This post continues the visit of *Sudoku Satisfaction* readers to Sysudoku with a demonstration of the slink marking bypass, line marking inscription, and basic subset spotting.

How was lunch? . . . Good. OK, we’re ready to continue. Here is the grid of our puzzle, Saturday 2, after Sysudoku box marking.

This box marking, as is customary in Sysudoku now, was performed in two phases. The first phase, *slink marking bypass*, is essentially box marking without writing the slotting pencil marks for slinks and aligned triples. We do mark naked pairs (twin pairs) or other subsets (partnerships).

New York Post Sudoku puzzle maker Wayne Gould is the inspiration for the Sysudoku bypass. He advised “shaking free of pencil marks” in order to see the true beauty of Sudoku. After the bypass, we do put the pencil marks in, but it is a joy to go as far as possible without them.

As it turned out here, the bypass produced all of the clues (uno’s). The pencil marks above, except for the naked pairs Enp14 and c2np59, were added afterwards. Later, on your copy of Saturday 2, read the bypass trace, accounting for every effect. In the bypass, the numbers are still taken in increasing order, but combined into one list.

Many of the bypass effects depend on the slinks shown above, but pictured mentally while doing it.

In Sysudoku box marking, we bypass more than the slink marking. We do not attempt crossing or c-u-c on lines of more than three unresolved cells. It’s not that we’re lazy. It’s because we have a very efficient lazy man’s inscription process, that incorporates partnerships.

Before taking that up, let’s talk about the kind of partnership we can handle in box marking, i.e. before inscription. Partnerships are more generally known as subsets. A *subset* is a set of *n* cells of the unit containing candidates of exactly *n* numbers. In the solution, each of the n cells will contain one of the n numbers.

The naked pairs (twin pairs) we’ve been discovering are naked subsets with n = 2. They are marked when two slinks of different numbers converge on the same two cells. Naked (plain)subsets are so named because there are no other numbered candidates present. In box marking that can happen with higher values of n, as long as we know that other numbers cannot be added later to the subset cells. So we can have naked triples, naked quads, etc.

Of course, when you spot a naked subset, you can remove any of its numbers from all other cells of the unit. If there are any other cells. If not, it’s not exactly a subset, is it?

After inscription, when all remaining candidates are known, hidden subsets are also possible. This occurs when candidates of numbers other than the n numbers required are present. How do you spot them? Look for *n* numbers are confined within *n* cells. To confirm difficult cases, Sysudoku provides a scratchpad algorithm to find them. When you find one, all the other numbered candidates are removed from the subset cells. These cells are spoken for, and the extras are locked out.

Let’s move on to line marking the Saturday 2 grid above. As we go through it, I’ll point out the best time to look for subsets of both kinds. After that, new ones can appear when removals are made.

At first reading of Satisfaction, I thought that inscribing was done to end basic solving with all candidates, and partnerships came after that. Then I came to realize that inscribing and line marking have the same input and output except for the c-u-c and crossing done before that. Then it finally dawned, why do the harder c-u-c beforehand? You’ll see in line marking a better way to do it. I crossed off crossing ahead of line marking very early on, when first announcing line marking in 2011. I showed that no naked single escapes the normal line marking. And the sweet part is that you can spot your other inscribed subsets in the line marking.

So let’s look at the grid above. Computers have a simple algorithm to do it, but most people number scan every unresolved cell. That means crossing each one, and the box slink pencil marks are of little help in doing that. Only marked unos and marked subsets are exempt.

OK, let’s at least do the units with fewer unresolved cells first. We may add unos and make eliminations on the harder ones. But we can do better. Let’s do lines only, in which the cells have largely the same needs. Every cell gets covered, right?

Heck, let’s write down the numbers needed to fill the line, and apply the list to every cell. If we start with a copy for every cell, we just take away those appearing in the crossing line and box of that cell, and we have the candidates to inscribe the cell.

Good grief! In ©PowerPoint I can place the fill needs in a string on the side or bottom of the line, make a copy with a Cntrl-C, paste a copy in the cell with a Cntrl-V and edit out the unwanted digits. In this systematic process, the slink marks help a lot. These are numbers already placed. Slinks along the line eliminate digits from the line’s fill string. They’re already marked. Numbers in three boxes, as clues or box slinks, are omitted from the fill string.

Now on top of all that, go back to that original thought and line mark first the lines with fewest unresolved cells. You could do it from here without any more coaching. Here is the trace:

You start with the 3f: list. The label means three free (blank) cells. I go down rows, then left to right on lines of the same number of free cells. The order can change when unos and subsets occur. You may have to refer to the finished product below the first time through, but it comes naturally very soon. You may want to write for a ©PowerPoint template (free). See the Tools link on the menu line above for the email.

The event on the row 9 marking is a box/line interaction where the lone pair of 5’s in r9 in the S box means no other 5’s can be in the box. The lone 5’s are slinked because one of them must be true, for r9’s sake.

The last list names the lines in the cross direction that are left when lines of one direction is completed. Even though all cells are marked, extra tasks performed in line marking remains to be done for these lines.

One of these is marking the cells containing two candidates. They play an important role in early Sysudoku advanced methods.

Also, there is the repositioning of marks to denote line slinks and triples. An example is the pair of 8’s in c1. In this, box marks retain their positions. A low corner mark is especially significant, and the partnering box mark is easily found.

And then there is the review of the marked line for subsets. If the line is the third to cover the bank or tier boxes, it is the right time to examine these boxes for subsets. This closing action on the SW box would have shown the alternate form of box/line when the 5 slot takes out 5r7c6.

The line marking ritual also includes another action for every new line slink. That is to spot a matching line slink in a parallel row. That would uncover a X-wing, the first fish you will find in many puzzles. That’s the case, for example, in Satisfaction’s Friday 2.

Break time. You get 30 minutes. Bathrooms are down the hall to the right.

We’ll be back (in next week’s post) for a quick look at some of the advanced methods and tools that follow line marking in the Sysudoku Order of Battle, and contribute to the solution of Saturday 2. We’ll also show how some of them are covered in the Satisfaction Solution Triangle methods.