This post describes the follow up to the candidate fill of each line in Sysudoku line marking.  This includes the marking of line slinks, checking for X-wings, identification of  naked and hidden singles, and marking of bi-value cells. It also includes checking for locked sets, both naked and hidden.

I hope you succeeded in finding all candidates of the Sue de Coq post puzzle.  Line marking trace is  unusually simple. No new clues were discovered for follow up box marking.  My line marking trace was

3f: r9, c9.   4f: r1, r3, r7.  5f: r6.  6f: r2.   Column closure: c1, c2, c3=>hp14, c7, c8.

We went from 3-fills (3 cells to fill), to one 6-fill.  There was no contest on which direction to go to complete the candidates.  When we finished the row of five free cells, there was only one row left, so we passed up the column fives and went for it.  On the closure of column 3 , . . . wait, did I mention closure?  Well, you’ll read about it below.  As I was saying, on the closure of column 3, both 1 and 4 appear in only two cells, so 2r7c3 is removed. This is called a hidden pair, and the removal produces a locked set of two cells, a naked pair.  It happens a lot, but usually when you’re completing the locked pair. The next post is doing the full frontal on closed sets.

For now, better find a quiet place and concentrate.  The previous post covered only the filling of lines with candidates.  This one is a checklist of follow up actions that wring the last drop of logical truth from each line.  As plodding as line marking is, on tough puzzles it is collecting the low hanging fruit. And from these head crackers, we need the full harvest.

Having just filled a line, we now have determined the remaining candidates in every cell of that line.  The following is a checklist of follow up actions:

1. Check for a hidden single, i.e. a fill number in only one cell.

A naked single pops out at you as all other numbers drop out of the fill list of a cell. When you finish filling the free cells, a number that appears only once in the line is a hidden single.  It may not be obvious among all the other candidates, but it’s a new clue.

That reminds me.  Did you box mark Dave Green’s 12/06/13 4-star ?  The trace is

Now you can checkpoint your box marked grid with this one.  Line marking will start with 3f: c2 with fill string 689.  It produces the hidden single   Whs8 and naked pair np69.

Next up is 4f: r4, with a fill string of 235. We get a naked single 2, whose effects include a hidden single 5!  This begins the collapse.

For next  time how about marking that much tracing far enough to show how the collapse gets under way.

2. a. Set slink marks for the line.

Move marks to the bottom cell corner for row or column slinks, add line marks.  In c2 above, we place the clue 8r5c2, tracing it as Whs8, but we place the naked pair string “69” in the lower right corner of the remaining two cells, marking the two column slinks.

Line slinks are usually found by checking each number in the fill string.

In review puzzle 40 from Tom Sheldon’s Sudoku Master Class, two row slinks are found  in r4, the first line marked.

2.b. As line slink marks are placed, check parallel marked lines for X-wings.

With the naked pair c4np48, found in box marking, the r4 slink forms a 4-wing.  Suddenly,  the two lines r4 and r6 have reserved columns 1 and 4 for their 4-candidates. One line takes c1 and the other, c2. No other 4’s are allowed in the two columns.  That is the X-wing law.

So far, there are no outlaw 4-candidates to remove, but since line marking isn’t finished, we just leave the “fish” icons pointing down the columns, to be reminded not to break the law.

Fishing is covered later in a series of posts. It is an advanced strategy, well beyond basic solving,  but this is the right time, and the most efficient way to find X-wings that currently exist.

3.a. Mark locked sets in the line.

Hidden and naked singles are one form of locked set. Hidden and naked pairs, triples, etc. restrict other candidates on the line, and can bring removals.

Detection of locked sets is the subject of my next post, but here is an example of a naked triple I found on a Wayne Gould puzzle from Train Your Brain Sudoku, Fiendish 75.  After marking c2, and while checking for more slinks, I noticed that numbers 3, 7 and 8 are alone in the three cells r136c2. It requires all of them to solve these three cells.  They cannot solve any other c2 cells so their candidates are removed from the other boxes.

The effect is to remove three candidates.  Here also is a second example of an X-wing found in line marking.  This time the 4-wing has a removal. The markers will be left in place until r5 and r8 are filled.

2.b. Mark locked sets in newly covered boxes.

That is, when the third line across a box is filled,  and candidates within the box are known,  look for hidden locked sets among those candidates.

We need only follow Sheldon’s Master Class 40 a little further for an excellent example.

The last filled line, r9, completed coverage of the SW, S and SE boxes. In SW,  numbers 4, 7 and 9 are confined to three SW cells, so some arrangement of these three numbers must solve these three cells.  Accordingly, other numbers are removed from these cells.

Before the victim diamonds are added to SW, it’s not easy to spot the hidden triple.  The key is n numbers being limited to n cells, without regard to other numbers in the same cells. The next post  has a scratch pad algorithm for deciding such difficult cases.

3.  Make boxline removals and promotions.

If new line marks are confined to one box,  remove other candidates of this number from the box.  This form is called a boxline.  It can also produce a  new dublex .

Another Sheldon review puzzle, Master Class 60, provides a nice example.  On the first 5f: line r3, the cells containing 2 are confined to the NW box.  One of them must therefore solve to 2, hence the removal and NW5.

4.  Border the line’s new bv cells.

A green square border is dragged in to border every  bi-value cell.

Early advanced solving methods depend on bv cells.

Line marking normally ends with line marking closure.  When there are few lines left to fill in each direction, you decide whether to finish with rows or columns.  When lines of the chosen direction are all marked, then all cells are filled with candidates, but the follow up actions are now done for unmarked lines in the other direction.    This is line marking closure.

Box marking and line marking are systematic and efficient ways to assemble all candidates for hard puzzles and devastate easy ones.  They do require concentration, but carry you along with steady progress.  A grid template is definitely helpful, and I recommend tracing on tough puzzles.