
SudokuHelpPlus Worked Example 2

SudokuHelp+ Puzzle No 80 (which appeared in ^{m}X as Puzzle No 207) is deceptively complex. It has 26 given values and 29 cells whose values can be immediately placed with the OnlySpotBoxes solving rule. But after that you are stuck and to unlock the whole puzzle and solve it with pure logic requires the joint application of several different Player and Expert class elimination techniques.
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.
This Puzzle highlights the application of the Player Class solving rule, "IntersectReject", as well as the "SingleValueChains" and "MultiValueChains" Expert Class solving rules. I hope you find this tutorial instructive.
Regards Greg Shalless 
SudokuHelp+  Novice Class Rules  Only Spot Boxes:
I well not dwell on the details of the cells solved by this rule because the primary purpose of this tutorial is to focus on some of the Player and Expert Class Elimination strategies.
Move the Mouse Over the picture to see what SudokuHelp+Solver does. 

SudokuHelp+  Player Class Rules  Locked Pairs:
The {48} Locked Pair in the BottomRight Box prevents other cells in Row 9 from taking either of these values. Unfortunately it doesn't lead to further solved cells. Move the Mouse Over the picture to see what SudokuHelp+Solver does. 

SudokuHelp+  Player Class Rules  Intersect Reject:
The Blue Rectangle shows the 3cell Intersection of the TopRight Box with Column 8. You will note that the only spots the value 8 (encircled in Red) can go in Column 8 are within this Intersection. If 8 were to be placed outside this Intersection in the TopRight Box we would not be able to solve Column 8, so 8 can be eliminated from all such cells. However we still can't solve any new cells. By the way the IntersectReject Rule would have also done the same eliminations as the LockedPair Rule above. We are not done with this rule yet, as it comes in pretty handy again later on.
Move the Mouse Over the picture to see what SudokuHelp+Solver does. 

As an aside, to further clarify what I mean by the 3cell Intersection between a Box and a Row or Column I present the diagram on the right. The Pink Cells represent a Row where the Mauve Cells are the 3cell Intersection between that Row and one of the Blue boxes. The Yellow Cells represent a Column where the Green cells are the 3cell Intersection between that Column and the other Blue box. (The Orange cell is the Intersection between the Row and the Column).
The IntersectReject rule effectively says this: If a Value in the Row is confined to the Mauve cells, it can't go in the Blue cells or you won't be able to place it in the Row. Similarly if a Value in the Box is confined to the Mauve cells, it can't go in the Pink (or the Orange) cells or you won't be able to place it in the Box. If a Value in the Column is confined to the Green cells, it can't go in the Blue cells or you won't be able to place it in the Column. Similarly if a Value in the Box is confined to the Green cells, it can't go in the Yellow (or the Orange) cells or you won't be able to place it in the Box.
Now back to our puzzle.


SudokuHelp+ Expert Class Rules  Single Value Chains:
One link in a SingleValueChain is two cells in the same Constraint Region (my generic term for a Row, Column or Box), which are the only spots in that Constraint Region that a particular Value can go. A twolink chain is formed when two such links on the same particular Value have a cell in common. In each such link one end is the value, the other isn't, but we don't know which yet. In any odd length SingleValueChain one of the endpoints must be the value and the other cannot be. In this puzzle we have a 3link SingleValueChain on the value 6, where each link is shown by two blue circles joined by a blue line. You will note that there are only 2 spots 6 can go in Column 7 (1st link in the Chain), only 2 spots it can go in the Middle Right Box (2nd link) and only 2 spots it can go in Row 5 (3rd link). The cell at Row 3 Column 5 is in the same Row as one of the endpoints of this Chain and the same Column as the other, and since we know that one of those two endpoints must be 6, the cell at Row 3 Column 5 cannot be 6. But we still can't solve any more cells!
Move the Mouse Over the picture to see what SudokuHelp+Solver does. 

SudokuHelp+  Player Class Rules  Intersect Reject:
The Blue Rectangle shows the 3cell Intersection of the Central Box with Column 5. You will note that (thanks to the 6 eliminated in the previous step by the SingleValueChain) the only spots the value 6 (encircled in Red) can go in Column 5 are within this Intersection. If 6 were to be placed outside this Intersection in the Central Box we would not be able to solve Column 5, so 6 can be eliminated from all such cells. However we still can't solve any new cells.
Move the Mouse Over the picture to see what SudokuHelp+Solver does. 

SudokuHelp+  Expert Class Rules  Multi Value Chains:
The work done previously has set up a number of additional bivalue (with only 2 possible values) cells which gives us the opportunity to deploy the MultiValueChains rule. There are 4 MultiValueChains shown in the image below and 3 of them serve to eliminate 3 different values from the cell at Row 3 Column 4, such that there is only one option left and that cell is thereby solved. The first and simplest Chain to follow is shown in Red and eliminates 7 from that cell. It is a twolink Chain and is otherwise known as the XYZap rule. 6 gets eliminated from the cell at Row 3 Column 4 by the 7link MultiValueChain shown in Blue. And finally 8 gets eliminated from that cell by the 6link Chain shown in Pink. Now there is another MultiValueChain there that eliminates 4 from the cell at Row 1 Column 6, where the two endpoints of that Chain are actually in the same column. This is a 4link Chain but because it involved a number of cells that already have multiple circles around them I decided to highlight it in a slightly different way using a Green box in the corner of each cell involved in the Chain. There may well be other MultiValueChains present but because the Solver is operating in a mode where it stops as soon as it finds a solved cell (in the hope that this is enough to solve the puzzle) we will stop looking as well and see how far the solved cell takes us. Move the Mouse Over the picture to see what SudokuHelp+Solver does. 

Finally some real progress, we can now solve a cell, which we will do with the OnlyValue rule. The trouble is that's as far as we can go, but it does set up a couple more bivalue cells, so maybe MultiValueChains will come to our rescue, again! Move the Mouse Over the picture to see what SudokuHelp+Solver does. 

SudokuHelp+  Expert Class Rules  Multi Value Chains (again):
A new 5link MultiValueChain is shown in Blue which enables us to eliminate 6 as a candidate in the cells at Row 1 Column 4, and also at Row 9 Column 6, the latter of which solves that cell. So once again although there may be other MultiValueChains present, we stop looking for more and see how far we can go with the new information. Move the Mouse Over the picture to see what SudokuHelp+Solver does. 

From here it turns out that the puzzle is now a piece of cake to solve completely. I chose to do it with the Novice Class rule OnlySpotCols. Move the Mouse Over the picture to see what SudokuHelp+Solver does. 

SudokuHelp+  Expert Class Rules  XYZap:
As was suggested there were other MultiValueChains present at that final step that unlocked the puzzle, and some of them were considerably easier to find than the one we showed you, which was the first one the SudokuHelp+Solver happened to find. However had we used the XYZap rule instead, we would have found a number of 2link MultiValueChains, which is after all what the XYZap rule really is. Each one is shown in a different colour. Move the Mouse Over the picture to see what SudokuHelp+Solver does. 
