dailysudoku.com

Glassman · Joined: 21 Oct 2005 Posts: 50 Location: England

If you have corners of a rectangle with candidates (ab), (bc), (cd), and (da), there are only two possible solutions. These will eliminate all other b's on side 1, c's on side 2, d's on side 3, and a's on side 4.

What is the proper name for my Exclusive Rectangle?

Or is it a special case of some more general technique?

TIA

Glassman Cool

Steve R · Posted: Mon Jun 11, 2007 8:39 pm Post subject:

Hi, Glassman!

I don’t know of a name for the specific rectangular pattern you describe but the underlying concept is an xy-chain. Rather confusingly they are also sometimes called y-chains or y-wing chains: xy-chain seems to be the prevalent term today but nothing is guaranteed in sudokuspeak.

The characteristic of such chains is that the nodes consist of cells with just two candidates and that adjacent cells have a candidate in common, which forms the link. Thus your chain could be written:

[ab] – b- [bc] – c- [cd] –d- [da] – a- [ab]

Here [ab] stands for the cell with a and b as candidates and –d- represents a link.

When, as here, the left end cell is the same as the right hand cell, the chain becomes a “loop.” In fact your example goes one better. You can start with any of the four cells and get a chain with the exactly the same logic behind it. This feature makes the loop continuous. A continuous loop consisting of an xy-chain is termed an xy-ring or an xy-cycle.

In any such ring, the labels (a, b, c etc) can be eliminated from any common associate of the cells which the label links. It does not matter how many links the xy-ring contains.

Wide ranging extensions have been developed. xy-chains are described here and nice loops (the ultimate extension for the moment) here but this is a vast subject.

Steve

Glassman · Joined: 21 Oct 2005 Posts: 50 Location: England

Steve — Thanks. I found this in a most unexpected source, last week's Radio Times. Most are easy, but every now and again the RT includes a gem that makes your head hurt!

This xy-ring just leapt out, as there were very few other cells with just two candidates, although it took me some time to work out its significance and how to utilise it.

Glassman Cool