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[Guest]

Is there any logical step from here?

[David Bryant]

[george woods1]

another way of looking at this 5 star constellation is to note that whichever of the 2's is chosen in block 7, it leads to a 9 in the "correct" position

[keith]

There is a method to solve this puzzle without chains:

[TKiel]

Keith,

Good spot on that combination. I think maybe you can claim naming rights on that.

[Steve R]

A very fine observation, Keith.

I’m not too sure about priorities, though. In this particular example the pattern seems to match an AUR:

“Almost Unique Rectangle (AUR): a set of four cells populated in such a way that if one or two of them does not have a specific candidate 'x', then those four cells form a unique rectangle.”

Perhaps your thoughts will develop in a different way.

Steve

[keith]

Tracy and Steve,

Thank you very much for the kind words.

Tracy,

If I had naming rights, I would call it something like a "Diagonal Pair in an X-wing". The real problem with naming is that the existing names are not good: An X-wing is really a fish, not a wing (etc.).

Steve,

The AUR sounds a little open-ended to me. I will do some research in the forums for solution methods and algorithms before I stake my claim! (And, to be pedantic, the pattern is a non-unique rectangle. Unique rectangles are quite welcome!)


Anyway, it was a thrill to find this. I have no idea how common this pattern may be, because the software I use to learn does not spot it! In fact, it will not even show the X-wing, because it is not "useful" (in that it does not lead to eliminations). It also does not identify the Rectangle.

Best wishes,

Keith

[keith]

I have spent some time pondering this. I think there are two observations:

1. There are useful variations of Unique Rectangles in which there is a "Diagonal Pair". For example, consider the following two blocks:

[Steve R]

Good stuff, Keith.

Have you considered the effect of weakening the X-wing assumption in your second example? It seems to me that, if there are just two places for 4 in the top row, the bottom left <246> reduces to <26>. Also, the top right <246> can contain any number of candidates as long as a 4 is amongst them.

If there is anything in this, the result is much less elegant than the X-wing but possibly of wider application.

Regards

Steve

[keith]

Steve,

I think you are definitely on the right track. This "Unique X-wing" is certainly cute, but I think it is more important to recognize that diagonal pairs might be useful to identify possible Unique Rectangles and, perhaps, X-wings. Once identified, there are a number of logical steps to be considered.

I believe the masters have declared Unique Rectangles with a diagonal pair to be "Type 5", and Ruud has proposed that the "Unique X-wing" be "Type 6". Take a look at (thank you, Tracy!)

http://www.sudoku.com/forums/viewtopic.php?t=3709

Keith's view:

This "Unique X-wing" is certainly a special case of something, for it enables you to immediately place values in squares.

Unique Rectangles are a way to eliminate possibilities, and these eliminations often result in values forced in some squares.

The next abstraction is "Almost Unique Rectangles", which seem to be mostly useful to define interesting cycles which may lead to possibility eliminations. If you follow the URL above, you will see discussions on "strong" and "weak" links in patterns and chains, which I think is exactly your point about "weakening the X-wing assumption".

Beat wishes,

Keith

[keith]

Today's SudoCue Nightmare puzzle has another example of this "Unique X-wing". You can see my message on the Nightmare discussion forum:

http://www.sudocue.net/forum/viewtopic.php?t=109

Best wishes,

Keith