## 20090712

### Sudoku

I’ve been doing sudoku for a couple of years now and I have developed a simple notation that keeps track of what numbers can go where in the 3×3 blocks. I do not track all combinations, only 6 basic “shapes” and their various permutations (rotations and reflections). I have found these simple shapes are all that is needed to work almost any puzzle. The more complex patterns you may think are important eventually simplify to one of my basic shapes as your solution for the puzzle progresses. The notation is very simple, easy to learn, and easy to manage (alter/edit as the solution progresses). A key property of any notation is that it needs to be easy to interpret and unambiguous in its meaning. I’ve seen a lot of people working puzzles by filling each block with all of the possible numbers that can go in each square. That is hideously busy, difficult to read, and impossible to manage. It can be an effective technique near the end of solving some of the most difficult puzzles though, so I won’t completely deny its value because I have resorted to using it exactly twice.

First a bit on how to read my figures. The black numbers are the numbers that the puzzle either started with or we have added in trying to solve the puzzle. The blue numbers are the ones that limit where in a 3×3 block that same number can be placed. The possible locations are denoted by numbers in gray. Finally, my notation is indicated in red. When working a puzzle I use a very fine mechanical pencil. I prefer 0.3mm lead, but that can be hard to find. A good clean soft eraser is also a must. With that out of the way, here are the six basic shapes I use to work sudoku puzzles.

The first I call Adjacent. When a number’s location is limited to only two cells that are adjacent, either in the same row or column, I place that number in the center of the line shared by the two cells. In this example I’m considering the possible locations of the number 2 in the Left Middle 3×3 block. The bottom two rows are blocked and the top row already has one value filled. The only two possible locations are next to each other. So, I place a small 2 on the line between those cells.

The next shape I call Line. It is an extension of the Adjacent shape in that it is comprised of three cells in the same row or column instead of just two as found in Adjacent. If the pattern falls in a column, I place the number centered vertically and near the left edge of the center cell. If it is a row, I’ll place the value centered at the bottom of the center cell. It is important to apply the notation consistently since the primary meaning is derived from the location of the values. It will be hard at first to remember where everything is supposed to go. It will also be difficult to read what the notation is telling you. As with anything, practice makes a big difference and I hope you’ll find this can be learned quickly.

The third pattern which contains cells within a single row or column I call Ends. It is exactly what is sounds like. The possible locations are at opposite ends of a row or column. Since the pattern spans the full length of a row or column, I’ve chosen to place the value in the center block. If the pattern is found in a column, place the value centered on the right side of the center cell (yes, in the cell where the number cannot actually go). If it is in a row, place the value centered at the top of the center cell. You should note that this is exactly opposite from where you’d place the value if the pattern was the Line and this demonstrates the importance of applying the notation consistently.

The next pattern is the Block. It is formed by a 2×2 sub-square in the larger 3×3 sudoku cells. The notation places the value on the intersection of the lines where the four cells of the block share a common corner. Each of the 4 cells does not have to be empty for the notation to be valid. If one of the cells already had a value filled in, you would still use this notation to indicate that the value could go in any of the remaining three cells. There is also one way to have two of the cells already filled in and still use this notation. If the two empty cells touch only at the common corner, like two white squares on a checker board, then this notation applies. The other pattern of two squares is actually the Adjacent shape.

The Corners pattern is somewhat similar to the Block. For this pattern, the locations are found in the 4 corners of the 3×3 sudoku cell. As was the case with the Block pattern, one or two of these cells can be filled with a value already and the notation is still valid. The two empty cell form has the two possible cells in different rows and columns (kitty corner, catty corner, cady corner, … whatever you call it). The notation goes in the outer most corner of one of the corner cells. I prefer to place the notation in the corner of a cell that is on the outside edge of the whole 9×9 puzzle grid. If that is not possible (e.g., the center 3×3 grid) then I go with the upper left if it is empty. Which cell you select is not too critical for this pattern.

The last pattern is a three cell pattern that has many permutations. It is called the L. It is a combination of the Adjacent and Ends patterns where one of the cells is common to both. This means that the long axis of the two patterns forms a right angle, hence the name L. The notation places the value in the cell at the corner of the pattern. The value is placed in the outermost corner of that cell which is also between the two touching cells (of the Adjacent pattern). Just look at the image.

As you use the notation, you will eventually need to place multiple numbers in the same pattern (e.g., both 2 and 8 need to be in the same Adjacent). Simply place the numbers close to each other, either side-by-side or one over the other, you’ll quickly learn which way works best in each situation. Being able to write small is a big help. The real power of the notation is that it allows you to see at a glance that some of the cells are not available. If two numbers are assigned to the same Adjacent (or Edge) pattern (e.g., 2 and 8 as in the previous example), then the cells of that pattern cannot be used to make any other pattern for another number. Four values assigned to the same Block means those four cells are out of play for any other numbers/patterns. You may not yet know where each number goes in these full patterns, but you do know that they do end up in there somewhere. The beauty of the notation is that there are no conflicts in position so the placement and interpretation of the notes is unambiguous.

So that is my notation. Try it out and let me know how it works for you. I’m always interested in hearing about more tips and tricks too.

I received a book of Sudoku puzzles for giftmas 2007. The title is “X-treme SUDOKU” by the editors at Nikoli Publishing. The book offers 160 difficult puzzles and 160 more very difficult puzzles – 320 in all. I found the book to be a fabulous waste of time when I had no choice other than to waste time (e.g., waiting in the doctor’s office). I just finished the book and feel like I have accomplished something. There were two puzzles that stood out as being deviously difficult – at least for me. (Hint: try solving the puzzle in my first figure!)

## fatima said,

August 14, 2009 @ 1819

hello,

i like your notation through shapes. it’s what i use too. i find the system of penciling in possible numbers too cumbersome. using shapes came naturally to me and i only discovered the numbers system recently when i tried to get some help with some shapes that have been plaguing me for a while. maybe you can help …

i have to coin two new terms to explain my predicament: using your idea of adjacents i add to it leftward adjacents and rightward adjacents. Leftward adjacents would be two blocks which include the center block of any row in a 3×3 square and the block immediately to the left of that block. Rightward adjacents are then simply the same configuration with the exception that the two blocks are the center block of the row of the 3x3square and the block immediately to the right of it. (the adjacent you have in your description above is a rightward adjacent. if the twos were moved one block to the left they would be leftward adjacents. hope this makes sense)

now here’s my problem … i frequently come across a situation where in row of three 3x3squares i find that one number can go into the ends of a line in one square, into leftward adjacents of a line in another square and into rightward adjacents of a line in a third square.

i think there may be a logical way to solve this and determine where the number would fall in each of these 3x3squaress, but i can’t figure it out.

any thoughts?

## JMZ said,

August 14, 2009 @ 2227

I draw no distinction between left, right, top, or bottom adjacents. I occasionally note adjacents that span two 3×3 blocks. The situation you describe (multiple squares spanning two or three 3×3 blocks) I ignore. There is no point in trying to define such a complicated notation for that situation. As you solve the puzzle, it will eventually simplify into one of the notations I’ve already discussed. Not every number (1-9) in every 3×3 block can be “noted” unless the puzzle is trivial.

Have fun.

## fatima said,

September 2, 2009 @ 0922

thanks!

## richardgoodrich said,

December 9, 2012 @ 1918

Joe,

I REALLY like your mark-up scheme. I had been using John Welch’s SLINK marking scheme. I “think” I have found one case of a “slink” make that perhaps your scheme does not cover? If you use the conventional row, column, block (3×3 mini-grid) numbering from left-right, top-down then if I had a DIGIT that went into b1s34 (r1c3 & r2c1) I do not see that covered in your system. When playing paper-and-pencil. I circle my answers and the slinks (when marked with your scheme) and that helps if I make a mistake. I have been discussing my “trace” method with John’s and so far like mine a lot better. Also bought “The Hidden Logic of Sudoku” by Denis Berthier. It is TOUGH to read, but I am still digging into it. I have been trying to program a Sudoku Player with C# in Visual Studio. Recently started learning Python and was able to duplicate Peter Norvig’s solver which I was quite impressed with. Thanks, I REALLY like what you have done here! Richard Goodrich