Sudoku-Help-Plus Multi Value Chains
	Sudoku-Help+ Puzzle No 78 (which appeared in mX as Puzzle No 159) is beautiful example of the Multi Value Chains solving rule. This is the killer rule for solving difficult Sudoku Puzzles. To find it you only need to look at bi-value cells so those solvers who don't like to do full mark-up can deploy it. We are looking for a Chain of bi-value cells where each link in the chain is two cells in the same Constraint Region with a value in common. You start with a bi-value cell, say {az}, and look for another bi-value cell in a common Constraint Region with a value in common with it, say {ab}. After that we look for {bc}, then {cd} and so on. The aim is to find a cell whose other value is z, for example {dz}. Now {az} might be z but if it’s a, then {ab} is b, {bc} is c, {cd} is d and {dz} is z, so any cell in the same Constraint Region as both {az} and {dz} cannot be z since one of them must be. A one link Multi-Value Chain is a Locked Pair. A two link Multi-Value Chain is a Locked Triple if all three cells {az}, {ab} and {bz} are in the same Constraint Region, and if {az} and {bz} aren't in the same Constraint Region then it is an XY-Zap. Longer Chains will often enable eliminations not discernible with any other Solving Rule. To get most value out of this tutorial in solving Sudoku it is strongly recommended that you attempt to solve the puzzle independently yourself first.
Sudoku-Help+ - Expert Class Rules - Multi Value Chains:
This shows the result of applying the Novice Class solving rules to the puzzle, which you should be able to arrive at fairly easily yourself. If not then you probably need to learn some of the Sudoku solving basics before revisiting this tutorial. Move the Mouse Over the picture to see what Sudoku-Help+Solver does.
This image shows how the XY-Zap rule (which is none other than a 2 link Multi Value Chain), can be used to solve the cell at Row 5 Column 6. In Multi Value Chain terms the {23}-cell is {za}, the {39}-cell is {ab} and the {29}-cell is {zb}, where z (ie 2) can be eliminated from any cell in the same Constraint Region as both {za} and {zb} since one of them must be z. From here the puzzle can be completed with Novice Class rules, but the point of this tutorial is to show you the full power of the Multi Value Chains rule, so let's look into the puzzle a bit deeper. Move the Mouse Over the picture to see where XY-Zap applies.
The puzzle is rich in bi-value cells (those with only 2 possible values) and as a result there are Multi Value Chains all over the place. In fact there are enough Multi Value Chains present for you to be able to reduce virtually any unsolved cell in the puzzle to a single value, that is to completely solve the puzzle. One such chain is shown but you should be able to find heaps of them. Each link in the Chain is shown with a different colour circle around the common value forming the link and a line joining the linked cells. Move the Mouse Over the picture to see the 5 link Multi Value Chain that solves the cell at Row 1 Column 5 by eliminating the 8 from the {68} there. It should also be clear that this same Multi Value Chain also eliminates the 8 in {18} at Row 6 Column 4.
And for good measure here's another much less convoluted 3 link Multi Value Chain that eliminates the 3 as a candidate in the cells at Row 3 Column 6 and Row 4 Column 4. Move the Mouse Over the picture to see it.
Finally this image shows you the 5 eliminations we have highlighted above and what the Multi Value Chains rule programmed into the Sudoku-Help+Solver does with this puzzle. It actually reduces 18 of the 19 unsolved cells to a single value. The only reason it wasn't able to solve the cell at Row 3 Column 3 was that I made a decision not to look at Multi Value Chains that start with a Locked Pair. Move the Mouse Over the picture to see what Sudoku-Help+Solver does.

Here endeth the lesson! We trust you learned something from it. We also hope you liked what you saw so much that you can't wait to get your hands on Sudoku-Help-Plus. All you have to do is click this PayPal link and it's yours. It will be emailed to you on receipt of your payment. You can find out more about Sudoku-Help-Plus here! There's also a third tutorial here highlighting the Intersect-Reject, Single-Value and Multi-Value Chains rules.