Prof, maybe wikipedia.org's Sudoku entry, and/or sudopedia.org, can help with answering the Bowman's Bingo question.

Jeff

My sense of Bowman Bingo is that it is a rather complicated method (compared to today's advanced methods) that can be looked at as a combination of tabling & forcing nets/chains. It dates back to 2005 at a time when one had to resort to forcing chains and trial & error methods such as Nishio & Ariadne's Thread to solve more advanced puzzles and by today's standards is considered to be under the broad umbrella of trial & error methods.

In any event, those of us who are into using the most elegant methods to solve the most difficult puzzles these days would not think of using a method such as Bowman Bingo. It is pretty much part of Sudoku yesteryear.

Extreme #71 is rated 9.0 by Sudoku Explainer which indicates it most likely requires some form of multi-inference AIC to solve.  I ran it through my solver which in addition to a number of ALS and Grouped AICs used two multi-inference AICs (a Kraken Box (in this case the strong inference set is box 1 contains 1) and a Kraken Blossom (in this case r7c6 must contain a digit)).  There are some players (or at least one) that would argue that these are no better than trial and error, but I don't agree.  The eliminations are fairly straight forward, manually finding them is the challenging part (okay so I'm known for understatement).  As far as how the eliminations work, they are basically the same as an AIC, but where all nodes in an AIC link to only one node before and one node after, multi-inference nodes link to one node before and two or more nodes after.  The easiest place to see this is with the Kraken Blossom which, in this case, is a trivalue cell.  In a simple AIC links would be to bivalue cells.

Unfortunately my solve doesn't have an option to output in Eureka notation and this listing would take a while to convert to Eureka notation.  I've converted the kraken eliminations.  If you are having a problem with the rest, I would be happy to convert specific eliminations or to take your pencil mark grid and suggest a next step converted to Eureka notation.  Note I didn't show single eliminations in the summary.

Sudopedia is a great source for descriptions of techniques.  If your interested in reading many of the original posts developing the techniques you can try the Collection of Solving Techniques

 2 . . | 1 . 9 | 4 . .
 . 8 . | . . 3 | . . .
 . . 7 | . 8 . | . . .
 ------+-------+------
 9 2 . | 5 . . | . . .
 6 . . | . . . | . . 1
 . . . | . . 4 | . 2 5
 ------+-------+------
 . . . | . 2 . | 8 . .
 . . . | 9 . . | . 7 .
 . . 8 | 4 . 1 | . . 3

+-------------------------+------------------+---------------------+
|     2      356     356  |   1    567    9  |     4   3568   678  |
|    15        8    1569  | 267      4    3  | 15679   1569  2679  |
|  1345   134569       7  |  26      8   56  | 13569  13569   269  |
+-------------------------+------------------+---------------------+
|     9        2     134  |   5   1367   67  |   367   3468  4678  |
|     6     3457     345  |   8    379    2  |   379    349     1  |
|     8      137      13  | 367  13679    4  |  3679      2     5  |
+-------------------------+------------------+---------------------+
| 13457  1345697  134569  | 367      2  567  |     8  14569   469  |
|  1345    13456       2  |   9    356    8  |   156      7    46  |
|    57     5697       8  |   4    567    1  |     2    569     3  |
+-------------------------+------------------+---------------------+

Locked Column Line/Box: r56c2 => r79c2<>7
Row Finned X-Wing: r3c2789|r9c28 => r2c8<>9
Column Finned X-Wing: r27c3|r237c9 => r2c7<>9
3-element Grouped Advanced Colouring: r4c5 =1= r6c5 =9= r6c7 =6= r6c45 ~6~ r4c5 => r4c5<>6
3-element Strong Nice Loop: r4c6 -6- r6c45 =6= r6c7 =9= r6c5 ~7~ r4c6 => r6c5<>7
4-element Strong Nice Loop: r4c6 -6- r4c789 =6= r6c7 =9= r6c5 =1= r4c5 ~7~ r4c6 => r4c5<>7
UR+2C/2SL (r46c35=13)  => r4c5<>3,r4c3<>1,r6c5<>1
UR+3C/2SL (r23c49=26)  => r2c9<>6
A=3 cell ALS xz-rule: r2c178 -7- r456c7 => r3c7<>6
A=3 cell ALS xz-rule: r2c178 -7- r2378c9 => r1c9<>6

4-valued/2-element Kraken Box (r13c2|r12c3=6) => r1c5<>6
(6)r1c23
||
(6)r3c2 - r12c3 = r7c3 - r7c46 = r89c5
||
(6)r2c3 - (6=7)r2c178 - r1c9 = (7-6)r1c5
+------------------------+-----------------+--------------------+
|     2     356*   356*b |   1  57-6c   9  |    4   3568    78c |
|    15c      8   1569*b | 267     4    3  | 1567c   156c  279  |
|  1345  134569*      7  |  26     8   56  | 1359  13569   269  |
+------------------------+-----------------+--------------------+
|     9       2      34  |   5     1   67  |  367   3468  4678  |
|     6    3457     345  |   8   379    2  |  379    349     1  |
|     8     137      13  | 367   369    4  | 3679      2     5  |
+------------------------+-----------------+--------------------+
| 13457  134569  134569b | 367b    2  567b |    8  14569   469  |
|  1345   13456       2  |   9   356b   8  |  156      7    46  |
|    57     569       8  |   4   567b   1  |    2    569     3  |
+------------------------+-----------------+--------------------+

XY-wing => r5c5<>7
3-element Nice Loop: r9c1 -7- r9c5 =7= r1c5 =5= r3c6 ~5~  => r3c1<>5
A=2 cell ALS xz-rule: r5c58 -4- r4c6789 => r5c7<>3
A=3 cell ALS xz-rule: r2c178 -7- r5c3578 => r2c3<>5
4-element Strong Nice Loop: r3c6 =5= r1c5 =7= r9c5 -7- r9c1 -5- r2c1 =5= r2c78 ~5~  => r3c78<>5
Column Finned X-Wing: r2789c1|r28c7 => r8c2<>5
3-element Nice Loop: r2c1 -5- r2c7 =5= r8c7 =1= r7c8 ~1~  => r7c1<>1,r2c8<>1
3-element Nice Loop: r7c8 =1= r8c7 =5= r2c7 -5- r2c8 ~6~ r7c8 => r7c8<>6
Overlap 4-element Grouped Nice Loop: ALS:r9c15 -6- ALS:r568c5 -5- r8c7 =5= r2c7 -5- r2c1 =5= r789c1 ~5~  => r9c2<>5
+------------------------+-----------------+-------------------+
|     2     356     356  |   1    57    9  |    4  3568    78  |
|    15*      8     169  | 267     4    3  | 1567*   56   279  |
|   134  134569       7  |  26     8   56  |  139  1369   269  |
+------------------------+-----------------+-------------------+
|     9       2      34  |   5     1   67  |  367  3468  4678  |
|     6    3457     345  |   8   39*b   2  |   79   349     1  |
|     8     137      13  | 367  369*b   4  | 3679     2     5  |
+------------------------+-----------------+-------------------+
| 3457*c 134569  134569  | 367     2  567  |    8  1459   469  |
| 1345*c   1346       2  |   9  356*b   8  |  156*    7    46  |
|   57*c   69-5       8  |   4   567*   1  |    2   569     3  |
+------------------------+-----------------+-------------------+
3-element Grouped Nice Loop: r9c2 -9- ALS:r29c8 -6- r1c8 =6= r1c23 ~6~  => r3c2<>6
+------------------------+----------------+-------------------+
|    2     356*c   356*c |   1   57    9  |    4  3568*   78  |
|   15        8     169  | 267    4    3  | 1567   56*b  279  |
|  134  13459-6       7  |  26    8   56  |  139  1369   269  |
+------------------------+----------------+-------------------+
|    9        2      34  |   5    1   67  |  367  3468  4678  |
|    6     3457     345  |   8   39    2  |   79   349     1  |
|    8      137      13  | 367  369    4  | 3679     2     5  |
+------------------------+----------------+-------------------+
| 3457   134569  134569  | 367    2  567  |    8  1459   469  |
| 1345     1346       2  |   9  356    8  |  156     7    46  |
|   57       69*      8  |   4  567    1  |    2  569*b    3  |
+------------------------+----------------+-------------------+

3-valued/2-element Kraken Blossom (r7c6=567) => r3c7<>9
(5)r7c6 -5- r3c6 -6- r3c49 -9-
||
(6)r7c6 - r89c5 = (6-9)r6c5 = r6c7
||
(7)r7c6 - r4c6 = r4c79 - (7=9)r5c7
+-----------------------+----------------+-------------------+
|    2     356     356  |   1   57    9  |    4  3568    78  |
|   15       8     691  | 267    4    3  | 1567    56   279  |
|  134   13459       7  |  26b   8   56b | 13-9  1369   269b |
+-----------------------+----------------+-------------------+
|    9       2      34  |   5    1   67d |  367d 3468  4678d |
|    6    3457     345  |   8   39    2  |   79d  349     1  |
|    8     137      13  | 367  369c   4  | 3679c    2     5  |
+-----------------------+----------------+-------------------+
| 3457  134569  134569  | 367    2  567* |    8  1459   469  |
| 1345    1346       2  |   9  356c   8  |  156     7    46  |
|   57      69       8  |   4  567c   1  |    2   569     3  |
+-----------------------+----------------+-------------------+

Locked Column Box/Box: r379c8|r237c9 => r5c8<>9
Continuous 3-element Strong Nice Loop: r4c3 =3= r4c78 -3- r5c8 -4- r5c23 =4= r4c3 => r6c7<>3
A=2 cell ALS xz-rule: r78c9 -9- r12459c8 => r7c8<>4,r4c9<>4
UR+2D/1SL (r45c38=34)  => r5c3<>4
B=2 cell ALS xy-rule: r34c6 -5- r1c59 -8- r56c7|r4c9 => r4c7<>7
4-element Grouped Nice Loop: r3c7 -3- ALS:r4c679 -8- ALS:r1c59 -5- r3c6 =5= r3c2 ~1~ r3c7 => r3c2<>1
+-----------------------+----------------+-------------------+
|    2     356     356  |   1  57*c   9  |    4  3568   78*c |
|   15       8     691  | 267    4    3  | 1567    56   279  |
|  134  3459-1*      7  |  26    8   56* |   13* 1369   269  |
+-----------------------+----------------+-------------------+
|    9       2      34  |   5    1  67*b |  36*b 3468  678*b |
|    6    3457      35  |   8   39    2  |   79    34     1  |
|    8     137      13  | 367  369    4  |  679     2     5  |
+-----------------------+----------------+-------------------+
| 3457  134569  134569  | 367    2  567  |    8   159   469  |
| 1345    1346       2  |   9  356    8  |  156     7    46  |
|   57      69       8  |   4  567    1  |    2   569     3  |
+-----------------------+----------------+-------------------+
Overlap 4-element Grouped Nice Loop: r3c7 -3- ALS:r4c679 -8- ALS:r1c59 -5- ALS:r3c46789 ~1~  => r3c1<>1,r2c7<>1
+-----------------------+----------------+---------------------+
|    2     356     356  |   1  57*c   9  |     4   3568   78*c |
|   15       8     691  | 267    4    3  | 567-1     56   279  |
| 34-1    3459       7  | 26*d   8  56*d |   13*d 1369*d 269*d |
+-----------------------+----------------+---------------------+
|    9       2      34  |   5    1  67*b |   36*b  3468  678*b |
|    6    3457      35  |   8   39    2  |    79     34     1  |
|    8     137      13  | 367  369    4  |   679      2     5  |
+-----------------------+----------------+---------------------+
| 3457  134569  134569  | 367    2  567  |     8    159   469  |
| 1345    1346       2  |   9  356    8  |   156      7    46  |
|   57      69       8  |   4  567    1  |     2    569     3  |
+-----------------------+----------------+---------------------+
3-element Grouped Advanced Colouring: r7c3 =9= r2c3 =1= r2c1 =5= r789c1 ~5~ r7c3 => r7c3<>5
5-element Grouped Nice Loop: r7c8 =1= r3c8 -1- r3c7 -3- ALS:r4c679 -8- ALS:r1c59 -5- r3c6 =5= r7c6 ~5~ r7c8 => r7c8<>5
+----------------------+----------------+------------------+
|    2     356    356  |   1  57*c   9  |   4  3568   78*c |
|   15       8    691  | 267    4    3  | 567    56   279  |
|   34    3459      7  |  26    8   56* |  13* 1369*  269  |
+----------------------+----------------+------------------+
|    9       2     34  |   5    1  67*b | 36*b 3468  678*b |
|    6    3457     35  |   8   39    2  |  79    34     1  |
|    8     137     13  | 367  369    4  | 679     2     5  |
+----------------------+----------------+------------------+
| 3457  134569  13469  | 367    2  567* |   8  19-5*  469  |
| 1345    1346      2  |   9  356    8  | 156     7    46  |
|   57      69      8  |   4  567    1  |   2   569     3  |
+----------------------+----------------+------------------+
Continuous 4-element Strong Nice Loop: r1c5 =5= r3c6 -5- r7c6 =5= r7c12 -5- r9c1 -7- r9c5 =7= r1c5 => r8c1<>5
6-element Nice Loop: r4c7 =3= r3c7 =1= r8c7 =5= r8c5 -5- r1c5 -7- r1c9 -8- r4c9 =8= r4c8 ~3~ r4c7 => r4c8<>3
4-element Nice Loop: r7c8 =1= r3c8 -1- r3c7 -3- r4c7 =3= r4c3 =4= r7c3 ~1~ r7c8 => r7c3<>1
7-element Nice Loop: r3c7 =1= r8c7 =5= r8c5 -5- r1c5 -7- r1c9 -8- r4c9 =8= r4c8 =4= r4c3 =3= r4c7 ~3~ r3c7 => r3c7<>3
4-element Nice Loop: r8c2 =1= r6c2 -1- r6c3 =1= r2c3 =9= r3c2 -9- r9c2 ~6~ r8c2 => r8c2<>6
5-element Grouped Advanced Colouring: r8c2 =1= r6c2 -1- r6c3 =1= r2c3 =9= r3c2 =4= r3c1 =3= r78c1 ~3~ r8c2 => r8c2<>3
XYZ-wing: r8c12, r3c1 => r7c1<>4
3-element Nice Loop: r7c2 =4= r7c9 =9= r9c8 -9- r9c2 ~6~ r7c2 => r7c2<>6
4-element Advanced Colouring: r6c3 =1= r6c2 -1- r8c2 =1= r8c1 =3= r8c5 -3- r7c4 =3= r6c4 ~3~ r6c3 => r6c3<>3
Naked Column Pair: r38c1 => r7c1<>3
Naked Block Pair: r79c1 => r7c2<>5
Locked Row Box/Box: r1c235|r3c26 => r1c8<>5
5-element Grouped Advanced Colouring: r7c4 =3= r8c5 -3- r8c1 =3= r3c1 =4= r3c2 =5= r3c6 =6= r23c4 ~6~ r7c4 => r7c4<>6
Advanced 2-line BUG Lite (SL:r6c7 =9= r6c5, SL:r6c2 =7= r5c2): r56c257 => r6c7<>7
+----------------+----------------+-----------------+
|  2   356  356  |   1   57    9  |    4  368   78  |
|  1     8   69  | 267    4    3  |  567   56  279  |
| 34  3459    7  |  26    8   56  |    1  369  269  |
+----------------+----------------+-----------------+
|  9     2    4  |   5    1   67  |    3   68  678  |
|  6   357*  35  |   8   39*   2  |   79*   4    1  |
|  8    37*   1  | 367  369*   4  | 69-7*   2    5  |
+----------------+----------------+-----------------+
| 57   349  369  |  37    2  567  |    8    1  469  |
| 34     1    2  |   9  356    8  |   56    7   46  |
| 57    69    8  |   4  567    1  |    2  569    3  |
+----------------+----------------+-----------------+
B=1 cell ALS xy-rule: r156c2 -6- r2c3 -9- r7c1346 => r7c2<>3
B=4 cell ALS xy-rule: r3c4689 -3- r1249c8 -9- r1569c2 => r3c2<>5
Hidden Pair in a box: r23c4 => r6c4<>6
Naked Row Pair: r6c24 => r6c5<>3
Row Finned X-Wing: r6c24|r7c34 => r5c3<>3
Hidden Pair in a box: r56c2 => r3c2<>3
Hidden Pair in a box: r23c9 => r2c9=29,r3c9=29

Prof,

Yeah, I've used a solving technique on Extreme #71. heh heh

I refer you to postings made by Jeff Loeb in the Second Crop Sudoku? thread under the Sudoku Addicts topic. The product I allude to has helped me immensely.

Jeff

Re: Sudoku Explainer as a solver

Sudoku Explainer's main use these days is that of giving a general puzzle-difficulty rating. Other than that, it has not been updated to keep abreast of more advanced solving methods. FWIW: newspaper Diabolicals rarely get much above SE=7.1; the Extremes (on this site) tend to vary between 7.5 and 8.3, but as with #71 can sometimes get up to SE=9. Andrew Stuart's famous Unsolvables vary roughly from 8.3 to above 9.

As for a solver that gives a good idea as to which more advanced methods are used in a solution, Ruud's SudoCue is pretty good in that department.

As for the best computer solver for the more advanced puzzles when it comes to using solving methods that those of us in Eureka are using and being able to print out the resulting chains etc. (as opposed to the brute force methods used in most other solvers), I think that Mike Barker's solver wins that trophy. Sadly, I haven't been able to find a download site for it! ;)

Professor:

• Main site: diuf.unifr.ch/people/juillera/Sudoku/Sudoku.html

This is the SE homepage download site (appears to be the same as above). The problem may be with your system- I've never had a problem using SE from this site on 3 different WinXP systems:

http://diuf.unifr.ch/people/juillera/Sudoku/Sudoku.html

Don: Maybe Mike's solver is at tucows.com, or download.com, or some such.

You can plug the name into google.com or ask.com, and you might get a URL out of that.

Jeff

Jeff, I was being facetious and having a little fun with Mike (hence the winking sign). He keeps his solver under lock and key. :)

1.  I had not heard of Sudoku Explainer before.  Now that I have seen it in action, it seems that it uses a lot of techniques (Nishio Forcing Chains, Dynamic Contradiction Chains) that should be called 'trial and error.'  I am not sure how useful it is in giving difficulty ratings either.  Sudocue will give a detailed difficulty rating if you turn everything on.

2.  I have been able to solve a number of unsolvables (including Extreme 71) what I call Nested Conditional AIC's.  All the techniques are described in 'A Study in Chain Forging' on the sudoku.com.au site.  The trick to extreme 71 seems to be to notice the eliminations that can be made in box 9.  It is relatively easy to eliminate the 4's and 6's in H7 and G8 if you allow nesting of AIC's.  You can then place the 7 in J1 and it is smooth sailing.

3.  Sudocue can solve almost anything if you turn on Table Conflict.  You need to copy log long, otherwise you will not get a detailed description.  Personally, I would call this technique 'trial and error.'

4.  Newsflash: the pattern Overlay Method is a form of 'trial and error' no matter Andrew thinks.

5.  Mike Barker's program would seem to be the best there is for solving hard sudokus.  Unfortunately, I cannot follow the steps and do not know what 'Kraken Blossom' even is.  Nevertheless, I do thank him for the solution and hope to be able to follow it someday.  I could use a hint.

6.  There are now a large number of sudokus on sudoku.com.au that cannot be solved using Andrew's techniques.  It was only a metter of time before he made the Extremes harder.  But he cannot have it both ways:  up until now, the Extremes could nto be called hardest in the world; sudoku.com.au puiblishes the hardest in the world.  But now the blurb on his back cover is false: you CANNOT solve any Extreme using the techniques in his book.  So, he will need to either update his book or go back to the 'easy' Extremes.  

Tom, Bob Hanson has a Sudoku Assistant/Solver. It is listed among the solvers in http://sudokulinks.com . Other solvers are listed at the sudokulinks.com site.

I think the URL for Bob Hanson's solver is http://www.stolaf.edu/people/hansonr/sudoku

Jeff

The Hanson program did not thrill me.  I put in one of my 61 unsolvables and one of the steps spanned over four pages.  Why not just use brute force, which will solve anything.

The Sudoku Explainer is better than I originally thought it was.  Many of the chains are nothing more than AIC's under another name.  It also uses a number of methods that Andrew does not recognize (cell forcing chains, region forcing chains, dynamic forcing chains OR almost AIC's) but that I have been using for about six months on the rally hard sudokus published at sudoku.com.au.  I think the drawback is that the solutions are not terribly efficient.  Mike Barker's solution are in some instances far shorter.  FWIW there is an especially elegant solution to unsolvable number 32 on sudoku.com.au.

Another problem is that for super tough sudokus, the program essentially uses trial and error.  I out in the Easter Monster and the first step alone required 17 separate chains.  I never got a rating for the puzzle as a whole, which apparently requires over an hour to compute.  I could hardly this a deductive solution.

It would be interesting to see whether Mike Barker's program can handle the hardest of the hard:

7.....4...2..7..8...3..8..9...5..3...6..2..9...1..7..6...3..9...3..4..6...9..1.35

(hardest known sudoku according to Susser)

7.....4...2..7..8...3..8..9...5..3...6..2..9...1..7..6...3..9...3..4..6...9..1.35

(hardest known sudoku according to Hanson)

1.......2.9.4...5...6...7...5.9.3.......7.......85..4.7.....6...3...9.8...2.....1

(easter Monster - solution on sudoku.com.au using matrices)

I know of no program that uses matrices, but I know of no other way of solving the Easter Monster. 

Newsflash: the pattern Overlay Method is a form of 'trial and error' no matter Andrew thinks.

The Pattern Overlay Method is a means of converting a Sudoku puzzle into a set of simultaneous Boolean equations, and then solving for those equations using a visualizable form of variable substitution.  It's primary shortcoming is that it is a much more mathematical attack on the puzzle with far more starting variables (36x36x36x9 compared to normal 9x9x9 for the more familiar candidates) than most people would ever want to deal with.  I am not sure how that equates to "trial and error" in your book, but I would be interested in finding that out.

Myth,

Regarding Pattern Overlay Method, I did not think the point would be that controversial.  It is true that the method may be viewed systematically, but so many other methods that are more powerful.

Brute force may be viewed as a logical, systematic method, but I would not want to try solving sudokus that way.

I think the distinction between logical or deductive, on the one hand, and trial and error on the other hand is best viewed in terms of elegance of the method.  That is how Andrew views 'logical' in his book.  POM fails dismally.  It takes many sheets of paper, no matter how systematic the method is.

ANY sudoku that is solvable may be solved using some combination of pigeonhole matrices, triangular matrices, and block triangular matrices.  The proof of this point is available elsewhere and I will assume you are familiar with it.  In one sense, the method is as systematic as POM.  It is not, however, intuitive, although it is an improvement over brute force and vastly more powerful than POM.

I am new to this forum, so I am not familiar with methods that have been described herein.  Some of us have tried to solve hard sudokus by thinking outside the box: almost unique rectangles, almost hidden pairs, dual forcing collapsable AIC's, etc.  Every sudoku I have seen that Andrew's solver requires POM can be solved more intuitively using these other techniques.  POM is best left on the slag heap or reduced to the trial and error section.  Its principal use is to explain the Sashimi observation.  Otherwise I find it completely useless to anyone wishing to solve a puzzle by hand within a reasonable period of time.  Many other methods exist that are more intuitive or more powerful or both.

I have been able to solve a number of unsolvables (including Extreme 71) what I call Nested Conditional AIC's.  All the techniques are described in 'A Study in Chain Forging' on the sudoku.com.au site....The trick to extreme 71 seems to be to notice the eliminations that can be made in box 9.

Why not just call them Almost AICs which is what Stephan Kurzhals, the author of 'A Study in Chain Forging' calls them. Why not just show us your Extreme 71 solution? For one thing, references to Box 9 are hard to evaluate when we don't know at what point in the puzzle you are referring to.

This puzzle:
7.....4...2..7..8...3..8..9...5..3...6..2..9...1..7..6...3..9...3..4..6...9..1.35

is actually #77 from the Top1465 with an extra clue in r9c8.  Its Sudoku Explainer rating is 9.8 and I haven't been able to solve it with the techniques I've implemented, although I'm sure longer multi-inference nice loops/AICs could do the job.  By the way, although this was once considered the toughest known puzzle, it only rates as moderately hard when compared to the Easter Monster or other extremely hard puzzles.

To answer a question raised by Bill, most of my technique development has resulted from extending techniques I've observed others use or questions others have raised (Kraken fish came from a solution by Anne Morelot as I tried to understand Jeff's and Carcul's Almost fish.  Death Blossom from a group effort started by SHuisman which turned out to reinvent a technique proposed by Bennys).  I guess the key is that they are all not just theorectical, but do have a basis in actual solutions.

Thomas,

I don't know that your definition of T&E really correlates with a correct T&E definition.  You are certainly correct in your assessment that POM is user unfriendly.  You might note that Medusa/3D Coloring was also considered user unfriendly when it was first proposed as well.

It perhaps will come as a shock to you that you can replicate the deductions of every pattern, past, present, and future, just by analyzing the simultaneous POM equations from the cells that make up the pattern.  Thus, I doubt anything other than pure T&E is more powerful.  POM can be used to explain any deduction, and future deductions and perhaps patterns could certainly be found by reverse engineering from POM if anyone were so inclined.  Partial POM is also a potential option.

The matrix methods you mention have a lot in common with the solving the simultaneous equations of POM.  The fact that POM divides up a candidate digit into its possible rookeries just means that POM doesn't have to use as many cells in general to find a deduction as a method that operates with the more coarsely granulated candidates.  Thus an equivalent POM deduction to a can consist of many more steps, but the steps can be significantly shorter.  Fairly often, I could find a pair of twp, three, or four-cell POM deductions that performed the same or more reduction as an AIC spanning seven or more cells.

So put POM on your own personal slag heap if you wish, but I wouldn't be surprised if somewhere down the road it rears its ugly head again :)

Here is Mike's first Kraken deduction for Extreme 71--a rather difficult puzzle.

4-valued/2-element Kraken Box (r13c2|r12c3=6) => r1c5<>6
(6)r1c23 
||
(6)r3c2 - r12c3 = r7c3 - r7c46 = r89c5
||
(6)r2c3 - (6=7)r2c178 - r1c9 = (7-6)r1c5
+------------------------+-----------------+--------------------+
|     2     356*   356*b |   1  57-6c   9  |    4   3568    78c |
|    15c      8   1569*b | 267     4    3  | 1567c   156c  279  |
|  1345  134569*      7  |  26     8   56  | 1359  13569   269  |
+------------------------+-----------------+--------------------+
|     9       2      34  |   5     1   67  |  367   3468  4678  |
|     6    3457     345  |   8   379    2  |  379    349     1  |
|     8     137      13  | 367   369    4  | 3679      2     5  |
+------------------------+-----------------+--------------------+
| 13457  134569  134569b | 367b    2  567b |    8  14569   469  |
|  1345   13456       2  |   9   356b   8  |  156      7    46  |
|    57     569       8  |   4   567b   1  |    2    569     3  |
+------------------------+-----------------+--------------------+

Now a reasonable question is how does one go about finding such a deduction without reverse engineering an assumption.  My answer is that a 3D-Medusa colored backdrop gives you a fighting chance to notice these animals.  Lets put in some relevant 3D coloring tags.

+------------------------+-----------------+--------------------+
|     2     356     356  |   1  57c6    9  |    4   3568    78  |
|    15       8    1569a |2b67C    4    3  | 1567    156  2B79A |
|  1345  134569A      7  | 2B6b    8   56  | 1359  13569  2b69  |
+------------------------+-----------------+--------------------+
|     9       2      34  |   5     1   67  |  367   3468  4678  |
|     6    3457     345  |   8   379    2  |  379    349     1  |
|     8     137      13  | 367   369    4  | 3679      2     5  |
+------------------------+-----------------+--------------------+
| 13457  134569  134569  | 367     2  567  |    8  14569   469  |
|  1345   13456       2  |   9   356    8  |  156      7    46  |
|    57     569       8  |   4   567    1  |    2    569     3  |
+------------------------+-----------------+--------------------+

From this we can see that a piece of our color molecule is going to look like...

a = A - B = b - C = c

Now when the 3D coloring stalls out, one tends to look for things like hinges, ALS, and AHS to help make new coloring connections or extend existing ones to new candidates on the grid.  One such marker is box 8.  Notice that any unsolved digits in box 8 will have to form a hinge.  Transporting the sixes in that box gives us...

(6)r89c5 = (6)r7c46 - (6)r7c3 = (6)r12c3

...which looks like a dead end since the (6)r12c3 group doesn't look like it can "turn the corner" and affect anything in the direction of column 5 or box 8.  However, if you are having a good day, you might notice the conjugate group "a" sharing a cell with one of those sixes.  You might see the already noted color AIC which can be coupled into your transport from the hinge as follows...

(6)r89c5 = (6)r7c46 - (6)r7c3
                       ||
                      (6)r2c3 - a = A - B = (6b)r3c4
                       ||
                      (6)r1c3   => r1c5 <> 6

...or also...

(6)r89c5 = (6)r7c46 - (6)r7c3
                       ||
                      (6)r2c3 - a = A - B = b - C = (7c)r1c5
                       ||
                      (6)r1c3   => r1c5 <> 6

The point being that it doesn't necessarily require a computer to make a Kraken deduction.  A little advance preparation can make many Kraken deductions both reasonable and well within the realm of human techniques.

Bill, please refrain from obfuscating threads by replacing the perfectly understandable common Eureka notation with your own mishmash.  While for some reason it might help you to figure things out; no one else is going to benefit from it, so keep it to yourself.

We could of course write this as an AIC (as I've advised David to do in reverse-engineering his GEM Nets into AICs)....But that's kinda ridiculous: it's better to embrace Kraken - BR -

This statement doesn't really make sense.  I found the deduction by building an AIC (also an AAIC as well as a Kraken) exactly as I described; and so I wrote it as an AIC exactly as I found it, which is not in any way "ridiculous". 

Threadbump, I forgot that you are impervious to common sense.  Obviously, your beloved fictional GSP was relying on you to substitute in your arcane mishmash for the generally accepted common notation.  Seriously, without all that extra insight provided by your amply padded spam, certainly no one would have a chance at understanding anything.

Was it a lack for attention as a child that made you such a friggin' troll?

Another misquote by Threadbump; who would have ever guessed....

The whole point of the change to a network based strong link definition which allows more than two elements to be strongly linked in AICs was to bring patterns such as naked triples, swordfish, and especially Mike's Kraken and Death Blossom deductions, into the AIC fold.  For quite some time I have represented AICs with these trinary and beyond strong links with aligned vertically oriented strong link symbols.  This has been discussed elsewhere, not that you would ever lower yourself to care.  So, yes, I represented what was discovered as an AIC, and what actually is an AIC as well as a Kraken deduction, as an AIC, so go pound away at something else, troll.

"...which eliminates 6 in A5 because of the "Kraken logic" I explained" - Bill Richter

You didn't explain anything, you merely repeated what I had already worked out and represented in your own twisted vernacular and then tried to claim credit for "explaining it".  Quit trying to kiss my butt while stabbing me in the back, troll.

Here is the last post of any significance on this thread.  I doubt anyone should be interested in the garbage that has been posted in between.  As soon as Bill erases his contributions, I'll erase mine.  Till then, let this serve to get the thread back on track.

Here is Mike's first Kraken deduction for Extreme 71--a rather difficult puzzle.

4-valued/2-element Kraken Box (r13c2|r12c3=6) => r1c5<>6
(6)r1c23 
||
(6)r3c2 - r12c3 = r7c3 - r7c46 = r89c5
||
(6)r2c3 - (6=7)r2c178 - r1c9 = (7-6)r1c5
+------------------------+-----------------+--------------------+
|     2     356*   356*b |   1  57-6c   9  |    4   3568    78c |
|    15c      8   1569*b | 267     4    3  | 1567c   156c  279  |
|  1345  134569*      7  |  26     8   56  | 1359  13569   269  |
+------------------------+-----------------+--------------------+
|     9       2      34  |   5     1   67  |  367   3468  4678  |
|     6    3457     345  |   8   379    2  |  379    349     1  |
|     8     137      13  | 367   369    4  | 3679      2     5  |
+------------------------+-----------------+--------------------+
| 13457  134569  134569b | 367b    2  567b |    8  14569   469  |
|  1345   13456       2  |   9   356b   8  |  156      7    46  |
|    57     569       8  |   4   567b   1  |    2    569     3  |
+------------------------+-----------------+--------------------+

Now a reasonable question is how does one go about finding such a deduction without reverse engineering an assumption.  My answer is that a 3D-Medusa colored backdrop gives you a fighting chance to notice these animals.  Lets put in some relevant 3D coloring tags.

+------------------------+-----------------+--------------------+
|     2     356     356  |   1  57c6    9  |    4   3568    78  |
|    15       8    1569a |2b67C    4    3  | 1567    156  2B79A |
|  1345  134569A      7  | 2B6b    8   56  | 1359  13569  2b69  |
+------------------------+-----------------+--------------------+
|     9       2      34  |   5     1   67  |  367   3468  4678  |
|     6    3457     345  |   8   379    2  |  379    349     1  |
|     8     137      13  | 367   369    4  | 3679      2     5  |
+------------------------+-----------------+--------------------+
| 13457  134569  134569  | 367     2  567  |    8  14569   469  |
|  1345   13456       2  |   9   356    8  |  156      7    46  |
|    57     569       8  |   4   567    1  |    2    569     3  |
+------------------------+-----------------+--------------------+

From this we can see that a piece of our color molecule is going to look like...

a = A - B = b - C = c

Now when the 3D coloring stalls out, one tends to look for things like hinges, ALS, and AHS to help make new coloring connections or extend existing ones to new candidates on the grid.  One such marker is box 8.  Notice that any unsolved digits in box 8 will have to form a hinge.  Transporting the sixes in that box gives us...

(6)r89c5 = (6)r7c46 - (6)r7c3 = (6)r12c3

...which looks like a dead end since the (6)r12c3 group doesn't look like it can "turn the corner" and affect anything in the direction of column 5 or box 8.  However, if you are having a good day, you might notice the conjugate group "a" sharing a cell with one of those sixes.  You might see the already noted color AIC which can be coupled into your transport from the hinge as follows...

(6)r89c5 = (6)r7c46 - (6)r7c3
                       ||
                      (6)r2c3 - a = A - B = (6b)r3c4
                       ||
                      (6)r1c3   => r1c5 <> 6

...or also...

(6)r89c5 = (6)r7c46 - (6)r7c3
                       ||
                      (6)r2c3 - a = A - B = b - C = (7c)r1c5
                       ||
                      (6)r1c3   => r1c5 <> 6

The point being that it doesn't necessarily require a computer to make a Kraken deduction.  A little advance preparation can make many Kraken deductions both reasonable and well within the realm of human techniques.

Kraken equals AAIC equals AIC Net equals AIC (because who really wants to add "Net" to it).  It has already happened.  As stated, Kraken deductions were one of the main forces behind the change in definition of AIC strong links that happened quite some time ago.

As I described it, I really did find this deduction as an AAIC since technically I didn't start from the branching point.  But they really amount to the same thing and I like paying a little homage to Mike for introducing us to the multi-tentacled critters.  You might note that Mike also referred to his Kraken deductions as AICs, so this is nothing new.  If you cannot figure out from the nice big 2-dimensional diagrams that these are nets, then adding the little "net" after "AIC" probably won't make that much difference anyway.

BR: Good!  Once we agree on that, we can retire the terms continuous & discontinuous, and say Nice Chain and Nice Loop.

No we can't. The use of the term Nice Loop as an all-inclusive term to include the sub-groups eg. continuous & discontinuous Nice Loop or as Andrew does in his book: Nice Loop Rule 1 (continuous) Rule 2 & 3 (discontinuous) is pretty much standard terminology. After all, although the purist may look on a Nice Loop as only a chain that is completed because it is continous (thus forming a loop), there is a sense in which a chain that is completed to include the target cell whether continuous or discontinous can be considered a loop. So, I understand what Mike is saying, but I don't think he is advocating a change in terminology.

BR: although I haven't done any Kraken: I probably should (I did add an AAIC)).

I think an AAIC is a Kraken.

BR: Don, what do you think of my AAIC above?  I can give you a lot more tips on how to find them...You're practically a member of the GSP who I want to teach, and I'd like to practice my stunts on you. 

Bill, I think you need a re-evalutation of your perspective: Let's go back in history a bit: In early 2007, Stephen first published an AAIC solution on this forum at a time when he was developing the subject with other graphics on his blog. People responded positively and somewhat in awe, because Stephen was putting his own original (and very welcome) spin on it. I immediately saw a manual solving advance and spent weeks studying his (very) few AAIC solutions here and his more numerous examples on his blog. The use of advanced AAICs became part of my manual solving for SE >7.6-8.0 puzzles.

While I was actively using AAICs in my own solving, I waited over several months to see others come out with manual AAIC solutions, assuming that it would just be a matter of time. Virtually no one did and so I created what was essentially an AAICs-101 thread (to try to kick-start interest in them) in September 2007 here:

http://www.sudoku.org.uk/SudokuThread.asp?fid=4&sid=9693&p1=2&p2=10

In that thread, Mike B very graciously commented: Now granted I used my solver to find the examples I've posted so you and Steve get the credit for showing that Kraken techniques can be used by real life human solvers. (And he also pointed out that 'what you are referring to as a new AAIC technique I've been using to solve puzzles for a year now'.)

In any event, I think I can say that the above thread contained the first graphics of basic AAICs that clearly show the various elements of an AAIC in a way that people unfamiliar with them can understand. I followed that thread with another which also used some advanced AAIC solving:

http://www.sudoku.org.uk/SudokuThread.asp?fid=4&sid=9761&p1=2&p2=10

Following those threads, there started to appear for the first time manually-derived multi-inference solutions. Perhaps it was pure coincidence, but I don't think so. It unequivocably was not due to the fact that I had somehow instructed advanced solvers here on AAIC solutions, but I do think that my threads did have the effect of reminding people that these were legitimate and useful manual solving alternatives.

All of the above said, it should be pointed out that Mike B, as I quoted him above, has been making use of, as he calls, them Kraken Blossom and Kraken Unit techniques in his solver for some time so it's not as if even Stephen was the first one to recognize their value. The way I look at it- Jeff (I believe) first described multi-inferences in general, Mike B first recognized their practical solving value but more for the purposes for use with his solver and Stephen first recognized their true value in manual solving. And I helped spread Stephen's message.

In any event, for you, having come up with one AAIC (6 months after I first published my first one), to think that, in view of the above and the fact that, at the very least I have matched you in solving the SE 8 puzzles, you are going to not only teach me how to find AAICs, but also lump me in with the GSP who you want to teach is a marked self-serving dissociation from the reality of the situation.

Bill, I think you missed the point.  Not all Sudoku terminology is perfect, but except in a few cases any reasonable person can understand if Jeff calls a discontinuous nice loop a "loop", or if Myth refers to a Kraken reduction as an "AIC", or if someone refers to an X-wing as a "fish".  The real problem is when people use different names for the same thing or use the same name for different things.  You are guilty of both.  My advice to you would come from Matt 7:5: "You hypocrite, first take the log out of your own eye, and then you will see clearly to take the speck out of your brother's eye."  If you really want to clean up Sudoku terminology stop calling a grouped strong link a bilocal or a hinge, stop calling grouped advanced coloring an MC-hinge, stop using a "+" to refer to a strong inference, etc.

BR: I thought it was me who was adding complicated OR statements to the b/b plot.

As far as I've been able to discern you've added assumptivity to b/b plots.  Myth added the idea of using "what is" eliminations to find Kraken eliminations in his recent post (my apologies to those that did it before).

BR: help heal a feud between Myth & I, having to do with the meaning of assumptive or pattern-based

More likely that I'd discuss reversibility with you than get caught up in the current hostility involving assumptivity.  I will say this, that any algorithm that starts with "solve the puzzle and determine the backdoors" is not going to make it in my book.  If you have an intelligent way to link chains in a b/b plot then that's worth discussing, but if you really want someone else to review your work then you should present it in their language.  I know I don't have time to translate all of your many posts.

A backdoor is a placement in a cell after which the puzzle is solvable with singles.  (2)r7c2 is a backdoor to U27 which you indicate you used in U#27:

BR: Now I'm working on U#27, and I noticed this simple algorithm can give conclusions that shouldn't be describes as AICs.  BTW U27 is easy to solve if we assume 2*G2. 

As far as U#26 I get as far as:

BR: In the Spaghetti grid (SG), I'll make sure every true candidate is a target, and in the Bivalue Grid (BVG), I'll make sure every right value is true, by transposing if necessary, and you can see that in the grid above, as I've written 63*D7, 73*E3, and 31*G7.

And you've failed my reasonableness test.  To be elegant and usable you can't assume you know the backdoors or what the answer is.  If that weren't the case then I'll give you a really elegant solution to U#27 - set (2)r7c2 and solve with singles or even better since you know the solution plug it in!

I did get a little farther:

BR: To follow my solution, you'll have to maintain the SG yourself as the solution progresses.  Start with arrows drawn between conjugates, large-font the bivalues, and add arrows between single candidates as we go on.  Here's the first useful one I find: Start with the conjugate G5 -1- H6, and assume the source 1*G5 is true.

This sounds like plug in values until you find something that works.  Again this fails my reasonableness test.  Start without a priori knowledge of the solution or backdoors, and tell my why you chose (1)r7c5 and we have something to discuss.  For example, I choose the trivalue cells I did in Challenge #9 because of the links to those cells.  Not perfect, but at least rational. 

As far as AICs, smart people have continued to find ways to expand the use of nice loops and AICs.  Thus, whereas, both began as bivalues and bilocals, they have expanded to include grouped and multi-inference nodes.  If you need proof just read Myth's post in which he refers to a Kraken elimination as an AIC.  QED

BR: Don, if you don't want my help, that's fine, but this is ridiculous:
DM> And he also pointed out that 'what you are referring to as a new AAIC technique I've been using to solve puzzles for a year now'.
...And now you want to say is due to you years ago?

Bill, be accurate about what is being written before responding: In the quote above, it is Mike B who is saying that what I (Don) is referring to a new AAIC technique, he (Mike) has 'been using to solve puzzles for a year now'.

So, I (Don) am not claiming to have done anything 'years ago'.

BR: I think Carcul had been doing it for a long time before that.

Yes, that's correct although Carcul didn't post any AAICs (which is the multi-inference subject at hand); he posted exclusively in nice loop notation. However, if one is going to give credit where credit is due on the general subject of Almost-this chain and Almost-that chain, he should be mentioned as being one of the very first when it comes to posting manual solutions.

One general point I want to make so that there is no misunderstanding: I think that credit for the overall development and use of multi-inference techniques as we know them to the present is due to Jeff, Carcul, Mike and Stephen. I look on my contribution as teeny-tiny in comparison, but helpful nonetheless: the posting of graphics and descriptions in a tutorial manner that should help those who haven't been able to understand how multi-inference can work in practical solving.  And come to think of it- also presenting a suggested one-line Eureka form of AAIC notation that I notice some others have used since with occasional minor variations.

MB> Start without a priori knowledge of the solution or backdoors, and tell my why you chose (1)r7c5 and we have something to discuss. 
I've explained this before, and it's explained in the purple prose. I choose every single candidate in the SG.  That means everything in the b/b plot, like 1*G5 , plus everything I can add in. 
MB> For example, I choose the trivalue cells I did in Challenge #9 because of the links to those cells.  Not perfect, but at least rational.
I didn't think that was much of an explanation, but I'm not looking for "rational".  I'm looking for algorithms.  If you can tell me what algorithm you use to find Kraken solutions manually, I'll be very interested, provided it's fast.

Bill, the fact that you view this as a competition is probably a major contributor to the problems that exist on this site.  This forum would be a much better place if you would stop trying to score points by tearing down other's ideas,  creating artificial distinctions between your ideas and other's, and trying to gain acceptance of your ideas by repetition.  Work with others with a common language and you'll find that your ideas will receive the credit they deserve. 

Myth, Bill's failings are obvious to anyone reading his posts.  Reiterating them only serves to further pollute the forum.

BR: I didn't think that was much of an explanation, but I'm not looking for "rational".  I'm looking for algorithms.  If you can tell me what algorithm you use to find Kraken solutions manually, I'll be very interested, provided it's fast.

I'll save you time and trouble. There is no algorithm for finding Kraken solutions and there will be never be an algorithm. You are trying to apply rigid mathematical thinking to a puzzle game. The human mind does not use finite absolute algorithms to determine all the 'moves' that are made in brain games such as Sudoku. That's not to say that there are not certain features of our search methods that can be described finitely such as transport in finding AICs, but overall a 'one size fits all' algorithm is never going to happen. Trust me.

DM> The human mind does not use finite absolute algorithms to determine all the 'moves' that are made in brain games such as Sudoku.
True, and that was one of the points that Steve seemed to be making, when he wrote that he frankly did not care how someone found a proof of an elimination.