The New Sudoku Players' Forum

[angusj]

Ocean, could you please explain how this pattern is to be applied and how it is related to the conjecture.

Perhaps I can help ...

Let r1c3 = A, r1c9=B, r8c3=C, r7c9=D, r9c1=E, r9c7=F
A&B are conjugates
E&F are conjugates

Code: Select all

 *-----------*
 |..A|...|..B|
 |...|...|...|
 |...|...|...|
 |---+---+---|
 !...|...|...|
 |...|...|...|
 |...|...|...|
 |---+---+---|
 |...|...|..D|
 |..C|...|...|
 |E..|...|F..|
 *-----------*   

Implicitly A&C are conjugates as are B&D because ...
if A=true => C=false => E=true => F=false => D=true => B=false
also if A=false => B=true => D=false => F=true => E=false => C=true
Hence, all other candidates in the same unit as AC (col3) and BD (col9) can be excluded.

[Jeff]

Thanks Angus,

What you are effectively saying is that C & D are conjugates and this is true if and only if C-E-F-D is a conjugate chain, ie both CE and DF must be conjugates also. Once the conjugate relationship of C & D is established, the x-wing requirement is met. Correspondingly, the conjecture describes AB and CD as conjugate chains, not just conjugate links. However, I gather that these conjugate chains ought to have odd number of conjugate links for the conjecture to work.

[angusj]

Once the conjugate relationship of C & D is established, the x-wing requirement is met.

Yes, that's a much simpler way of putting it .

However, I gather that these conjugate chains ought to have odd number of conjugate links for the conjecture to work.

Yes, I think that's right. For x-wings the link count is one, and for Ocean's unusual example above it is three for the lower chain.
Edit: On further reflection, I think the conjugate chains must either both have an odd number of links or both have an even number of links.

[Jeff]

I think they have to be both odd.

[Ocean]

I think it might be appropriate with a short note on notation - don't know if the terminology is standard, so it might be misunderstood.

Define conjugate chain something like this:

Conjugate chain A-B (for candiate x): A conjugate chain with cells colored A or B in such a way that either all A's are equal to x, or all B's are x. The rule "A<=>not B" (if A's are x, then B's are not, and if B's are not x, then A's are) is valid for all cells in the chain. Also, the rule "B<=>notA" applies.

The definition of "Conjugate chain C-D" is similar.


Hope this may clarify the discussion of "odd/even number of links". In the conjecture, A and B do not refer to 'ends' of the chain, they apply to any cells with 'opposite colors' A and B.

Note also that the conjecture mentions only ONE weak link between conjugate chains. The special cases/patterns of 'x-wing' and 'x-wing-lookalike', with TWO weak links, lead to a join of the two chains, producing one single conjugate chain from the two.




---Turbot Fish?

Thanks to Jeff for pointing out the connection to "Turbot Fish". It seems that 'x-wing lookalike' patterns are rather equivalent to a "Generalized Turbot Fish", with no restriction of the number of edges.

I also had a brief look at Doug's "Fishy Cycles". It might be interesting to generalize that concept to a "Fishy Mesh", by using the whole "conjugate chain" instead of only "conjugate pairs". Maybe these structures are worth a study. With N chains each with K cells, we might get some unexpected results, even for a restricted grid of 9x9.


Also, thanks to Angus for explaining the pattern example.

[angusj]

I think they have to be both odd.

Yes, you're absolutely right. No sooner than I'd turned my PC off for the night when I realised that this was the case too. The ends of each chain must be conjugates not the same.

[Jeff]

It seems that 'x-wing lookalike' patterns are rather equivalent to a "Generalized Turbot Fish", with no restriction of the number of edges.

Ocean, This is the way I look at it from the viewpoint of a generalised turbot fish. A turbot fish has 5 links (3 of them are non-conjugates) whereas a generalised turbot fish has 'n' links (3 of them are non-conjugates), where n is odd. The x-wing lookalike pattern is a special case of the generalised turbot fish where 2 non-conjugate links overlap in one unit. The effect is to reduce the total number of links 'n' by 1 and cause the 2 remaining non-conjugate links to become symmetrical, such that the exclusion of candidates is equally applicable if the pattern is reflected.

[Ocean]

Jeff:

Agrees with you! This stuff is actually all about Turbot fishes! It's the same thing.