Links: Not in MY House!

Advanced methods and approaches for solving Sudoku puzzles

Postby champagne » Tue Oct 02, 2007 10:13 am

Keith, wrote

Incorrect. The pincers of an XY-wing cannot both be true
.

And later on

Multi-coloring:
#==#--#==#==#==#
Possibilities are:
a==A--a==A==a==A
A==a--A==a==A==a
A==a--a==A==a==A
Any other cell that sees both ends of the chain cannot be A.
Extra credit: Note that the ends of this multi-coloring chain are a W-link.


Something as “This is not true and I’ll prove it is true”

Keith wrote

W-coloring:
#==#++#==#==#==#
Possibilities are:
a==A++a==A==a==A
A==a++A==a==A==a
a==A++A==a==A==a

Any other cell that sees both ends of the chain ... You can make no statement!!


A valid “XYWing” type alternate chain has the general form

#==#--#==#--#==#--#==#
Both ends are in what you call W-link.
At least one is true, maybe both.

Any intermediate sequence of the form #==#--#==# is a W-link.

So the chain can be shortcut in one of these forms:

#==#--#++#--#==#
#==#--#++#
#++#--#==#

Reversely, any valid chain including W-link can be extended to a standard chain using (distant) strong links and (distant) weak links.

Your chain is not a “XYWing” type chain.
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Postby ronk » Tue Oct 02, 2007 11:02 am

keith wrote:Notation:
# Any value
== Strong Link
-- Weak link
++ W-link
a (lower case) Is not true
A (upper Case) Is true

As you say, there are three types of links, but their names and definitions have been problematic. That's why many of us have adopted Jeff's strong inference and weak inference terms. These newer terms don't have the historical baggage.

I I had my druthers, your three links above would be conjugate link, weak link and stronk link, respectively ... but few people think of a bivalued cell as a conjugate link. -- If wishes were horses, then beggars would ride.:)

Reference: Forcing chains: Terminology and Definition.
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Postby keith » Tue Oct 02, 2007 11:05 pm

Ruud wrote:
keith wrote:
Ruud wrote:Keith,
the connection between these candidates is similar to the one between the pincers of an XY-Wing. Both can be true, but we cannot deduce anything from that fact because it doesn't always happen. Therefore, we can omit this "useless" piece of information from our definition.

Incorrect. The pincers of an XY-wing cannot both be true.

XZ-XY-YZ

Only one can (and one must) be Z.


Since the pincers never belong to the same house (otherwise we would have a naked triple), they can have the same value.

Possible resolutions to XZ-XY-YZ are:
Z-X-Y
Z-X-Z
Z-Y-Z
X-Y-Z

Ruud


Ruud is absolutely correct here, and I apologize for the sloppy statements in my post on XY-wings, quoted by Ruud and re-quoted above.

Keith
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