3) Yes (if a Rule 3 would eliminate the double weakly linked digit in a unit, there must be a Rule 1 loop with possibly more eliminations.)
4) please forget the nonsense "(as strong links)" (deleted it above). If i understood you right, the last grouped x-cycle sample in sudowiki is an example having only group nodes.
5) still don't get you, for building a chain there is no difference, if links are grouped or not. They either are weak links or strong links (which serve as weak links too).
I should state, that this sudowiki page is very old (about 2006). Nice loops are completely out of fashion, and usually were replaced by AIC's.
(A popular reference for solving strategies now is Bernhard Hobiger's
hodoku site. But to my surprise it has not described and used grouped x-chains, it just shows them as grouped AIC's.)
Using AIC's, (grouped) x-cycles were replaced by (grouped) x-loops and (grouped) x-chains.
AIC's are "alternating inference chains" of strong and weak (possibly grouped) links, starting and ending with a strong one.
They are loops, if the last node would eliminate the first one ("sees" it, is linked to it), because in that case you could continue the chain with the first one and repeat forever.
All digits, which see the first and last node (have a link to both) then can be eliminated.
If you have a loop, it is said, that "weak links become strong links", i.e. (in the 1-digit case) you can eliminate the other digits in that units.
E.g. the AIC for the first 'Rule 1' example would be as follows ('=' is a strong, '-' a weak link). Here it does not matter, where you start the loop.
8C2 = G2 - H3 = H9 - J7 = 8C7, loop => -8G3,G9,C39
(without using the loop, you would eliminate 8C39, because they see both 8C2 and 8C9, and starting with other nodes you would get the other eliminations too)
These AIC loops are equivalent to the 'Rule 1' x-cycles, the AIC's without loop can make all eliminations of the other types. Instead of closing an alternating chain with 2 links, stop the chain and look if there are digits which see both ends.
In the sample for Rule 2 you have
1E3 = E7 - J7 = 1J1 => -1G3,F1 (forcing J1 with the strong link) (thats a "skyscraper")
Alternatively you could start the AIC in J1 and extend it to a cycle
1J1 = J8 - E8 = 1E3 - G3 = 1J1 => 1J1
But stopping at 1E3 already would give you the eliminations -1G3,F1.
Rule 3:
1C7 = G7 - G2 = 1H3 => -1C3 (thats a "kite")
8B1 = C3 - E3 = 8E7 => -8B7 (kite)
The last sample to grouped x-cycles is a loop with additional eliminations:
4AB7 = C89 - C12 = AB3 - HJ3 = G12 - G89 = 4HJ7, loop => -4C5,DE3,G56,EF7
There are many ways to find and describe single digit eliminations, and many names for them.
Grouped x-loops and x-chains will catch all the eliminations of x-cycles, as well as named patterns like (grouped) skyscrapers/kites/empty rectangles, finned and sashimi x-wings, turbot fish, 222 swordfish.
Also some simpler exotic fish eliminations can be done, e.g. in this sample solvers make the elimination of 7 in r9c4 with a "finned franken swordfish" or a "grouped swordfish", but a grouped x-chain also does the job.
- Code: Select all
.---------------.-----------------.---------------.
| 6 379 79 | 137 137 5 | 8 4 2 |
| 5 8 1 |f37 4 2 |e37 9 6 |
|c37 24 24 | 8 6 9 |d357 d1357 13 |
:---------------+-----------------+---------------:
| 13 237 27 | 4 18 68 | 9 167 5 |
| 149 79 5 | 126 19 3 | 247 1267 8 |
| 149 6 8 | 1259 159 7 | 234 123 13 |
:---------------+-----------------+---------------:
|b79 5 3 |a79 2 1 | 6 8 4 |
| 8 1 469 | 3569 359 46 | 235 235 7 |
| 2 47 467 | 356-7 3578 468 | 1 35 9 |
'---------------'-----------------'---------------'
)!
