by **Ocean** » Thu Sep 01, 2005 4:56 pm

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.

Last edited by

Ocean on Thu Sep 01, 2005 6:32 pm, edited 1 time in total.