This page is for absolute Sudoku beginners and anyone else wanting to evaluate Systematic Sudoku. It begins a series of pages explaining what Sysudoku methods are, why they work, and how and when to do them. These pages are a more accessible version of the methods reported in first year posts of 2011 and 2012, along with innovations added since. They also contain many links to Sysudoku posts over the interim years that illustrate methods especially well.
Systematic Sudoku, or Sysudoku, is a human engineered system for solving Sudoku puzzles of all levels of difficulty, and for documenting the solving process. It uses constructions and visual tools in place of exhaustive searching, for a more humanly effective, and enjoyable solving experience. As a reader, you might stop well short of the techniques you see in the current Sysudoku reviews of puzzle collections, and still find yourself logically solving most of the puzzles you encounter.
On the other hand, you might become a sysudokie, someone interested in improving and promoting systematic methods required for the human solving of very difficult puzzles. Experts are welcome to listen in, and contribute comment, but the blog is not another forum for discussions.
Like many strategic games, a Sudoku solution is earned by accumulating small advantages. “Sudoku” literally means “number placement”. Solving is placing numbers 1 – 9 in the grid cells according to the rule published with every puzzle. You are to place these numbers only as you become certain that they belong there. The beginning clues, the givens, define the puzzle. In addition to clues, the solving process adds candidates, the known possible placements in each unsolved cell. The phase of solving in which candidates are still being identified is Sysudoku Basic. Many Sudoku are solved in this phase.
Once all remaining candidates are known and posted to the grid, advanced methods gradually remove candidates and add more clues. These methods are based on relationships among candidates. The solving process eventually reaches collapse, a critical point at which normal follow up to a removal leads directly to the solution. The extreme stage applies only to puzzles you are unable to solve with advanced methods. It involves trials, methods testing sets of candidates logically determined to be true or false together, “true” meaning “in the unique solution”.
This page takes you through a detailed solving of a 4-star Dave Green puzzle, to a collapse point, in the earliest stage of Sysudoku Basic, called– for reasons described later – the bypass. In itself, this page will enable you to solve most newspaper and magazine puzzles. It will also give you an idea of the challenge and dynamic nature of the Sudoku. Finally, these basic pages explain most of the engineering choices made in the design of Sysudoku Basic.
Here is how boxes and lines are named in Sysudoku. Box names are compass direction from a center box C. Rows are numbered North to South, and columns, West to East.
Sysudoku Basic begins by restricting possible placements within boxes. In the bypass, we use the fact that a box can contain only one copy of a number. Clues and aligned candidates outside the box “sweep” the number from cells of the box. Not only that, but clues on different lines can be coordinated to force a clue in the box, in two ways:
One bread-and-butter move of box marking, the double line exclusion, or dublex. In the dublex, the same number sweeps two lines of two boxes in the same direction, forcing a placement in the third line of the box they cross.
Of course, new clues join right in, eliminating clues in other boxes. In all Sysudoku diagrams, added clues and pencil marks are in a script font, like the 1 above. Formal script numbers, like the other clues above, are givens, which define the puzzle.
The other bypass mainstay move is the crosshatch. Clues of the same number in two crossing lines simultaneously sweep a box. Here, E (East) and SW (yes, SouthWest) 1’s sweep the W box, leaving only one cell for the W 1-clue, or as we name it in Sysudoku traces, W1. Where is W1?
Did I mention traces? A trace is a record of the solving actions. In Sysudoku traces, space is saved by not saying why or exactly where in the box the change occurs. Instead, the trace reader knows, or figures out, why the change occurs. She posts each placement to the grid as she reads. In this case W1 goes into r6c1. On partial grids, you have to look where it is cut off. Was that your answer?
OK, let’s do the bypass with trace on a 5-star puzzle. This one was offered in Dave Green’s the Akron Beacon Journal on December 10, 2017, a Sunday.
We sweep all boxes of each number, in 1 through 9 not already containing the number. When there are more than two clues of a number on the grid, we do this box by box, going West to East in each bank, North bank through South bank.
With only two candidates there can only be one dublex or two crosshatches. We need only look at the crosshatch targets NE and S. The S sweeps leaves two aligned candidates, which form a dublex with N1, confining C1 to the intersection of C and c6. This is not enough restriction for a clue, but it shows how aligned candidates can define clues, a very important principle in the bypass.
The bypass goes better if we account for the effects of aligned candidates, without marking them on the grid. In the next phase, box marking, we do write them in. In the bypass the objective is sweeps leaving a single cell for a clue. In the grid above, see you can find a 2-clue.
It would be easier without the distraction of sweep arrows, but look at each box not containing a 2-clue, left to right and top to bottom. It doesn’t quite happen. Sorry for the tease, but the search is systematically necessary.
We have better luck with the 3’s, hitting NW3 immediately. Before looking at more boxes, however, we look for the effect of the newest clue. It so happens that NW3 generates NE3 and W3, both by dublex.
We can start the bypass trace.
NW3 is a cause, and indented below are listed its immediate effects. We continue by treating each listed effect as a cause. Each event is described as a clue value in a box. The reader is expected to post events to the grid as she reads, keeping the grid up to date. Beyond that, the reader supplies the where and why. It’s a very compact record. Reading traces is a great learning tool.
For the effect list, we also check for any new placements of earlier numbers 1 and 2, ignoring effects on later ones, because we will get to them, and it is not efficient to check for the same event twice. Efficiency promotes happiness in Sudoku.
Note that W3 is posted in a smaller font. That’s a visual aid, to mark that W3 is still an effect, as the first effect becomes a cause. What are the W3 effects? The dublex with E3 leaves two free cells. Two aligned free cells can be significant, but not here. But looking in the other direction, there is something. W3 leaves three free cells in r1, for three missing numbers. It’s a 3-fill.
A 3-fill is a box or line having only three cells free for placements. Here is the plaque on the Sysudoku office wall governing 3-fills.
Let’s call the first condition, the two-number rule; the second, the two-cell rule. These conditions produce surprising clues. Convince yourself that they work every time. The 3-fill rule was adopted into Sysudoku basic in 2016, five years into the blog. Why?
The 3-fill gets to the free cell squeeze sooner. One free cell means a clue, two free cells, a pair of cells reserved for two numbers. The 3-fill was taken from the third basic stage, line marking.
You can extend the squeeze further with a 4-cell rule, with a three-number rule and a three-cell rule. That would require much more searching, for much less frequent clues. I think this puzzle tells us that we have the right balance.
In this case, the missing numbers are 4, 8 and 9. Neither condition is met, but what we can do is put a string of the missing numbers next to r1, the 3-fill unit, to alert us to any action that satisfies the 3-fill condition. It is represented in the trace by a 3-fill marker showing the unit and the missing numbers.
Still on the grid above, we turn now to W3, promote it to a cause, and find it has no immediate effects. Continuing the box scan, two free cells are left in C, two in S, and two in SE. On the 4’s, the single W4 has a chance because of the SE “wall”, SEr9 . W4 leaves an unwritten slink in SE. The hidden dublex leaves another in SW.
Going on to the 5’s, a crosshatch N5 leaves a 3-fill in N. Missing numbers are 4, 8 and 9 again. Then as the box scan reaches SW, the SE wall’s unwritten aligned triple dublex/NW5 crosshatch produces SW5, taking one of those free celsl for SE4. You could interpret it as SW5 creating a 3-fill r9, and the given 4 seeing two of its cells. We jump on the two remaining free cells of r9, and N1 places the missing 1.
Here is an unfair pop quiz. Who is the cause of the two-cell squeeze that now produces S7 and S3? The current trace holds the answer.
Now S1 is considered the cause of S7 and S3, because C8 is already posted as an effect when S1 becomes a cause, S3 forms two dublexes for C3 and SE3.
Notice how the trace at this point becomes a guide to the next marking action, in a way that avoids overlooking any effects.
When the C3, SE3 and then S7 bringing no clues, we step back up to the last effect listed, but not examined as a cause for effects. We pull it to the right, where there is space for an effects list, and continue. In this case, S8 is promoted to cause status, and the c4 3-fill is noted, with a fill string on the grid, and a bracketed version in the trace. Since the 3-fill has no immediate effect, the “step up” is actually repeated, with the search for the next “pull out” reaching the top line of the trace, where original causes are added, and we are directed to look to see if SE5 is there yet.
Now let’s get real. Managing a trace like that is easy in the comment section of standard ©PowerPoint page. It’s harder in pencil or pen, and I don’t trace when I’m solving just for myself. It’s for sysudokies, or other experts who have cause to question whether or not they have made the best solution against some standard. But we all know, if you’d like to become a true expert at something, that is exactly how to do it.
But that’s writing a trace. Reading a trace is for learner’s particularly. I’m treading carefully to make this readable without it, but getting out a blank grid, posting the givens, and adding the clues and pencil marks as you read items off the trace is the only way to have the state of the puzzle before you on every step. Then make sure that no action is posted without you understanding exactly why it works.
Going to the 6’s, place NW6 in the grid above, and you see three immediate effects, SW6 and 3-fills in NW and NE. The clue of the same number comes first. To list the effects first, then treat them in turn as causes, we list the 3-fills in unexpanded form:
SW6 completes the 6’s and has no effect.
In the NW 3-fill, the first condition of the two-number rule applies, for NW1. This leaves two cells for two missing numbers, 4 and 8, in NW. Whenever this occurs, and the placement are not determined by existing clues, as in the S box here, the cells are reserved for these numbers by writing them in as pencil marks. The marks are placed in the top corners, in a cell marking system called slink marking. More on that in the Basic Overview page.
Pencil marks to reserve cells for missing numbers are allowed in the bypass, when other pencil marks are deferred, because they directly restrict placements.
The N9 effects list ends with a NEr3 3-fill, actually a 3-fill in both NE and r3.
Here is the trace, which you can now follow to the next grid.
We not finished exploring the effects of the SE5 list, and have left behind many unexplored branches of the depth first tree.
As often happens, the 3-fill lists have stretched the trace across the page. Ordinarily, we could abandon the trace and fill in the remaining cells freestyle, but we one this one we may have trouble bridging the blank space in the middle bank.
The trace is continued here to the solution. Bypass generation of causes and effects continues in the same manner, clearly in collapse mode. You can continue solving and tracing from the diagram above, or continue by reading the trace as you post to the grid above.
This Dave Green 5-star featured a box configuration that is helpful to know about – the wall. Sometimes a number builds is own wall by sweeping the single cell missing from it.
As you read Sysudoku basic traces, you’ll become familiar with many more of these memorable box configurations. Here are two more.
A 2 by 2 square of clues creates slinks or clues from sweeps from either direction. Look along the passing row and column for clues or marks of numbers missing from the square. The square may include a pair or triple subset.
A sweep across a four-corner box of clues creates a slink in the other direction. Two such clues in a line creates a naked pair. Interchange the first and second rows or columns, and you have a cup.
An understanding of this page, and its Sysudoku Basic example, is sufficient for mastery of newspaper and magazine puzzles, up to a 5-star level.
When you have confirmed this, you’ll be ready to try your bypass skills on this example, from a review of A.D. Ardson’s Sudoku Diabolical. It’s puzzle 350. Trace your solution and compare with the trace posted on 2/13/2018.
This page is the first of a series of pages on Sysudoku Basic, Advanced, and Extreme methods. After the bypass, the goal of Sysudoku Basic is to prepare for Advanced methods, and puzzles fairly labeled as Hard, Very Hard, Tough, Super Tough, Diabolical, Extreme, etc.
Collections of such puzzles are available in books and websites. Many of these collections are already reviewed in the Sysudoku blog. You can locate review posts on the Titles page, or the category side bar .
Next in the Basic series is Box Marking page.