almost-locked sets: extension to XY-cycles/3D Medusa

Advanced methods and approaches for solving Sudoku puzzles

almost-locked sets: extension to XY-cycles/3D Medusa

Postby Bob Hanson » Wed Dec 07, 2005 5:48 pm

Well, if almost-locked sets (which are basically Y-cycles) can be transferred to X-cycles, I suggest that then the two can also be combined in a 3D mode, the way 3D Medusa or "advanced coloring" combines X- and Y-cycles.

Here we go. I don't have an example, only an idea.

Code: Select all
   |---c1--|---c2--|---c3--||---c4--|---c5--|---c6--||---c7--|---c8--|---c9--
-----------------------------------------------------------------------------
r1 |   3   |       |   3   ||  578  |       |       ||       |       |       
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r2 |       |       |       ||       |       |       ||       |       |       
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r3 |       |       |       ||       |       |       ||       |       |       
===========================||=======================||=======================
r4 |       |       |       ||       |       |       ||       |       |       
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r5 |       |   36* |       ||  678  |       |       ||       |       |     
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r6 |       |       |   3   ||  358  |       |       ||       |       |       
===========================||=======================||=======================
r7 |       |       |       ||       |       |       ||       |       |       
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r8 |   3   |   3   |   3   ||  57   |       |       ||       |       |       
---+-------+-------+-------||-------+-------+-------||-------+-------+-------
r9 |       |       |       ||       |       |       ||       |       |       
-----------------------------------------------------------------------------
      A        A      A         B
-----------------------------------------------------------------------------

(Please don't look to closely at this. We have to presume other values are there that would make this possible. It's just hypothetical.)

Here A is {18 58 168}, on candidate 3, and B is {578 678 358 57} in column 4.
A is weakly linked to B in row 6 via candidate 3.
The starred 6 is weakly linked to B, that's clear. As for A, it is linked to the
only candidate 3 in row 5 (the "5" in {18 58 168}). That constitutes
a weak link, too.

The starred 6 can be eliminated, because if it were NOT, then B would have to
include 3 at r6c4, but the 3x3 grid that forms then for A disallows that placement.

The point is simply that weakly linked almost-locked sets work
exactly like weakly linked "strong chains" (combinations of 2-valued cells
and conjugate pairs). They can be X-based, Y-based, or combinations
thereof. If they have multiple links, it is not uncommon for one of the links
to be able to be eliminated by inspection, as here.

Overall, the point is that everything anyone has done involving X-cycles, Y-cycles, XY-wings, XYZ-wings, 3D Medusa, coloring, and such is subsumed under almost-locked sets. I'll present that in the next posting.
Bob Hanson
 
Posts: 75
Joined: 04 December 2005

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