As you finish box marking, the grid contains pencil marks of box slinks, aligned triples, and subsets within boxes or 3-fills. Line marking fills in the line slinks and subsets, and all remaining candidates.
That makes line marking the last phase of Sysudoku Basic. In a puzzle surviving line marking, all candidates that can be identified, based only on the defining Sudoku placement rule, are pencil marked on the grid. Also, box slinks and line slinks, the strong links created when exactly two candidates are present in a unit, are distinguished from other candidates by placement location along cell edges. Also, every bi-value cell is marked.
The grid is ready for advanced methods, those hidden implications of the grid structure and placement rules that advanced solvers discover.
For a detailed walkthrough of Sysudoku line marking, we’re repeating the review solution of puzzle 25 from Train Your Brain Su Doku: Fiendish, collection by Wayne Gould, published in 2008.
In my review starting November 4, 2014, I disagreed with Wayne’s stand on written pencil marks, but his idea became the inspiration for the slink marking bypass, as a way of generating more clues with fewer candidates.
Box SW has the only bv on my grid. I keep a green square on the template to drag over for that. Ask yourself why the other cells with two candidates are not bv.
Yes, “bv” , pronounced “bee-vee” is both singular and plural, like “deer”.
Four lines are decorated with fill strings. Well almost. The primary action of Sysudoku line marking is to copy the fill string into each unmarked cell of the line, then eliminate from the copy any number sharing a unit with the number.
Therefore, there’s no reason to include a number that is eliminated from every copy. As we make up fill strings, we omit those. For fill strings from 3-fills, we just delete (scratch out on paper) such numbers.
Now that you are reading Sysudoku traces, we can start with the grid entering line marking. To find your own way here, just pick out the givens. Then compare routes.
You can even request a free ©PowerPoint grid template from the Tools page. I’ll include some advanced method templates with it.
So what lines do we start line marking with? The one’s with the fewest free cells. Easiest first. In case of ties, rows downward, then columns rightward. Our marking list starts as 3f: r1, r7, c2, c3. 3f: ? That’s our label for a list of units having three free (open for placement) cells.
Here’s what we do on rows r1 and r3. Three new bv come with the markings.
Got it? Your turn on r7, then check below.
Earlier posts still have duplicating marks to show both box and line slinks, but I gave that up. The grid was too cluttered. It works better to stick with one mark per candidate in a cell, giving precedence to box slink marking over line slink marking.
On the columns, 4 and 6 are scratched from the fill strings. In c2, only one cell to fill, and a 5 is shifted to show the first column slink, in the right corner. In c3, 2r4c9 removes a 2 from the copy.
As each line and box is completely filled, we check for subsets. That is a set of n cells containing all candidates of n numbers, where n is less that the free cells. All of the units filled so far fail that second condition.
Now to see if there are any 4f:’s. Actually it’s easier to look for 5 closed cells.
The first one is r9. Of the 4 numbers, two are discarded, leaving 29. You can account for the fill string numbers that way. Then c5 and c9.
Remember to check for line slinks and subsets on each line fill. To cover the grid, you only need to line mark all rows or all columns.
On to the 5 free list. 
After marking r2, the North boxes are filled, and we check them for subsets. Perhaps NE? Are 4 numbers confined to 4 cells? 3? No cigar. Row r4 is a bit of a subset question in itself, but again, no.
The South boxes are filled with r9, but again, no subsets.
We’ll wind up the Sysudoku Basic series with a page on subset spotting and confirmation. And a crumb we leave for math people and programmers.
Now we have a typical line marking decision. Do we go to 6 free cells with r5, or do columns 7 and 8 first. I vote for r5, because after completing the 5: c7, c8 list, we may still have a 6: c4 and 7: c6 to do. A little planning doesn’t hurt.
Good choice! Not only is the final fill string short, but we get two Gould-on gifts – naked triples in c4 and c8. Since 2 and 6 are absolutely required for the triples, they are eliminated from other c4 and c8 cells.
This is a good place to note that complementing hidden subsets accompany the more easily spotted naked subsets. In c4 the hidden triple 349 is present as well. You can spot it by realizing that 3,4 and 9 are confined to three cells, r4c348. Got that? One row and three columns.
The terms “naked” and “hidden” are not about subsets. They’re about spotting subsets. Once the doomed candidates are removed, we have two subsets, dividing up the cells and numbers. Quickly, what is the hidden subset in c8?
Here is the trace of the line marking so far. Line marking traces show only the order of lines marked, and of course, the hanging trees of effects.
Even though all rows have been marked, line marking is not finished. Line marking is closed by checking the unmarked lines of the other side – in this case, the columns – for line slinks and subsets. We call that the close of line marking. With all the removals on the above grid, we may not get to the close on this one. You can check out the format of the Close list on almost any current post.
Sysudoku is unique in making the solving path accessible to all readers. The reward is, readers find my many mistakes, and tell me about them. If you got this far, you will probably join them.
When the Guide series gets to fish, I’ll add another line marking chore carried over to the close. It’s checking line slinks for X-wings. That’s also covered in the Line Marking Follow Up post of October 2011.
As for Fiendish 25, after removing the diamond doomed candidates above, you can follow up with the collapse on your own, or follow this trace as long as any discipline is required.
A breakout like this in the middle of a line marking demands special care, because you can’t trust the unmarked cells. No worries here.
The last two Sysudoku Basic pages explain more about box/line exclusion and subsets.
Systematic Sudoku 