both candidates in the line restricted common to an ALS:
Code: Select all
+--------------------------+--------------------------
|                         a,b
| (a)(b)X (a)(b)Y (a)(b)Z=====ALS(a,b,...) (a)(b)U ...
|   abW       -       -    |
|    -        -       -    |
+--------------------------+--------------------------

where "(a)" implies "a" is optional and "-" implies "a" and "b" are not candidates in the cell. Then the cell in the box, "abW", must be either "a" or "b". In addition, "a" and "b" can be eliminated from all other cells, "(a)(b)U", common to the box-line and the ALS. In the above example r4c6<>5,8 and r6c1<>7 where ALS(a,b,...)=r4c5="58". Interestingly this is the same box structure the RW utilized in his UR+2rx elimination.

I'm hoping that this is an overly restrictive view of the elimination and that something exists which allows for more general eliminations - we'll see.
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Postby RW » Tue Jul 04, 2006 8:32 am

I'm almost done with checking your submitted puzzles, and then I find a new bunch...

Mike Barker wrote:Again I've included techniques already submitted, but which here only require singles. Let me know if this is of interest.


As this thread is intended for people who wish to learn to find advanced eliminations and use them in manual solving, more puzzles are always welcome. However, they do not need to contain only singles apart from the advanced move. If I had a scoring system, I would probably give more points for the allowed "non-advanced" moves also (locked candidates, subsets up to triples). If you wish to improve your puzzles, aim for symmetry. In fact, as the amount of submitted puzzles is a bit overloading for me right now, I'll require some kind of symmetry from now on (I need some time off to watch football also:) ).

I'll add the puzzles to the list and give feedback on them all when I've checked them all.

I wrote:The easier this "advanced" move is to recognize for a human solver, the better.


Ocean, I cannot imagine a puzzle that fits this description better than your first XY-ring puzzle! Good work, I'll add it to the top of the XY-ring category.:D

Mike Barker wrote:Interestingly this is the same box structure the RW utilized in his UR+2rx elimination.


I suppose it is because structures like that are easier to spot without pencilmarks. I've noticed that I have a different view on many techniques as I never use pms in my own solving (but I do use them when I check the puzzles for this thread).

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Postby RW » Wed Jul 05, 2006 7:05 pm

I've added most of the puzzles to the list. Again, some failed the test and some were placed in another category. I found some with really interesting (and actually quite simple) BUG+3 solutions and even a BUG+4 that could be used to solve the puzzles, so I put those puzzles in the BUG category. At the moment there's already quite a lot of XY-chain puzzles, so I preferred the BUG solutions, even though they might not be that much easier than the XY-chains.

The puzzles that didn't make it to the list were:

Code: Select all
14 Empty Rectangle
4....3.8.5.....6.9.1...7...6..3..5.8..26..........1....765.........8..9.......175


31 6-node XY-chain
...95...2872..6.....34.....4......6..........231.4..5...8.2....9..7......5...86..


34 UVWYXZ-wing
3.....1..7.5....2.......4.6..8.....193..........51.34.4.32.9.....21...6.....47...
..8....17.......9.164....5.....32..8....7.4....34.1.2.8....5...5.....236.96..3...

32 5-node XY-ring
.2...16....79.8.......5.4.........37381.......65...8.1..6872....4......8..9....2. >

44 7-node XY-chain
.34...6..8.5.......2.5.79.....865...7..2..5.1........2.596..2....6....4.3........


There's still some that I'm a bit confused with, but I'll look at them again later, now I must go watch the French get their asses kicked by the Portuguese! Go Carcul!!

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Postby JPF » Wed Jul 05, 2006 9:17 pm

RW wrote:... but I'll look at them again later, now I must go watch the French get their asses kicked by the Portuguese! Go Carcul!!
RW
:D
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Postby tso » Thu Jul 06, 2006 4:18 pm

It doesn't appear there have been any "Effortless Extemes" that are labeled as standard Swordfish, just the finned and franken versions.

I'm not completely clear on what qualifies as an EE, but this puzzle:

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


... posted minutes ago in a thread called "Best Ever Swordfish Sudoku?" requires only Swordfish and singles. However, the Swordfish is needed as the very first move, making it not-so-effortless. I infer from the rules that a puzzle is supposed to need more than merely the one advanced tactic to qualify, but many of the puzzles in the thread contradict this. Someone else decide if this belongs -- and either way, we need a Swordfish on the list.
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Postby daj95376 » Thu Jul 06, 2006 6:16 pm

tso,

This is a great diagonal puzzle that's nicely solved by multiple Swordfish and Naked/Hidden Singles. Too bad it violates Rule #1 for this thread.

BTW: I've contacted RW about some of the original puzzles that were improperly classified or failed to meet Rule #1. He's been updating his original post to make corrections. You might want to review them again.
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