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Introducing Chains and Links
From sudokuwiki.org, the puzzle solver's site
|In the first diagram I show a distribution of the candidates for number 3.
Drawn in blue and green are the possible links between certain candidates. The commonality is that some of these rows, columns and boxes there are only two candidates left. This means that one or other of these candidates will be the solution. We don't know which but each link described an EITHER / OR relationship. I've shown in green the links within a box, but often the link could be in a row or column AND a box, but it's not relevant which type of unit if the link is in both.
Links of this type are called Bi-location links, because the link is across two cells.
Some 3s on the board
In the next diagram I have picked a random Sudoku from my solver examples and highlighted all the cells with two remaining candidates. These cells are extremely useful and will play a huge part in most chaining strategies. Because each yellow cell has only two candidates left, we know one or other will be the solution and exactly the same EITHER / OR relationship is present.
Such cells are known as Bi-value links. (Both bi-value and bi-location candidate pairs are also known as conjugate pairs (or complimentary pairs) but these are term I am moving away from).
I don't need to draw a link between each pair value in each cell, it is implied by the nature of the two candidates in each cell.
|The beauty of a chain is that it will resolve into one of two states. In this diagram I have a simple 5 cell chain. The links could be across rows, columns, boxes (bi-location) or within a cell (bi-value), it doesn't matter. The point, that when we get to the solution either the bottom left part of the diagram will come true or the bottom right: that is, one set of cells will be ON (the solution) and set will be OFF (not the solution). Note that the ON/OFF alternates. Pretending to set the state of any of the chain candidates to ON or OFF automatically sets the state of the rest of the chain.
The ON/OFF states of a chain
|Chains, Loops and Nets
Chaining strategies can be divided into two types: whether they use a linear chain or a spreading network-like pattern. Linear chains have a start and end and do not branch out. A 'net'-like strategy such as 3D Medusa takes advantage of all the chains possible on the board. Obviously a net of chains contains lots of small linear chains in many combinations.
Chain, Loop and Net
|The strategies listed to the right use only bi-location links. Effectively that means they stick to one number and ignore all other numbers. You can think of them as working within one plane or dimension.||
|These strategies use bi-value AND bi-location links. Whereas the strategies before stick to one number these can move between number plains using bi-value cells and are hence 'two dimensional'.||
|These strategies use strong and weak links to bust out of the constraints of bi-value and bi-location only link connections.||
|These strategies use, in addition to bi-value and bi-location links, other more sophisticated links, although they are still two dimensional.||* Also uses exotic types of links such as Grouped Cells and Almost Locked sets.|