## Expanded XY-Chain Patterns

Advanced methods and approaches for solving Sudoku puzzles
RonK wrote:You make it sound as if the "extra power" comes from the ALS. That is just not true. The extra power comes from the continuous loop. Your example is a continuous loop even with r7c1 and r6c1 treated separately.

- r2c1 -9- r1c5 -3- r7c5 -1- r7c1 -5- r6c1 -3- r2c1 - continuous loop

PIsaacson wrote:Being a devotee of ALS chains, I would have said, "The extra power comes from the continuous loop as well as the dual-linked ALS chain.OWTTE

From another angle, dual linked ALS and continuous loops share a common property, which might be considered the source of such extra power.

Any group of cells, or other strong sets, whose candidates can be contained in an equal number of links will eliminate all other candidates residing in the links. Here "link" is used in the general sense to mean sharing the same house or cell.

This is a general property that can be applied to all logical Sudoku solving methods.

One example might be this simple mixture of ALS with other logic here.
Allan Barker

Posts: 266
Joined: 20 February 2008

Allan Barker wrote:From another angle, dual linked ALS and continuous loops share a common property, which might be considered the source of such extra power.

Any group of cells, or other strong sets, whose candidates can be contained in an equal number of links will eliminate all other candidates residing in the links. Here "link" is used in the general sense to mean sharing the same house or cell.

Allan
Superb observation.
You are saying that the two brightest stars in the sudoku constellation (doubly-linked ALS and nice loops) share the common property of having rank 0, that is (sets-link sets) or (base sets-cover sets) = 0.
Last edited by aran on Tue Mar 24, 2009 8:32 am, edited 1 time in total.
aran

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Joined: 02 March 2007

### Expanded XY-Chain Patterns

There is also a third more basic technique that can be used to make the cell eliminations for the root XY-chain that results by removing the 3 from r7c1 in the example. The XY-chain then becomes an XY-chain for every digit in the chain and thus cell eliminations are possible for every digit. I think this can be generalized as follows. Any XY-chain with N cells and N digits which is also an XY-loop is a multidigit XY-chain or a continuous loop or a doubly-linked ALS-XZ, and cell eliminations are possible for all of the digits. I didn't notice Allan Barker's reply when I first wrote this. Maybe this is the same thing that he is saying from a different viewpoint.
Bud

Posts: 56
Joined: 24 August 2008

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