The New Sudoku Players' Forum

[Mike Barker]

I admit I'm frightened by colouring, forcing chains, or anything else which could result in chasing down a bunny trail. One technique I've been using is basically a simple forcing chain, but occurs often enough that I'm wondering if it has a name of its own. The situation is shown below. A value exists in a row common to a block only once outside of the block (for example, a) and at least once inside the block for that row. In addition the value exists in a column common to the block only once outside of the block (for example, b) and at least once inside the block for that column (a=b=x<>c). Finally it does not exist in the cell common to the row and column (d<>x). In this case the value can be eliminated from the cell at the intersection of the column containing the row element and the row containing the column element (e).

Code: Select all

  ...|xxx
  cac|xdx
  ...|xxx
  ---+---
  ...|.c.
  .e.|.b.
  ...|.c.


As I said, its a simple forcing chain (if "e" is the value then "a" is not, etc.), but its as easy to identify as an X-wing and I've found it very useful in solving puzzles. So the question is, "Does it have a name?"

[Jeff]

.........So the question is, "Does it have a name?"

Welcome to the forum, Mike. The answer is yes, if I understand your description correctly. One of the names is a grouped x-cycle. Refer here for a full description of the technique or refer here for a similar example.

[Mike Barker]

I guess a better name for it would be a Grouped X-wing. A grouped X-wing is an X-wing with a block as one node such that at least one cell along the row and one cell along the column within the block contains the number (x=v and at least one of (a,b,c)=v and one of (d,b,f)=v and "?" may or may not).

. . .|. . .|?d?
. . .|. v.|abc
. . .|. . .|?f?
----+---+---
. . .|. . .|. . .
. . .|. x.|. v.
----+---+---
. . .|. . .|. . .
. . .|. . .|. . .
. . .|. . .|. . .

If the intersection, b, does not contain the number then the number is rejected from x,a,c,d,f and accepted in both v's. If the intersection contains the number then it is rejected from a,c,d,f. Further inferences will depend on what the "?s" are.

[Myth Jellies]

This looks very similar to rules 4 & 5 for coloring. Check out the example in this thread!

[Jeff]

I guess a better name for it would be a Grouped X-wing. In this light it includes the Turbo Fish, the Fillet-of-Xwing, and the Broken X-wing, but with a very simple inference.

Hi Mike, I think any term related to x-wing would be inappropriate because elimination from an x-wing is outside the chain, but elimination of your pattern is within the chain.

[Mike Barker]

I agree, its not really an X-wing and as stated it is, in fact, coloring 4 and 5. However, if I go back to the original post for the case without a value in the intersection and only make the inference that "x" is not the value, then the result is actually more general than coloring because this conclusion does not require the v-x column or row to be conjugate pairs.

[Mike Barker]

The pattern is close to a Turbo Fish, but has possible extra elements in the diagonal block:

Code: Select all

 . . . | . . . | . . .
 . . . | B---------C .
 . . . | | . . | . | .
-------|-------|----
 . . . | A . . | . | .
 . . . | A . . | . | .
 . . . | . E E-----D .
------------------------
 . . . | . . . | . . .
 . . . | . . . | . . .
 . . . | . . . | . . .

Because of the extra elements, there are three consecutive weak sides and thus no Turbo Fish exclusions. However, its possible to group the extra elements. In this case all of the Turbo Fish rules apply! I guess I'd refer to this as a Grouped Turbo Fish.

[Jeff]

I guess I'd refer to this as a Grouped Turbo Fish.

Hi, Mike. That is exactly what I have been thinking.

[Mike Barker]

Just as an interesting observation, it turns out that a Sashimi Fillet-o-fish X-wing is also a Grouped Turbo Fish.

[Jeff]

Just as an interesting observation, it turns out that a Sashimi Fillet-o-fish X-wing is also a Grouped Turbo Fish.

Exactly.

[Myth Jellies]

...Or you could say that this particular grouped turbot fish is a filet-o-turbot fish, at least for the A's and E's . It works for the A's and E's no matter how many cells in box 5 are filled.

And don't forget that there will be some broken wings there as well.

X-Wings, Colors, Turbot Fish, X-Cycles, Broken Wings....choose your favorite method and there is at least one extension (groups or fillets), if you need them, that will enable you to make the reduction for this setup.

[Mike Barker]

Taking things one step farther the sashimi of a 2x2x2 Swordfish is a 7-sided grouped Turbo fish. (Does a 7-sided discontinuous grouped or ungrouped x-cycle have a name?) I don't know if its been posted, so I went ahead and tried to catalogue the possible reductions for this configuration. What I came up with follows. In the case with one weak side (6 strong sides):

Code: Select all

 . . . | . . . | . . .
 . . . | B-----------C
 . . . | | . . | . . |
---------|-----------|
 . . . | | . . | . . |
 . . . | | . . | E---D
 . . . | | . . | | . .
---------|-------|----
 . . . | A . . | | . .
 . . . | . . . | | . .
 . . . | . . G---F . .

If the number is in A, then it is also in G which is a -><- so it is not in A, C, E, or G; and is in B, D, and F. In addition, the guardian which creates the weak side also contains the value.

Three possibilities exist with two weak sides. If they are contiguous:

Code: Select all

 . . . | . . . | . . .
 . . . | B-----------C
 . . . | | . . | . . |
---------|-----------|
 . . . | | . . | . . |
 . . . | | . . | E---D
 . . . | | . . | | . .
---------|-------|----
 . . . | A . . | | . .
 . . . | . . . | | . .
 . . . | . . G | F . .

If the number is in G, then it also in in F which is a -><- so it is not in G, nor in the union of the guardian cells for this puzzle and subsequent ones.

If they are separated by one strong side:

Code: Select all

 . . . | . . . | . . .
 . . . | B-----------C
 . . . | | . . | . . |
---------|-----------|
 . . . | | . . | . . |
 . . . | | . . | E---D
 . . . | | . . | . . .
---------|------------
 . . . | A . . | . . .
 . . . | . . . | . . .
 . . . | . . G---F . .

If the number is in A, then it is also in G which is a -><- so it is not in A, C, or E and is in B and D.

If they are separated by two strong sides:

Code: Select all

 . . . | . . . | . . .
 . . . | B-----------C
 . . . | | . . | . . |
---------|-----------|
 . . . | | . . | . . |
 . . . | | . . | E . D
 . . . | | . . | | . .
---------|-------|----
 . . . | A . . | | . .
 . . . | . . . | | . .
 . . . | . . G---F . .

If the number is in G, then it is also in F which is a -><- so it is not in G or E and is in F.

There are four possibilities with 3 weak sides. In the first, all three weak sides are contiguous. In this case no reduction can be made. If two weak sides are contiguous, and the third weak side is separated with two strong sides, then again no conclusion can be reached. Finally if one weak side is separated from two contiguous weak sides by one strong side:

Code: Select all

 . . . | . . . | . . .
 . . . | B-----------C
 . . . | | . . | . . |
---------|-----------|
 . . . | | . . | . . |
 . . . | | . . | E . D
 . . . | | . . | | . .
---------|-------|----
 . . . | A . . | | . .
 . . . | . . . | | . .
 . . . | . . G | F . .

Then if the value is in G, then it is also in F which is a -><- so it is not in G.

Finally if none of the weak sides are contiguous then:

Code: Select all

 . . . | . . . | . . .
 . . . | B-----------C
 . . . | | . . | . .   
---------|------------
 . . . | | . . | . . .
 . . . | | . . | E---D
 . . . | | . . | . . .
---------|------------
 . . . | A . . | . . .
 . . . | . . . | . . .
 . . . | . . G---F . .

If it is in A, then it is in G which is a -><- so it is not in A or C and is in B.

The case of four weak sides has only one interesting configuration that is two contiguous weak sides separated from the other two noncontiguous sides by one strong side each. This is equivalent to the Sashimi for the 2x2x2 Swordfish with (A,G) the filleted cell:

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 . . . | . . . | . . .
 . . . | B-----------C
 . . . | . . . | . .   
----------------------
 . . . | . . . | . . .
 . . . | . . . | E---D
 . . . | . . . | . . .
----------------------
 . . . | A . . | . . .
 . . . | . . . | . . .
 . . . | . . G---F . .

In this case if A is true, the G is true which is a -><- so it is not in A. As for the grouped Turbo Fish. The seven sided fish can also be grouped:

Code: Select all

 . . . | . . . | . . .
 . . . | B-----------C
 . . . | . . . | . .   
----------------------
 . . . | . . . | . . .
 . . . | . . . | E---D
 . . . | . . . | . . .
----------------------
 . . . | A . . | . . .
 . . . | A . . | . . .
 . . . | . G G---F . .

and everything still applies.

[Jeff]

I don't know if its been posted, so I went ahead and tried to catalogue the possible reductions for this configuration.

Hi Mike, Very nice layout. All these configurations can be summarised by the following statements.

• A strong link can be used as a strong link or weak link
  
• Apart from the 2 links connected by the node where a deduction is to be made, all links must propagate alternatively, ie. ....strong-weak-strong-weak....
  
• At the node where a deduction is to be made, if the adjacent links are both strong, the candidate can be placed in the node.
  
• At the node where a deduction is to be made, if the adjacent links are both weak, the candidate can be eliminated from the node.