dailysudoku.com

[storm_norm]

the simple forcing loop, A.K.A xy-loop




very very importantly... not all xy-loops are going to be nice and rectangular like the one in the image above. xy-loops can take on different shapes and involve many cells. I like to call them "pools" or "ponds" since most big patterns involved fish names.

in the image is a puzzle that was discussed in the forum.

Normally, the circled candidates would represent your average, run of the mill, xy-chain. However, this particular xy-chain has a potency for damage beyond what a normal xy-chain can do.

#1... a particular xy-chain which has its pincer ends sharing the same house can be considered a simple forcing loop, xy-loop.

#2... the green arrows show that the direction of the chain is indeed valid in both directions. what this means is that regardless of the cell you start in and which ever candidate you substitute in for that cell, the loop stays true in either direction you travel. keep this in mind...

#3...Imagine this as a multi-sided polygon. each two cells represent a "side" of the polygon. the row, column, or box that the "side" is in is where the eliminations take place. the eliminations come from the candidate that the "side" has in common.

in other words, each side contains two cells. Those two cells automatically become pincer cells (using the rules of a xy-chain). The cells that form the "side" of the loop share one candidate and can therefore eliminate any other candidates they see.
(this will sound sort of like describing a xy-chain)

example...
consider the cells, r6c2{8,9} and r6c4 {4,8}, these cells share the number 8. the rules of a xy-chain says that the 8 either goes in r6c2 or r6c4. since the 8 has to go in either of the two ends of the chain, you can eliminate 8 from r6c3.
This applies to all sides of the loop and makes eliminations as shown by the red boxes in the image.
it might be best to get a handle on what a xy-chain is first.
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this next example is alternative logic to the forcing loop. BUT!! it is highly specialized. how specialized you ask??

#1... it can only be a 4 cell xy-loop
#2... it only involves eliminations on two sides of the xy-loop
#3... the "roof" and "floor" candidates must be conjugate to the row, column, or box they reside in.
I'd say that is pretty special.



now before I go any further, I have to site the source of my research.
http://www.scanraid.com/Multivalue_X_Wing_Strategy
in the link is two examples, and both are exactly 4 cell xy-loops.

in the image above is an example of a simple forcing loop. it eliminates the 4 in r6c3.

this particular and very specialized pattern is what Andrew Stuart calls a multivalue x-wing. consider rows 3 and 5... candidates 5 and 8 are conjugate to those rows respectively. focus on column 5. what if both the 5 and the 8 are true? this would force 4 into two cells in column 3. Can't have that.
It also elminates the 7's in col 5. can't have that either.
this forces a 4 to exist either in r3c3 or r5c3 and eliminates the 4 in r6c3

edit: to change some of the wording. thank you to Steve R who has suggested changing some of the wording in the second example.

[storm_norm]

I welcome and encourage any thoughts on how to clean up the wording.
I probably tried a little too hard in making the connection between multivalue x-wings and xy-loops and made some wording errors that, I hope, can be stated clear enough for the readers of the forum to understand.

[Marty R.]

Do I understand correctly that those eliminations of 8 in rows 68 only occur in an XY-Loop? I'm no theoretician, but if the answer is yes, then why? Why wouldn't they occur in an ordinary XY-Chain where the pincer ends aren't in the same house? For example, if the 34 were in r6c7 rather than r6c8, the 8s are still there.

[storm_norm]

Marty, hello there...

ignore for a moment that we are discussing xy-loops and form the xy-chain from r8c2 going clockwise around to r8c8. this chain says you can eliminate the 8's in row 8.
However, if you move the (3,4) cell to r6c7, this breaks that chain. now the (3,4) cell is no long connected to the (3, cell in r8c8 and then doesn't eliminate the 8's in row 8.

the same can be said about the 8 in r6c3

except this time you are starting your xy-chain in r6c4 and going around connecting to r8c2... again, moving the (3,7) cell would break the ordinary xy-chain that this forms.

a side of a loop is comprised of two adjacent cells of the loop. any two adjacent cells of the loop can act as pincer cells.

[Victor]

Marty wrote:

[Marty R.]

I guess I just assumed the 8s would still be true since the XY-Chain is still valid if one of the pincer cells is moved, but after looking at Norm's answer and studying it, I realize I was wrong.