## POM Analysis of the X-Wing: The Filet-O-Fish Rule

Advanced methods and approaches for solving Sudoku puzzles

### POM Analysis of the X-Wing: The Filet-O-Fish Rule

Rather than force you to read how this came about and the reasoning behind it all, I am going to give away the ending. Collectively, the following are covered by the term "finned sealife".

Filet-O-Fish Rule

If you can form a swordfish/x-wing pattern by not considering candidates in cells (1..n), then you can keep any eliminations from that swordfish/x-wing pattern that share a group with all cells (1..n). The cells (1..n) have been called the fin.

Sashimi Observation

When using the Filet-O-Fish rule, it is possible for X-wings to be missing a candidate at a vertex point and not degenerate. Similarly, it is also possible to find stable 2-0-2 & 1-2-1 swordfish structures, etc., that otherwise could not exist without a fin.

Sushi Presentation Rule

For any NxN swordfish, the fin and the valid Filet-O-Fish eliminations all occur in the same box. (This only applies to a single instance of a Filet-O-Fish swordfish and its row-column conjugate. There may be multiple Filet-O-Fish swordfish in the grid with fins and deletions in other boxes)

(Note: While normal NxN swordfish follow the rule that the reduction lies in the same box as the fin, you can use filet logic to uncover headless or skinny swordfish, etc., that have an Nx(N-1) structure. These structure have a "fin" and also turn out to have multiple reductions in several different boxes just like regular swordfish do. You can then reapply filet logic to this to come up with another fin and a subset of reductions. These are very advanced, but if you would like to learn more, you can find them in the Big Fish thread.)

Filet-O-Fish Example:
Code: Select all
`+----------+-----------+-----------+| .  .  .  |  .  .  .  |  .  1  .  || . *1  .  | *1  .  .  |  .  .  .  || . -1  1  | -1  1  1  |  .  .  .  |+----------+-----------+-----------+| . *1  .  | *1 #1 #1  |  .  .  .  || . -1  1  | +1  1  1  |  .  .  .  || .  .  .  |  .  .  .  |  .  .  1  |+----------+-----------+-----------+| .  .  .  | -1  1  1  |  .  .  .  || .  .  .  |  .  .  .  |  1  .  .  || 1  .  .  |  .  .  .  |  .  .  .  |+----------+-----------+-----------+`

Sashimi Example:
Code: Select all
`+----------+-----------+-----------+| .  .  .  |  .  .  .  |  .  1  .  || . *1  .  | *1  .  .  |  .  .  .  || . -1  1  | -1  1  1  |  .  .  .  |+----------+-----------+-----------+| . *1  .  | *. #1 #1  |  .  .  .  || . -1  1  | +1  1  1  |  .  .  .  || .  .  .  |  .  .  .  |  .  .  1  |+----------+-----------+-----------+| .  .  .  | -1  1  1  |  .  .  .  || .  .  .  |  .  .  .  |  1  .  .  || 1  .  .  |  .  .  .  |  .  .  .  |+----------+-----------+-----------+`

If the candidate 1's marked with the hash sign (#) weren't there, then we could form an x-wing from the 1's marked with an asterisk (*) and eliminate all the 1's marked with plus and minus signs. Since the 1 marked with a plus sign shares a group with both of the hashed 1's, we can make that elimination in r5c4.

How this came about: A POM Analysis of an X-Wing

Let's start with a little review of what POM does. POM assigns labels to all of the potential solutions for a given digit. For example, if you have solved for a specific digit in six boxes, and the remaining three boxes form a right triangle, then you are left with something like this

Code: Select all
`+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  X    X    @    ||  X    .    .    |  .    X    X    |  X    X    X    ||  X    .    .    |  .    X    X    |  X    X    X    |+-----------------+-----------------+-----------------+|  X    .    .    |  X    X    X    |  X    X    X    ||  X    X    X    |  X    X    X    |  X    @    X    ||  X    X    X    |  X    X    @    |  X    X    X    |+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  @    X    X    ||  X    X    X    |  X    @    X    |  X    X    X    ||  @    X    X    |  X    X    X    |  X    X    X    |+-----------------+-----------------+-----------------+`

It should be fairly easy to see that there would be four potential solution patterns for this grid. The potential patterns are represented by 'a', 'b', 'c', and 'd' in the following grid.

Code: Select all
`+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  X    X    @    ||  X    a    b    |  cd   X    X    |  X    X    X    ||  X    c    d    |  ab   X    X    |  X    X    X    |+-----------------+-----------------+-----------------+|  X    bd   ac   |  X    X    X    |  X    X    X    ||  X    X    X    |  X    X    X    |  X    @    X    ||  X    X    X    |  X    X    @    |  X    X    X    |+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  @    X    X    ||  X    X    X    |  X    @    X    |  X    X    X    ||  @    X    X    |  X    X    X    |  X    X    X    |+-----------------+-----------------+-----------------+`

If cell r3c3 no longer has the digit in question as a candidate (it solved to another digit, or the candidate was removed due to a naked pair or something), then we can eliminate any patterns which use that cell. This would kill 'd' in this case and leave the following

Code: Select all
`+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  X    X    @    ||  X    a    b    |  c    X    X    |  X    X    X    ||  X    c    X    |  ab   X    X    |  X    X    X    |+-----------------+-----------------+-----------------+|  X    b    ac   |  X    X    X    |  X    X    X    ||  X    X    X    |  X    X    X    |  X    @    X    ||  X    X    X    |  X    X    @    |  X    X    X    |+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  @    X    X    ||  X    X    X    |  X    @    X    |  X    X    X    ||  @    X    X    |  X    X    X    |  X    X    X    |+-----------------+-----------------+-----------------+`

Now, the simplest case which can lead to an interesting x-wing is where a digit has been solved for in five boxes, and the remaining four boxes form a rectangle. In this situation, you wind up with something like this

Code: Select all
`+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  X    X    @    ||  X    .    .    |  .    .    X    |  X    X    X    ||  X    .    .    |  .    .    X    |  X    X    X    |+-----------------+-----------------+-----------------+|  X    .    .    |  .    .    X    |  X    X    X    ||  X    .    .    |  .    .    X    |  X    X    X    ||  X    X    X    |  X    X    X    |  X    @    X    |+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  @    X    X    ||  X    X    X    |  X    X    @    |  X    X    X    ||  @    X    X    |  X    X    X    |  X    X    X    |+-----------------+-----------------+-----------------+`

Note that rules of symetry and influence would imply that the potential solution pattern contents of any given box's cell must be evenly distributed to all of the cells in the diagonally opposed box. For example

Code: Select all
`+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  X    X    @    ||  X    abcd .    |  .    .    X    |  X    X    X    ||  X    .    .    |  .    .    X    |  X    X    X    |+-----------------+-----------------+-----------------+|  X    .    .    |  a    b    X    |  X    X    X    ||  X    .    .    |  c    d    X    |  X    X    X    ||  X    X    X    |  X    X    X    |  X    @    X    |+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  @    X    X    ||  X    X    X    |  X    X    @    |  X    X    X    ||  @    X    X    |  X    X    X    |  X    X    X    |+-----------------+-----------------+-----------------+`

Repeat this for all four open cells in box 1 and you get

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`+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  X    X    @    ||  X    abcd efgh |  .    .    X    |  X    X    X    ||  X    ijkl mnop |  .    .    X    |  X    X    X    |+-----------------+-----------------+-----------------+|  X    .    .    |  aeim bfjn X    |  X    X    X    ||  X    .    .    |  cgko dhlp X    |  X    X    X    ||  X    X    X    |  X    X    X    |  X    @    X    |+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  @    X    X    ||  X    X    X    |  X    X    @    |  X    X    X    ||  @    X    X    |  X    X    X    |  X    X    X    |+-----------------+-----------------+-----------------+`

Noting that each potential pattern must show up exactly once in every row, column, and box with an open cell, we can fill in the rest of the grid.

Code: Select all
`+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  X    X    @    ||  X    abcd efgh |  jlnp ikmo X    |  X    X    X    ||  X    ijkl mnop |  bdfh aceg X    |  X    X    X    |+-----------------+-----------------+-----------------+|  X    ghop cdkl |  aeim bfjn X    |  X    X    X    ||  X    efmn abij |  cgko dhlp X    |  X    X    X    ||  X    X    X    |  X    X    X    |  X    @    X    |+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  @    X    X    ||  X    X    X    |  X    X    @    |  X    X    X    ||  @    X    X    |  X    X    X    |  X    X    X    |+-----------------+-----------------+-----------------+`

Now, to form an x-wing, you just have to remove from consideration four cells, one from each box, which form a rectangle parallel with the grid. There are 16 such four-cell combinations which will form an x-wing. One four-cell combo is r2c3, r2c5, r4c3, and r4c5. These four cells contain the potential patterns 'b' through 'o' which are killed, leaving only 'a' and 'p' valid. This results in the following

Code: Select all
`+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  X    X    @    ||  X   *a    X    | *p    X    X    |  X    X    X    ||  X    --  #p    |  --  #a    X    |  X    X    X    |+-----------------+-----------------+-----------------+|  X   *p    X    | *a    X    X    |  X    X    X    ||  X    --  #a    |  --  #p    X    |  X    X    X    ||  X    X    X    |  X    X    X    |  X    @    X    |+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  @    X    X    ||  X    X    X    |  X    X    @    |  X    X    X    ||  @    X    X    |  X    X    X    |  X    X    X    |+-----------------+-----------------+-----------------+`

The dashes represent cells that no longer have a valid potential pattern associated with them. They are also the cells which get eliminated due to the x-wings shown by either the asterisked or the hashed cells. This is no surprise, since POM will find all nishio-type eliminations.

A perhaps more interesting result occurs when you remove just three cells out of any four-cell combination that leads to an x-wing. For example, if we only remove r2c3, r2c5, and r4c3 from consideration, POM shows us that we kill potential patterns c, d, e, f, g, h, i, k, l, m, and o; leaving only a, b, j, n, and p valid and the following grid

Code: Select all
`+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  X    X    @    ||  X    ab   X    |  jnp  X    X    |  X    X    X    ||  X    j    np   |  b    a    X    |  X    X    X    |+-----------------+-----------------+-----------------+|  X    p    X    |  a    bjn  X    |  X    X    X    ||  X    n    abj  |  --   p    X    |  X    X    X    ||  X    X    X    |  X    X    X    |  X    @    X    |+-----------------+-----------------+-----------------+|  X    X    X    |  X    X    X    |  @    X    X    ||  X    X    X    |  X    X    @    |  X    X    X    ||  @    X    X    |  X    X    X    |  X    X    X    |+-----------------+-----------------+-----------------+`

Thus, POM shows that you can make an elimination in cell r5c4 in this scenario. Since there are 4 of these three-cell removal combinations for every four-cell removal combination that leads to an x-wing, it could be very fruitful to find some method of recognizing when these partial x-wing or partial swordfish eliminations can be made.

Obviously we could resort to POM, nishio, or perhaps coloring with groups; but, could there be a more palatable method. Once again POM gives us a hint.

Note that when you remove a cell from consideration using the POM process, the potential patterns that you removed had no effect on the patterns for cells that shared a group with that removed cell. When, in the above grid, we remove cell r4c5 from consideration, killing patterns b, j, and n; there can be no other b's, j's, or n's in row 4, column 5, or box 5. Thus those cells are unaffected by the removal.

This is exactly what we need for the Filet-O-Fish rule.

Hope you find it useful.

(Edited to add new observations to the rules at top of the thread.)
Last edited by Myth Jellies on Tue Mar 21, 2006 12:29 am, edited 3 times in total.
Myth Jellies

Posts: 593
Joined: 19 September 2005

Hey, nicely observed! I assume it will work for any N-by-N (X-wing, Swordfish, Jellyfish, Squirmbag), as long as the "extra" candidates (fins? ;) ) only exist in *one* of the boxes (3x3), right?

I'm going to try to add this to my solver and see what it can do (after disabling some of the more exotic strategies). It wouldn't surprise me if it finds a lot of X-wings to "filet", at least.

(Not that my solver is very exotic anyway. I'm kind of limiting it do what I can do on paper, so that I can use it to generate interesting, but not unsolvable (for me) problems.)

Vidar
vidarino

Posts: 295
Joined: 02 January 2006

Yes, Filet-O-Fish should work for any NxN swordfish.

While it is not a requirement, the 1 or more extra cells which make up the "fins" probably need to be in the same box due to the way swordfish patterns and their deletions tend to line up.
Myth Jellies

Posts: 593
Joined: 19 September 2005

Hi MJ, One more fish in the aquarium; congratulations.

This Filet-O-Fish can also be expressed by the following nice loop:

[r5c4]-1-[almost x-wing:r24c24]=1=[r4c56]-1-[r5c4] => r5c4<>1

Last edited by Jeff on Sun Mar 12, 2006 2:57 am, edited 1 time in total.
Jeff

Posts: 708
Joined: 01 August 2005

Myth Jellies wrote:While it is not a requirement, the 1 or more extra cells which make up the "fins" probably need to be in the same box due to the way swordfish patterns and their deletions tend to line up.

Indeed. The "fins" need to be in one of the boxes, because the FoF rule allows you to keep eliminations that fall in the *intersection* of the extras and the regular elimination lines. If they weren't, there wouldn't be any intersection.
vidarino

Posts: 295
Joined: 02 January 2006

Myth Jellies wrote:Yes, Filet-O-Fish should work for any NxN swordfish.
While it is not a requirement, the 1 or more extra cells which make up the "fins" probably need to be in the same box due to the way swordfish patterns and their deletions tend to line up.

The following constraints apply:
1. Only one fin is allowed
2. The fin-candidate(s) and the candidate(s) to be excluded must be in the same box
Jeff

Posts: 708
Joined: 01 August 2005

Jeff wrote:This Filet-O-Fish can also be expressed by the following nice loop:

[r5c4]-1-[almost x-wing:r2c2|r2c4|r4c2|r4c4]=1=[r4c5|r4c6]-1-[r5c4] => r5c4<>1

Very good. Interestingly, we can also consider this particular Filet-O-Fish as a "grouped" Turbot Fish:

[r5c4]-1-[r2c4]=1=[r2c2]-1-[r4c2]=1=[r4c4|r4c5|r4c6]-1-[r5c4]
=> r5c4<>1.

Carcul
Carcul

Posts: 724
Joined: 04 November 2005

### Re: POM Analysis of the X-Wing: The Filet-O-Fish Rule

Myth Jellies wrote:Thus, POM shows that you can make an elimination in cell r5c4 in this scenario.

Hi MJ, I noticed that POM proved the deduction without bifurcation. Would this imply that all forcing chains can be proven by POM without bifurcation?
Jeff

Posts: 708
Joined: 01 August 2005

More sashimi

POM and the Filet-O-Fish concept seems to also imply that you don't need all of the points of the x-wing to make the elimination. If in our original example, the one is removed as a candidate from r4c4, we can still use sashimi to make the elimination in cell r5c4.

Code: Select all
`+----------+-----------+-----------+ | .  .  .  |  .  .  .  |  .  1  .  | | . *1  .  | *1  .  .  |  .  .  .  | | . -1  1  | -1  1  1  |  .  .  .  | +----------+-----------+-----------+ | . *1  .  | *. #1 #1  |  .  .  .  | | . -1  1  | +1  1  1  |  .  .  .  | | .  .  .  |  .  .  .  |  .  .  1  | +----------+-----------+-----------+ | .  .  .  | -1  1  1  |  .  .  .  | | .  .  .  |  .  .  .  |  1  .  .  | | 1  .  .  |  .  .  .  |  .  .  .  | +----------+-----------+-----------+  `
Myth Jellies

Posts: 593
Joined: 19 September 2005

Some more sashimi

Code: Select all
`+-----------+-----------+-----------+| 6   .   . | .   6   . | 6   6   6 || 6   .   . | .   6   . | 6   6   . || .   .   . | 6   .   . | 6   .   . |+-----------+-----------+-----------+| .   6   . | .   6   . | 6   .   . || .   6   . | .   .   . | 6   6   . || .   .   . | 6   .   . | .   .   6 |+-----------+-----------+-----------+| .   .   . | .   .   . | 6   6   . || .   .   . | .   .   . | .   .   . || .   .   . | .   .   . | .   .   . |+-----------+-----------+-----------+`

The above is a grid from Jeff's elementary x-cycle thread where he makes a ton of cool reductions using simple x-cycles. Filet-O-Fish/sashimi will also work on this grid as well. In fact, you can apply it in several different ways and places.

Let *6 and *. indicate the nxn swordfish to be fileted.
Let #6 indicate the extra fin.
Let -6- indicate the cells that can be deleted.

You can opt for some sashimi X-wing:
Code: Select all
`+-----------+-----------+-----------+| 6   .   . | .   6   . | 6   6  #6 || 6   .   . | .   6   . | 6   6   . || .   .   . |*6   .   . |-6-  .  *. |+-----------+-----------+-----------+| .   6   . | .   6   . | 6   .   . || .   6   . | .   .   . | 6   6   . || .   .   . |*6   .   . | .   .  *6 |+-----------+-----------+-----------+| .   .   . | .   .   . | 6   6   . || .   .   . | .   .   . | .   .   . || .   .   . | .   .   . | .   .   . |+-----------+-----------+-----------+`

If you don't like that flavor, you can try some with the fin repositioned:
Code: Select all
`+-----------+-----------+-----------+| 6   .   . | .   6   . | 6   6  -6-|| 6   .   . | .   6   . | 6   6   . || .   .   . |*6   .   . |#6  .   *. |+-----------+-----------+-----------+| .   6   . | .   6   . | 6   .   . || .   6   . | .   .   . | 6   6   . || .   .   . |*6   .   . | .   .  *6 |+-----------+-----------+-----------+| .   .   . | .   .   . | 6   6   . || .   .   . | .   .   . | .   .   . || .   .   . | .   .   . | .   .   . |+-----------+-----------+-----------+`

Here is another one, not as useful as the first two.
Code: Select all
`+-----------+-----------+-----------+| 6   .   . | .   6   . | 6   6   6 || 6   .   . | .   6   . | 6   6   . || .   .   . |*6   .   . |*6   .   . |+-----------+-----------+-----------+| .   6   . | .   6   . |-6-  .   . || .   6   . | .   .   . |-6-  6   . || .   .   . |*6   .   . |*.   .  #6 |+-----------+-----------+-----------+| .   .   . | .   .   . | 6   6   . || .   .   . | .   .   . | .   .   . || .   .   . | .   .   . | .   .   . |+-----------+-----------+-----------+`

But here is my favorite, the rare 2-1-2 sashimi swordfish. Lip smacking good.

Code: Select all
`+-----------+-----------+-----------+| 6   .   . | .   6   . | 6   6   6 || 6   .   . | .   6   . | 6   6   . || .   .   . |*6   .   . |*6  *.   . |+-----------+-----------+-----------+| .   6   . | .   6   . |-6-  .   . || .   6   . | .   .   . |-6- -6-  . || .   .   . |*6   .   . |*.  *.  #6 |+-----------+-----------+-----------+| .   .   . |*.   .   . |*6  *6   . || .   .   . | .   .   . | .   .   . || .   .   . | .   .   . | .   .   . |+-----------+-----------+-----------+`

And don't forget that in this grid, every sashimi x-wing has a sashimi 5x5 swordfish, marked with %, to go with it. (Is this a squirmbag?)
Code: Select all
`+-----------+-----------+-----------+|%6  %.   . | .  %6   . | 6  %6  %6 ||%6  %.   . | .  %6   . | 6  %6  %. || .   .   . |*6   .   . |*6   .   . |+-----------+-----------+-----------+|%.  %6   . | .  %6   . |-6- %.  %. ||%.  %6   . | .  %.   . |-6- %6  %. || .   .   . |*6   .   . |*.   .  #6 |+-----------+-----------+-----------+|%.  %.   . | .  %.   . | 6  %6  %. || .   .   . | .   .   . | .   .   . || .   .   . | .   .   . | .   .   . |+-----------+-----------+-----------+`
Myth Jellies

Posts: 593
Joined: 19 September 2005

I set up a POM grid with the final three boxes solved for a digit and found a whole bunch of sashimi/filet-o-fish swordfish similar to this...

Code: Select all
`+-----------+-----------+-----------+ | .   2   2 | .   2   2 | .   2   2 | | .   2   2 | .   2   2 | .   2   2 | | .   2   2 | .   2   . | .   2   . | +-----------+-----------+-----------+ | .   2   . | .   2   . | .   2   . | | .   2   2 | .   2   2 | .   2   2 | | .   2   . | .   2   . | .   2   . | +-----------+-----------+-----------+ | .   .   . | .   .   . | 2   .   . | | .   .   . | 2   .   . | .   .   . | | 2   .   . | .   .   . | .   .   . | +-----------+-----------+-----------+ `

In case you are having trouble finding it, the swordfish cells are marked with *, the fin with #, and the removeable candidates with -2- in the following...

Code: Select all
`+-----------+-----------+-----------+ | .  -2-  2 | .   2   2 | .   2   2 | | .  -2-  2 | .   2   2 | .   2   2 | | .  *2  #2 | .  *2   . | .  *2   . | +-----------+-----------+-----------+ | .  *2   . | .  *2   . | .  *2   . | | .   2   2 | .   2   2 | .   2   2 | | .  *2   . | .  *2   . | .  *2   . | +-----------+-----------+-----------+ | .   .   . | .   .   . | 2   .   . | | .   .   . | 2   .   . | .   .   . | | 2   .   . | .   .   . | .   .   . | +-----------+-----------+-----------+ `

That was good, but from my POM analysis, I also discovered that you could remove the same two candidates using the following grid.

Code: Select all
`+-----------+-----------+-----------+ | .  -2-  2 | .   2   2 | .   2   2 | | .  -2-  2 | .   2   2 | .   2   2 | | .   2   2 | .   2   . | .   .   . | +-----------+-----------+-----------+ | .   2   . | .   2   . | .   2   2 | | .   2   2 | .   2   2 | .   2   2 | | .   2   . | .   2   . | .   2   2 | +-----------+-----------+-----------+ | .   .   . | .   .   . | 2   .   . | | .   .   . | 2   .   . | .   .   . | | 2   .   . | .   .   . | .   .   . | +-----------+-----------+-----------+ `

It is interesting that you can replace the two holes in box 6 with the single hole in r3c8 and get the same deletions. I can't find any filet-o-fish that explains why you can remove candidates in r1c2 & r2c2 in this grid. A cut at using x-cycles with groups wasn't fruitful either.

So the challenge is, can anyone come up with a simple method or pattern recognition scheme that would catch these deletions?

Originally, the best I was able to do was some intersecting groups thing, something like the following:

Code: Select all
`+-----------+-----------+-----------+ | .   2AB 2B| .   2   2 | .   2   2 | | .   2AB 2B| .   2   2 | .   2   2 | | .   2Ab 2b| .   2B  . | .   .   . | +-----------+-----------+-----------+ | .   2a  . | .   2A  . | .   2   2 | | .   2a  2A| .   2a  2a| .   2a  2a| | .   2a  . | .   2A  . | .   2   2 | +-----------+-----------+-----------+ | .   .   . | .   .   . | 2   .   . | | .   .   . | 2   .   . | .   .   . | | 2   .   . | .   .   . | .   .   . | +-----------+-----------+-----------+ `

group A1 (r123c2) = not a2 (r456c2) = A3 (r5c3) = not a4 (r5c25689) => A5 (r46c5)

and

group B1 (r12c23) = not b2 (r3c23) = not B3 (r3c5)

and, noting that both B3 and A5 cannot both be true, we can invalidate cells that contain both an A and a B. Thus we can eliminate the 2's in r12c2.

I didn't think that this was a legitimate use of the method, though, due to the one way implication from a4 to A5 instead of a two way equals. After all, isn't nishio just a set of one way implications from a cell.

We could get around that, though, if we adopted the hinge/tee/cross rule which states that any true value outside a box containing a hinge/tee/cross, but on an arm of that hinge/tee/box, prohibits any other true values outside the box along the other arm of that hinge/tee/cross. Thus, the cross (not all cells of the hinge/tee/cross need to be filled in) in box 5 allows us to say that A3 and B3 cannot both be true. We have no one-way implications; so cells containing AB can legitimately be removed as candidates.

Code: Select all
`+-----------+-----------+-----------+ | .   2AB 2B| .   2   2 | .   2   2 | | .   2AB 2B| .   2   2 | .   2   2 | | .   2Ab 2b| .   2B  . | .   .   . | +-----------+-----------+-----------+ | .   2a  . | .  +2   . | .   2   2 | | .   2a  2A|+.  +2  +2 | .   2   2 | | .   2a  . | .  +2   . | .   2   2 | +-----------+-----------+-----------+ | .   .   . | .   .   . | 2   .   . | | .   .   . | 2   .   . | .   .   . | | 2   .   . | .   .   . | .   .   . | +-----------+-----------+-----------+ `

The hinge/tee/cross rule has probably been discovered many times. I've seen it recently in some posts by Rod Hagglund
Myth Jellies

Posts: 593
Joined: 19 September 2005

Myth Jellies wrote:
Code: Select all
`+-----------+-----------+-----------+ | .  -2-  2 | .   2   2 | .   2   2 | | .  -2-  2 | .   2   2 | .   2   2 | | .   2   2 | .   2   . | .   .   . | +-----------+-----------+-----------+ | .   2   . | .   2   . | .   2   2 | | .   2   2 | .   2   2 | .   2   2 | | .   2   . | .   2   . | .   2   2 | +-----------+-----------+-----------+ | .   .   . | .   .   . | 2   .   . | | .   .   . | 2   .   . | .   .   . | | 2   .   . | .   .   . | .   .   . | +-----------+-----------+-----------+ `

It is interesting that you can replace the two holes in box 6 with the single hole in r3c8 and get the same deletions. I can't find any filet-o-fish that explains why you can remove candidates in r1c2 & r2c2 in this grid. A cut at using x-cycles with groups wasn't fruitful either.

So the challenge is, can anyone come up with a simple method or pattern recognition scheme that would catch these deletions?

Hi MJ, I will give it a go. Could the pattern be an almost jelly fish nice loop?

My logic is that I know that there is a swordfish of 2-row formation (called a skinny) that works just as well as a swordfish of 3-row formation. From the grid above, the almost jelly fish is just an almost skinny jelly fish of a 3-row formation.
Jeff

Posts: 708
Joined: 01 August 2005

We could also find an alternative prove if we had the complete grid, and not just the "2s".

Regards, Carcul
Carcul

Posts: 724
Joined: 04 November 2005

Carcul,

The whole point here is that you have all the information you need in the layout of the 2's to remove the candidates in r1c2 & r2c2. The rest of the digits would be immaterial.
Myth Jellies

Posts: 593
Joined: 19 September 2005

Hi Myth Jellies.

Myth Jellies wrote:The whole point here is that you have all the information you need in the layout of the 2's to remove the candidates in r1c2 & r2c2. The rest of the digits would be immaterial.

I suspected that you would say that . In that case, I have analized your grid with the "2s" and have found that the eliminations are easy to understand in light of the following grouped X-cycle:

[r1c2|r2c2]-2-[r1c3|r2c3|r3c3]=2=[r5c3]-2-[r5c6]=2=[r1c6|r2c6]-2-
[r3c5]=2=[r3c2|r3c3]-2-[r1c2|r2c2], => r1c2,r2c2<>2.

One possible interpretation of this complicated chain can be, for example, that if "2" is in any of the cells r1c2 or r2c2, then both row 3 and column 6 would have necessarily their "2" in the same unit (box 2), which is a contradiction.

Regards, Carcul
Carcul

Posts: 724
Joined: 04 November 2005

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