zoltag,
For the grid that you show, I like Jean-Christophe's "closed loop" idea in his post. I would use "AIC loop" or maybe "AIC ring," but of course the name isn't particularly important as long as you understand the idea. I think AIC loops are much easier to spot than a sashimi swordfish. For some fish, though, there may not be a AIC loop handy, such as for an ordinary, unfinnned 3 x 3 x 3 swordfish.
Like Jean-Christophe, I'm going to deviate slightly from the specific question that you posed and consider a different approach. I recently started to get into exotic fish, by which I mean Frankenfish and mutant fish, and I notice that in your grid there is a Franken Swordfish for digit 9.
I remember that it took me a while to grasp the "sashimi" concept. So you may be muttering to yourself asking why I'm introducing even more new and possibly confusing terminology. Again, don't worry about the specific terms so much at this point. Initially, I'll try to avoid all the usual fishy jargon and explain things in terms of the basic defining constraints of Sudoku. If you bear with me, I think you'll find this interesting, as you can sometimes use this approach when no AIC loop is handy.
The following grid shows only digit 9 candidates in your full grid.
Code: Select all
·-------------·-------------·-------------·
| · X9 · | 9 · · | · 9 · |
| · · -9 | · · 9 | · 9 · |
| X9 X9 · | · · · | · 9 · |
·-------------+-------------+-------------·
| · · 9 | 9 · 9 | · · · |
| · · · | · · · | 9 · · |
| · X9 · | -9 X9 · | · · · |
·-------------+-------------+-------------·
| X9 X9 · | -9 X9 · | · · · |
| · · · | · · · | · · 9 |
| · · 9 | · · 9 | · · · |
·-------------·-------------·-------------·
Let's focus on the digit 9 candidates in columns 1, 2, and 5 of the Sudoku grid. These have been marked with an "X" in the grid above. Suppose we arrange these candidates in a matrix, as shown below. Initially, don't worry about the columns of this matrix, but only the rows. The candidates of column 1 of Sudoku grid have been listed in row 1 of the matrix. There are only two, r3c1 and r7c1. Since column 1 of the Sudoku grid must contain exactly one "9", exactly one of the candidate premises in row 1 of our matrix must be true. Similarly, in row 2 of our matrix, all candidates in column 2 of the Sudoku grid have been listed. There is a bit of a twist, in that r13c2 is a grouped candidate. r13c2 = 9 means that exactly one of r1c2, r3c2 is 9. Again, all possible candidates in column 2 have been listed in row 2 of our matrix, so exactly one of the candidate premises in row 2 of our matrix must be true. Finally, in row 3 of our matrix, all candidates in column 5 of the Sudoku grid appear, so exactly one of these candidate premises must be true. Since each row of our matrix must contain exactly one true premise, our matrix contains exactly three true premises.
Code: Select all
weakly linked weakly linked weakly linked
(box 1) (row 6) (row 7)
| | |
| | |
V V V
·--------------·-------------·-------------·
| r3c1 = 9 | | r7c1 = 9 | <-- exactly 1 premise must be true (col 1)
·--------------+-------------+-------------·
| r13c2 = 9 | r6c2 = 9 | r7c2 = 9 | <-- exactly 1 premise must be true (col 2)
·--------------+-------------+-------------·
| | r6c5 = 9 | r7c5 = 9 | <-- exactly 1 premise must be true (col 5)
·--------------·-------------·-------------·
Now consider the columns of our matrix. There is some method to the madness in how the candidates have been arranged in columns. Notice that in column 1 of our matrix, all of the candidates (r3c1, r13c2) there lie in box 1 of the Sudoku grid. Thus, they are weakly linked -- at most one of the candidates can be true. Similarly, all candidates which appear in the second column of our matrix lie in row 6 of the Sudoku grid, so at most one of those candidates can be true. Finally, for the third column of our matrix, all candidates which appear there lie in row 7 of the Sudoku grid, so at most one of those candidates can be true.
So each of the three columns of our matrix can contain at most one true premise. But we've established earlier that our matrix has exactly three true premises. Thus each column of our matrix must contain (exactly) one true premise.
From the first column of our matrix, we deduce that either r3c1 = 9 or r13c2 = 9. These are both in box 1, so any other candidate in box 1 can be eliminated. This gives the elimination of (9)r2c3, which was your original subject. But there's more -- from column 2 of our matrix, we deduce that either r6c2 = 9 or r6c5 = 9, so that any other candidate in row 6 can be eliminated. Thus, (9)r6c4 is eliminated. From the last column of our matrix, we deduce that one of r7c125 is "9", so any other candidate in row 7 can be eliminated. So, (9)r7c4 is eliminated. In total, we've found the same eliminations resulting from Jean-Cristophe's AIC loop.
If you think about this a bit, you'll see that what makes such a matrix possible, in this case, is that all digit 9 candidates in columns 1, 2, and 5 of the Sudoku grid are contained within box 1, row 6, and row 7. In fishy jargon, it's often said that box 1, row 6, and row 7 "cover" all the digit 9 candidates in columns 1, 2, and 5, or that box 1, row 6, and row 7 form a "cover set" for columns 1, 2, and 5. Whenever we have a set of N houses, all of whose candidates are covered by another set of N houses, then we have the potential for a (perhaps exotic) fish. The first set of houses (the one which is covered) is often called the "base set." If no two houses of the base set share a common candidate, then we do in fact have a fish structure. In this case that's clearly true since all houses in the base set are columns. If you consider the reasoning behind the eliminations derived from the above matrix, you'll see that any candidate which is in the cover set, but not in the base set, can be eliminated. Thus, if you can identify suitable base and cover sets for some fish, you don't have to construct explicitly a matrix like the one above -- you can work directly from that principle.
I'll spare you (and myself) the details here, but for general fish we can sometimes come up with even more eliminations. Some day you may want to take a look at
this post.
A final note: In the Sudopedia, you will find the terms "defining set" and "secondary set," respectively, instead of "base set" and "cover set."