simply subsets of almost-locked sets. As a result, all 3D Medusa business
is subsumed under a generalized almost-locked set concept.
The two sorts of components comprising strong chains are 2-valued cells
such as {12} or {47}, and "conjugate pairs" -- exactly two possible
placements ("marks") for a certain candidate in a certain row, column, or
block. I sometime write 2-values cells like this:
- Code: Select all
1
2
or this:
- Code: Select all
1
\
2
and I write conjugate pairs like this:
- Code: Select all
-1------1-
or this:
- Code: Select all
|
1
|
|
1
|
OK, so the way these work is that a candidate can be "weakly linked" to them -- the "buddy" of one of the chain nodes. I write that like this:
- Code: Select all
|
1........1*
|
|
1
|
So here, if the 1* is TRUE, then the top chain 1 is FALSE. And if the top chain 1 is TRUE, then the 1* is FALSE. That's a weak link.
I'll call these numbers -- the marks we pencil in -- "nodes."
OK, first thing, these are all just almost-locked sets. Obviously the Y-cycle 2-valued cells are; it may not be as obvious regarding conjugate pairs, but there the trick is to number the columns or rows -- say, "1 only in rows 3 and 5" amounts to {35} in almost-locked set notation.
Now, the weak link definition I gave yesterday applies here as well:
Node A is weakly linked to node B if they are "buddies" -- "in the same house" -- in the same row, column, or block. For single nodes, "Node A is weakly linked to node B" implies "Node B is weakly linked to node A."
A node A of candidate k is weakly linked to an almost-locked set if it is weakly linked to ALL of the nodes of that set.
An almost-locked set A is weakly linked to another almost-locked set B if ALL of the candidate k nodes of A are weakly linked to ALL of the candidate k nodes of B.
Notice that the concept of weak link is still reflexive. (Is that the right word?) A weakly linked to B implies B weakly linked to A.
Now, here's the kicker. We can write an almost-locked set like a branching chain:
{58 18}
is
- Code: Select all
1
|
/ \
5 8
and {58 18 186}
is
- Code: Select all
1
|
5--+--6
|
8
What's interesting to me (at least) is that these can now be plugged right
into the same sort of analysis I was doing with chains.
Here's an XY-wing in terms of simple Y-cycles:
- Code: Select all
2
.3....3
. 4
. .
. .
2 .
4.....*
* can't be 4.
And here it is in terms of almost-locked sets, where the left two
Y-cycle cells are combined into the almost-locked set A
where A={23 24}. B={34}, the mutual weak link is via 3.
- Code: Select all
3....3
/ 4
2--A .
\ .
4.....*
Same conclusion! (You can't have two weak links to the same
almost-locked set, because that would set too many nodes FALSE.)
Here's an XYZ-Wing:
- Code: Select all
3
.\
. 4
. .
. .
================
. .
. *
. .
2......2. .
\ 3 .
\ \.
4......4
A 4 at * can be eliminated because, based on Y-cycles,
a 4 there would force both the 24 to be 2 and the 34
to be 3, and no 4 for the 234 cell. But that would
leave nothing for the 234 cell.
In almost-locked sets, we have:
- Code: Select all
3....3
/ 4
2--A .
\ .
4.....*
again. The same code!
So, basically what is going on is we are inserting this
fork-like structure into our standard picture of chains.
This, of course, could be extended to any number of
almost-locked sets interacting weakly with any number
of strong chains. Because strong chains are, in fact, just
sets of almost-locked sets!
Whereas for a chain we have:
- Code: Select all
X........a--A
where X is a candidate that is going to force a false and A true,
for almost-locked sets we have the same, but with a twist:
- Code: Select all
A
/
Z..a--
\
A
BOTH candidates A and B go true, thus allowing continued logical
implications along BOTH forks, provided there is weak linking from those
candidates to others.
The implication is clear: There are many, many applications of this
almost-locked sets idea, and, as I will propose NEXT, this in fact is probably all one needs to solve Sudoku, if you want to do it that way.
