A Sudoku is a logic puzzle. The rules are simple. You have a 9 by 9 grid which is split into 9 equal sized boxes (3 by 3). In each row, column and box you have to write the numbers 1 to 9 (in any order). No mathematics is involved, so it is similar to a magic square puzzle but not the same.

You can't have the same number twice in a row, column or box. When the puzzles are created you are given some numbers to start with. Using these numbers you have to work out where the other numbers go.

For instance say I had a 2 in the top row of the grid and a 2 in the next box one row down. This means that the 2 can only go in the third box, along the bottom row.

It maybe easier to understand with a diagram:

I have a 2 in the top row and in the middle row. This means the 2 has to go where I've put the stars:

??2 | ?6? | ?89
?45 | ??2 | ?7?
?7? | ?56 | ***

Sudoku software can be found at www.sudoku.com. The software is sold by Pappocom (a one man software company run by Wayne Gould).

Sudoku puzzles can also be found in the Times Newspaper.

Note: There's a little mistake in the diagram. See if you can spot it.

Sudoko -- sometimes written Su Doko -- is the latest craze in newspaper puzzles in the UK. It was imported from Japan by The Times and quickly began to appear in most UK Newspapers. Of these, The Guardian claims its puzzles are superior since they are generated by hand and not, as they claim the others' are, by computer. Indeed, the Guardian's G2 suppliment on May 13, 2005 has a Su Doko puzzle on every on of its twenty pages.

The puzzles are simple in concept, but often very difficult to complete. They comprise a 9x9 grid organised as nine 3x3 grids with some of the squares containing a digit (1-9). The aim is to fill every square with a digit (1-9) such that each 3x3 grid, horizontal and vertical line contains each digit exactly once.

Let's have a look at an example starting grid:

-------------------------------------
|   :   :   | 9 :   : 8 |   :   :   |
|   :   : 4 |   :   :   | 6 :   :   |
|   : 9 :   |   : 3 :   |   : 7 :   |
-------------------------------------
| 9 :   :   | 2 :   : 4 |   :   : 6 |
|   :   : 1 |   :   :   | 5 :   :   |
| 3 :   :   | 8 :   : 6 |   :   : 2 |
-------------------------------------
|   : 6 :   |   : 7 :   |   : 3 :   |
|   :   : 8 |   :   :   | 2 :   :   |
|   :   :   | 1 :   : 5 |   :   :   |
-------------------------------------

Be aware, that they are not always -- in fact, not usually -- symmetrical like this. I chose this one because it looks nice.

The first square we can easily fill is the centre one. Initially any number could be in any square, so this square could contain any of these numbers:

1 2 3 4 5 6 7 8 9

It cannot be 2, 4, 6 or 8 since those numbers already exist in that 3x3 grid.

1 2 3 4 5 6 7 8 9

It cannot be 3 or 7 which already exist on that vertical line.

1 2 3 4 5 6 7 8 9

Finally, it cannot be 1 or 5, which already exist on that horizontal line.

1 2 3 4 5 6 7 8 9

The only number is 9, so we can safely enter that into that square on the grid.

-------------------------------------
|   :   :   | 9 :   : 8 |   :   :   |
|   :   : 4 |   :   :   | 6 :   :   |
|   : 9 :   |   : 3 :   |   : 7 :   |
-------------------------------------
| 9 :   :   | 2 :   : 4 |   :   : 6 |
|   :   : 1 |   : 9 :   | 5 :   :   |
| 3 :   :   | 8 :   : 6 |   :   : 2 |
-------------------------------------
|   : 6 :   |   : 7 :   |   : 3 :   |
|   :   : 8 |   :   :   | 2 :   :   |
|   :   :   | 1 :   : 5 |   :   :   |
-------------------------------------

There is another number we can fill in from the starting grid. If we look at the center horizontal section, the middle and right-hand 3x3 grids have sixes in, but the left one doesn't. The 6 in this section cannot be on the top or bottom row since there are already sixes on those horisontal row, and cannot be on the middle vertical row either. The right-middle square already has a 1 in, so the 6 must be on the left middle square. I've tried to illustrate this below; it may look a little confused but I hope you understand the meaning.

-------------------------------------
|   :   :   | 9 :   : 8 |   :   :   |
|   :   : 4 |   :   :   | 6 :   :   |
|   : 9 :   |   : 3 :   |   : 7 :   |
-------------------------------------
| 9 :~#~:~~~|~2~:~~~:~4~|~~~:~~~:~6~|
| 6 : # : 1 |   : 9 :   | 5 :   :   |
| 3 :~#~:~~~|~8~:~~~:~6~|   :   : 2 |
------#------------------------------
|   : 6 :   |   : 7 :   |   : 3 :   |
|   :   : 8 |   :   :   | 2 :   :   |
|   :   :   | 1 :   : 5 |   :   :   |
-------------------------------------

Not all parts of the puzzle are this easy, of course; there may not at a particular point be a definite next step. The only solutions in this case must be found by logically following possible paths, backtracking if a problem occurs. Feel your brain groan as you metally recurse through logical pathways.

Now you have learned the basic rules of the puzzle it should be a simple matter to complete the rest. I feel I must warn you, however. These puzzles do look easy at first glance, but you ofen find yourself overlooking something and don't realise you've gone wrong until much later.

Solution to the demonstration puzzle

If you've decided to try out the example above, and see if you could complete it, here's the solution for you to check your result not, of course, to cheat with!

-------------------------------------
| 7 : 5 : 6 | 9 : 4 : 8 | 1 : 2 : 3 |
| 1 : 3 : 4 | 5 : 2 : 7 | 6 : 8 : 9 |
| 8 : 9 : 2 | 6 : 3 : 1 | 4 : 7 : 5 |
-------------------------------------
| 9 : 8 : 7 | 2 : 5 : 4 | 3 : 1 : 6 |
| 6 : 2 : 1 | 7 : 9 : 3 | 5 : 4 : 8 |
| 3 : 4 : 5 | 8 : 1 : 6 | 7 : 9 : 2 |
-------------------------------------
| 5 : 6 : 9 | 4 : 7 : 2 | 8 : 3 : 1 |
| 4 : 1 : 8 | 3 : 6 : 9 | 2 : 5 : 7 |
| 2 : 7 : 3 | 1 : 8 : 5 | 9 : 6 : 4 |
-------------------------------------

Sudoku puzzles can trace their lineage from the magic square (carrés magiques) of 18th century mathematical genius Euler, via latin squares, through American and then Japanese versions in the 80s (from which the name arises- roughly translated, it means solitary number), to their current surge in popularity in the UK, fuelled by publication in most of the quality newspapers (having first appeared in late 2004, however, in the Daily Mail).

For a month or so now, a few of my fellow students have been whiling away time in our own mathematical square, plugging away at Suduko grids from brightly coloured Japanese puzzle collections (combined circulation of dedicated Sudoku books in Japan is around 600,000 a month). Since I like to fuel the illusion that I know a thing or two about numbers, but am in fact generally rubbish at puzzles, I avoided attempting any until earlier this week when I spotted a bunch in The Independent- the next thing I knew, I'd lost my Wednesday afternoon.

Of course, I mangled the first one horribly, so sat down to approach the second one in a more systematic fashion. The simplest trick, as described by Sir Norris above, is to find cells which, through a combination of pressures from the row, column and 3*3 block, can only take a single value. After getting bored with pencilling in possibilities, I knocked together a Maple worksheet that generated lists of possible values for each cell. The best case of this brute force approach is when singletons appear- by plugging those back in and repeating the process, I solved the elementary problem from the Indie entirely mechanically. Which, although a triumph for my feeble coding skills, wasn't particularly enlightening with regards to the subtleties of Sudoku. So I fed the most difficult problem in, and was rewarded with not one cell that was completely determined. Now I had a challenge :)

Some advanced Sudoku tactics

Possibly ruining all the fun, and listed in order of desperation- the further you get in this list, the more fiendish the original puzzle.

So, having carried out the entirely routine steps of identifying potential values (either through fancy computing or writing extremely small in pencil), and assuming that wasn't sufficient to solve the grid, we're in a position to apply some logic. The hope here is that through reasoning, we can reduce or eliminate entirely the need for guesswork and backtracking. Of course, a lucky guess will speed things up, so a combination of reason and intuition make for a true sudoku master. These methodical steps seem like an ideal starting point for a full-fledged solver program, should any of you coders fancy a distraction.

Effective singletons

If no singletons have been identified, then the best case is a collection of cells with only two options. If in any given row, column or block there are two such cells with identical options, then their fates are inextricably linked- determine one, and you know the other. Often, this can set up a chain reaction across the board as a whole. But they can also force the value of another cell, as the following example shows.


-------------------------
|(4,5) :    3    :       |
|  2   :         :       |
|      : (4,5,9) : (4,5) |
-------------------------

Here the values 2 and 3 are given, and we've worked out the options for three cells of interest. If we figure out which of (4,5) the top left cell is, then we get the bottom right for free. But we immediately know the value of the (4,5,9) cell, which may help us determine entries elsewhere if we're currently stumped by the (4,5) choice. This is because whichever the top left cell turns out to be, 4 or 5, the bottom right cell will be the other, 5 or 4. So really, no other cell in this block can be a 4 or a 5, so the (4,5,9) entry is in fact the singleton (9).

This works just as well in a row or column, but the block was a bit easier to display in ascii art. Furthermore, larger cases are theoretically possible- cells with the options (3,4,5), (3,4,5), (3,4,5), (3,4,5,6) force the 6 for the last cell; or (1,2), (1,2), (3,4), (3,4), (1,2,3,4,5) forces the 5 in a cell that seemed to have unassailably many options. But such intricate cases require more cells to be just-so, and hence are rarer.

The importance of once-and-only-once

So far, the approach of identifying singletons or effective singletons has relied on the fact that a given number may appear only once in the row/column/block- we generate lists of potential values by eliminating the ones that have been 'used up' by already making an appearance. But we have a two way condition- each value needs to appear precisely once. So instead of looking for cells that can only take a single value, we look for values that can only occupy a single cell. Here's another block to illustrate.


---------------------------
|(4,6) :    3    : (1,7,9) |
|  2   :  (1,4)  : (7,8,9) |
|(7,9) : (4,5,9) :  (4,5)  |
---------------------------

In this complicated looking list of options, we can in fact determine two of the values. The top left cell has to take the value 6, since no other can; and similarly the 8 must lie in the middle of the third column.

Proof by contradiction

Suppose all the singletons,both immediate and effective have been identified, and furthermore any forced choices from the second strategy above have been made. You've scanned all the affected rows, columns and blocks in light of these changes, and have no more such cases to consider. So a guess is needed. What makes a good gambit? Chances are, what you do have are several chains of cells who's fates are linked, as described earlier. A (4,5) choice may be linked to another such choice in a column- but also be connected to a (3,5) in its row. Now, if it turns out to be a 4, then we don't get any information about the (3,5) pair- but if it's a 5, we determine that (3,5) is really a 3. So, you may have some luck in considering one of the cases from a particularly popular pair. Chose one option, and chase it around the grid. If it generates a contradiction somewhere (two 3's in the same block, for instance), then it can't be a valid choice- so you know it's the other one from the original pair. But beware- absence of contradiction is not sound grounds to assume that your choice was correct, unless of course you finish the puzzle in the process of following its implications. It could just be that you haven't reduced enough cells to get the contradiction, rather than there not being a contradiction at all; and then you'll come unstuck later. The validity of a choice is only certain once you've found counterexamples to the validity of all the other choices- this, in simpler fashion, is what we've been doing from the start in dismissing possibilities.

Sudoku Variants

Sudoku seem to be spreading across the puzzle pages of Britain's quality compacts and broadsheets, and inevitably different papers will introduce gimmicks, such as the Daily Mail's Sudoku-X (which uses rectangles and places a condition on diagonals). The Independent seems particularly keen, having tried letters in place of numbers (with one of the rows or columns yielding an actual word), and on Saturdays offering a Super Sudoku version, which ups the ante to 16X16, 4 subgrids (in hex), clearly making it even harder to keep track of a comprehensive list of possible values, either in your head or on paper! They even ran a national tournament (I qualified for regionals but was moving that day. Oh well.) However, the tried and tested 3X3 version will probably emerge as the staple, since the amount of time it consumes can be tweaked to more or less manageable cases, whilst the super is always going to while away a weekend. Sticking with the format but changing the character set, the debutante has pointed out that Sidduroku, substituting hebrew letters for numerals, will 'fuzzle the brain'!

Complexity

Research at the university of Tokyo has shown that solving n-by-n belongs to the NP-complete class, which is (roughly) one where a solution is easily checked for validity but becomes exponentially more difficult to compute as n grows. There are, for instance, 5524751496156892842531225600 different 9-by-9 latin squares possible- but the additional sudoku requirement on the internal 3-by-3 boxes carves that number down considerably. Reminiscent of the eternity puzzle, specifying the position of numbers in the grid can actually make things harder for the solver since it reduces the number of possible solutions. This is partly how it is possible to adjust the difficulty of a sudoku grid.

themanwho has put a solver online which implements some of the above strategies, removing the donkey work and leaving the interesting stuff (if any; it cheerfully ripped through a maximum difficulty problem from the Independant I tried with it.) It also makes my maple version look shambolic. You can find it here.

I found Wntrmute's tactics quite similiar to what's going on in my troubled head solving the damn sudoku of the day, but I noticed one thing missing... The tactic I usually leave to the super-duper complicated puzzles, just before the guessing part of "proof by contradiction".

I call it by its lame title:

A row's safe value

It goes like this. Suppose we've tried all the other tactics over and over again. Now we have, in some box, a value that can only populate two or three cells, which happen to be on the same row or column.
I'll demonstrate:

---------------------------
|(1,4,6):    3    : (1,7,9) |
|   2   : (4,6,8) :  (7,9)  |
|(1,4,8): (4,5,8) : (1,5,7) |
---------------------------

You can see here that 5 could only exist in the two right cells of the bottom row. This could happen for many reasons, but what's important right now is that you can conclude that no 5s could exist in the same row... Eliminate the options of the other cells in this row and it might be the trick to finding your next number.
The same goes for 7 and 9 in my almost-impossible example.

This one tactic helped me solve some serious sudoku puzzles without having to take guesses. So there you go. Off with your Sudoku.

Solving a SuDoku Puzzle - Step By Step

Before starting, visit this URL and download the sudoku puzzle; we'll solve it together!
http://www.timesonline.co.uk/article/0,,18209-1739363,00.html

Above, several noders have provided nice examples on the rules of sudoku puzzles, as well as a few examples on how to fill in a square or two on a grid. This writeup serves to demonstrate how to solve, from beginning to end, a sudoku puzzle.

You see, my mother is a big fan of word and letter puzzles, such as crosswords. When I got into a big sudoku kick earlier this year, I attempted to show her how to solve one, yet she didn't really seem to understand what was to be done, or what techniques to use. The big reason is that her thought processes are generally less logical and more organic in nature.

My solution was to take a relatively simple example and show her step by step how to deduce additional squares, all the way to the end of the puzzle. She read through this a time or two and was then able to start solving the puzzles. Before long, she was nearly as skilled as I am at solving sudoku puzzles.

So, without further ado, here is the solution to a sudoku puzzle from beginning to finish. I have chosen to solve the "mild" difficulty puzzle from the August 18, 2005 edition of the London Times as an example. This puzzle is available at http://www.timesonline.co.uk/article/0,,18209-1739363,00.html; you are advised to print yourself out a copy, get out a pen, and solve along with me.

Su Doku: August 18, 2005
The London Times Online
No. 306 - Rating: Mild

-------------------------------------
|   : 1 : 3 |   :   :   | 4 : 8 :   |
|   :   :   | 4 :   : 2 |   :   :   |
| 9 :   : 7 |   :   :   | 2 :   : 6 |
-------------------------------------
| 3 :   :   |   : 2 :   |   :   : 5 |
|   : 2 :   | 7 :   : 1 |   : 6 :   |
| 8 :   :   |   : 4 :   |   :   : 3 |
-------------------------------------
| 4 :   : 5 |   :   :   | 6 :   : 8 |
|   :   :   | 1 :   : 5 |   :   :   |
|   : 8 : 2 |   :   :   | 5 : 3 :   |
-------------------------------------

The first thing to look for when starting a sudoku puzzle are each set of three rows (the first, second, and third rows; the fourth, fifth, and sixth rows; and the seventh, eighth, and ninth rows) and each set of three columns (again, the first, second, and third columns; the fourth, fifth, and sixth columns; and the seventh, eighth, and ninth columns). What you're looking for are numbers that appear in two of the members of the set, but not the third.

Let's go through each set of rows and columns one at a time.

In the first set of rows:
2 appears in the second and third rows, but not in the first
4 appears in the first and second rows, but not in the third

2 is a potential candidate to put onto the grid. We can see above that a 2 already occurs in the second set of columns (more specifically, the second row of the sixth column) and the third set of columns (more specifically, third row, seventh column). I've bolded these above to make it more clear.

-------------------------------------
|   : 1 : 3 |   :   :   | 4 : 8 :   |
|   :   :   | 4 :   : 2 |   :   :   |
| 9 :   : 7 |   :   :   | 2 :   : 6 |
-------------------------------------

Thus, a 2 cannot be placed in the second or third row, because there is already a 2 in that row, and also, a 2 cannot be placed in the second or third small square, because there is already a 2 in each of these small squares. Let's then X out all of the squares we cannot place a 2 in. Note: you would NOT actually X out all these squares if actually solving a puzzle. This is to help you visualize the eliminated squares.

-------------------------------------
|   : 1 : 3 | X : X : X | 4 : 8 : X |
| X : X : X | 4 : X : 2 | X : X : X |
| 9 : X : 7 | X : X : X | 2 : X : 6 |
-------------------------------------

That leaves only a single possible square where a 2 could be placed in the first row, so we can add that to the grid.

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 :   |
|   :   :   | 4 :   : 2 |   :   :   |
| 9 :   : 7 |   :   :   | 2 :   : 6 |
-------------------------------------

Let's repeat the process with the 4 that appears in the first and second rows, but not in the third:

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 :   |
|   :   :   | 4 :   : 2 |   :   :   |
| 9 :   : 7 |   :   :   | 2 :   : 6 |
-------------------------------------

-------------------------------------
| 2 : 1 : 3 | X : X : X | 4 : 8 : X |
| X : X : X | 4 : X : 2 | X : X : X |
| 9 :   : 7 | X : X : X | 2 : X : 6 |
-------------------------------------

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 :   |
|   :   :   | 4 :   : 2 |   :   :   |
| 9 : 4 : 7 |   :   :   | 2 :   : 6 |
-------------------------------------

Much like when the 2 was placed, you can observe that there can't be a 4 in the first or second row, nor the second or third small square. This leaves only one place you can put the 4; the second square in the third row.

In the second set of rows:
3 appears in the fourth and sixth rows, but not in the fifth
2 appears in the fourth and fifth rows, but not in the sixth

Putting the 3 on the grid is as simple as before. Taking the middle three rows...

-------------------------------------
| 3 :   :   |   : 2 :   |   :   : 5 |
|   : 2 :   | 7 :   : 1 |   : 6 :   |
| 8 :   :   |   : 4 :   |   :   : 3 |
-------------------------------------

-------------------------------------
| 3 : X : X | X : 2 : X | X : X : 5 |
| X : 2 : X | 7 :   : 1 | X : 6 : X |
| 8 : X : X | X : 4 : X | X : X : 3 |
-------------------------------------

-------------------------------------
| 3 :   :   |   : 2 :   |   :   : 5 |
|   : 2 :   | 7 : 3 : 1 |   : 6 :   |
| 8 :   :   |   : 4 :   |   :   : 3 |
-------------------------------------

... we can easily place a 3 in the very center square. Now, let's try placing the 2...

-------------------------------------
| 3 :   :   |   : 2 :   |   :   : 5 |
|   : 2 :   | 7 : 3 : 1 |   : 6 :   |
| 8 :   :   |   : 4 :   |   :   : 3 |
-------------------------------------

-------------------------------------
| 3 : X : X | X : 2 : X | X : X : 5 |
| X : 2 : X | 7 : 3 : 1 | X : 6 : X |
| 8 : X : X | X : 4 : X |   :   : 3 |
-------------------------------------

Hmm... there are still two possible places that we could put the 2 in the sixth row. Can we get another clue? Let's look at that whole column, leaving the "imaginary" X's in place.

-------------
| 4 : 8 :   |
|   :   :   |
| 2 :   : 6 |
-------------
| X : X : 5 | 
| X : 6 : X |
|   :   : 3 | <-- we want to put a 2 in this row
-------------
| 6 :   : 8 |
|   :   :   |
| 5 : 3 :   |
-------------

In the leftmost column, you'll see a number 2 in the third row above. Let's put Xs in the entire leftmost column as well:

-------------
| 4 : 8 :   |
| X :   :   |
| 2 :   : 6 |
-------------
| X : X : 5 | 
| X : 6 : X |
| X :   : 3 | <-- we want to put a 2 in this row
-------------
| 6 :   : 8 |
| X :   :   |
| 5 : 3 :   |
-------------

And we're left with only one possible square to put that 2 into!

In the third set of rows, there's nothing of note, so let's look at where we're at (solutions we've added are in bold):

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 :   |
|   :   :   | 4 :   : 2 |   :   :   |
| 9 : 4 : 7 |   :   :   | 2 :   : 6 |
-------------------------------------
| 3 :   :   |   : 2 :   |   :   : 5 |
|   : 2 :   | 7 : 3 : 1 |   : 6 :   |
| 8 :   :   |   : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 :   : 5 |   :   :   | 6 :   : 8 |
|   :   :   | 1 :   : 5 |   :   :   |
|   : 8 : 2 |   :   :   | 5 : 3 :   |
-------------------------------------

Let's try adding more numbers using the columns.

In the first set of columns:
4 appears in the first and second columns, but not in the third
3 appears in the first and third columns, but not in the second

Neither of these make it possible for us to add a number. With the 4, we know it has to go in the middle three squares of the third row. Since they're all empty and the only additional clue we can get is that it can't go into the bottom square in that row, we have to skip it. Here's a grid with Xs indicating squares where you can't put the 2 along with that little region in bold to help you visualize this.

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 :   |
| X : X :   | 4 :   : 2 |   :   :   |
| 9 : 4 : 7 |   :   :   | 2 :   : 6 |
-------------------------------------
| 3 : X :   |   : 2 :   |   :   : 5 |
| X : 2 :   | 7 : 3 : 1 |   : 6 :   |
| 8 : X : X | X : 4 : X | X : 2 : 3 | <- no new 4 in this
-------------------------------------         row because of
| 4 : X : 5 |   :   :   | 6 :   : 8 |         the 4 in the 5th
| X : X :   | 1 :   : 5 |   :   :   |         column
| X : 8 : 2 |   :   :   | 5 : 3 :   |
------------------------------------- 
  ^   ^
  there can't be another 4 in this column
   because of the 4 in the seventh row
      |
      |
      there can't be another 4 in this column
       because of the 4 in the third row

A similar problem occurs with the 3. We can only say that it goes in the second column in one of the bottom three squares, but that still leaves us with two possibilities and we can't reduce it any more than that.

In the second set of columns:
1 appears in the first and third columns, but not in the second
4 appears in the first and second columns, but not in the third
2 appears in the second and third columns, but not in the first

We can't put the 1 in for the same reason we couldn't put the 4 and the 3 in the first set of columns: there are two empty squares left that are possibilities. Can you see why? The possible squares are marked with a 0 below.

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 :   |
|   :   :   | 4 : 0 : 2 |   :   :   |
| 9 : 4 : 7 |   : 0 :   | 2 :   : 6 |
-------------------------------------
| 3 :   :   |   : 2 :   |   :   : 5 |
|   : 2 :   | 7 : 3 : 1 |   : 6 :   |
| 8 :   :   |   : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 :   : 5 |   :   :   | 6 :   : 8 |
|   :   :   | 1 :   : 5 |   :   :   |
|   : 8 : 2 |   :   :   | 5 : 3 :   |
-------------------------------------

With both the 4 and the 2, we have more success. First, the 4:

-------------
| X : X : X |
| 4 : X : 2 | <- there can only be
| X : X : X |         one 4 in each
-------------         small square
| X : 2 : X |              |
| 7 : 3 : 1 | <--------/
| X : 4 : X |
-------------
| X : X :   |
| 1 : X : 5 |
| X : X :   |
-------------
  ^   ^
  there can only be one 4 in each column


-------------------------------------
| 2 : 1 : 3 | X : X : X | 4 : 8 :   |
|   :   :   | 4 : X : 2 |   :   :   |
| 9 : 4 : 7 | X : X : X | 2 :   : 6 |
-------------------------------------
| 3 :   :   | X : 2 : X |   :   : 5 |
|   : 2 :   | 7 : 3 : 1 |   : 6 :   |
| 8 :   :   | X : 4 : X |   : 2 : 3 |
-------------------------------------
| 4 : X : 5 | X : X : X | 6 : X : 8 | <- only one 4 in this row
|   :   :   | 1 : X : 5 |   :   :   |
|   : 8 : 2 | X : X : 4 | 5 : 3 :   |
-------------------------------------

As you see, we can thus place a 4 in the bottom row of the sixth column. How about the 2?

-------------
| X : X : X |
| 4 : X : 2 |
| X : X : X |
-------------
| X : 2 : X |
| 7 : 3 : 1 |
| X : 4 : X |
-------------
|   : X : X |
| 1 : X : 5 |
|   : X : 4 |
-------------

-------------------------------------
| 2 : 1 : 3 |   : X : X | 4 : 8 :   |
|   :   :   | 4 : X : 2 |   :   :   |
| 9 : 4 : 7 |   : X : X | 2 :   : 6 |
-------------------------------------
| 3 :   :   |   : 2 : X |   :   : 5 |
|   : 2 :   | 7 : 3 : 1 |   : 6 :   |
| 8 :   :   |   : 4 : X |   : 2 : 3 |
-------------------------------------
| 4 :   : 5 | 2 : X : X | 6 :   : 8 |
|   :   :   | 1 : X : 5 |   :   :   |
| X : 8 : 2 | X : X : 4 | 5 : 3 : X |
-------------------------------------

And we can put the 2 into the seventh row of the fourth column.

In the third set of columns:
2 appears in the first and second column, but not the third
3 appears in the second and third column, but not the first
8 appears in the second and third column, but not the first

We can quickly place the three in the second square of the first of these columns, because it is the only possible square:

-------------
| 4 : 8 : X |
| 3 : X : X |
| 2 : X : 6 |
-------------
| X : X : 5 |
| X : 6 : X |
| X : 2 : 3 |
-------------
| 6 : X : 8 |
| X : X : X |
| 5 : 3 : X |
-------------

We can also place the 2:

-------------
| 4 : 8 : X |
| 3 : X : X |
| 2 : X : 6 |
-------------
| X : X : 5 |
| X : 6 : X |
| X : 2 : 3 |
-------------
| 6 : X : 8 |
| X : X :   |
| 5 : 3 :   |
-------------

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 : X |
|   :   :   | 4 :   : 2 | 3 : X : X |
| 9 : 4 : 7 |   :   :   | 2 : X : 6 |
-------------------------------------
| 3 :   :   |   : 2 :   | X : X : 5 |
|   : 2 :   | 7 : 3 : 1 | X : 6 : X |
| 8 :   :   |   : 4 :   | X : 2 : 3 |
-------------------------------------
| 4 :   : 5 | 2 :   :   | 6 : X : 8 |
|   :   :   | 1 :   : 5 | X : X : 2 |
| x : 8 : 2 | X : X : 4 | 5 : 3 : X |
-------------------------------------

Unfortunately, we can't place the 8 quite yet because it can be placed into two possible squares in the seventh column. Do you see which ones? (The fourth and the fifth from the top)

Let's look at our grid now:

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 :   |
|   :   :   | 4 :   : 2 | 3 :   :   |
| 9 : 4 : 7 |   :   :   | 2 :   : 6 |
-------------------------------------
| 3 :   :   |   : 2 :   |   :   : 5 |
|   : 2 :   | 7 : 3 : 1 |   : 6 :   |
| 8 :   :   |   : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 :   : 5 | 2 :   :   | 6 :   : 8 |
|   :   :   | 1 :   : 5 |   :   : 2 |
|   : 8 : 2 |   :   : 4 | 5 : 3 :   |
-------------------------------------

At this point, we can start taking advantage of the fact that each smaller square must have each number in it once. Let's start with the little one in the upper left.

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 :   |
|   :   :   | 4 :   : 2 | 3 :   :   |
| 9 : 4 : 7 |   :   :   | 2 :   : 6 |
-------------------------------------
| 3 :   :   |
|   : 2 :   |
| 8 :   :   |
-------------
| 4 :   : 5 |
|   :   :   |
|   : 8 : 2 |
-------------

In the upper left square, we know that we need to still put in a 5, a 6, and an 8. We can place the 8:

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 :   |
| X : X : 8 | 4 :   : 2 | 3 :   :   |
| 9 : 4 : 7 |   :   :   | 2 :   : 6 |
-------------------------------------
| 3 : X :   |
| X : 2 :   |
| 8 : X :   |
-------------
| 4 : X : 5 |
| X : X :   |
| X : 8 : 2 |
-------------

Now we've placed the 8 and only have the 5 and 6 to worry about in the upper left square, but we can't decide which square the 5 and the 6 goes into. So we move on:

            -------------
            |   :   :   |
            | 4 :   : 2 |
            |   :   :   |
-------------------------------------
| 3 :   :   |   : 2 :   |   :   : 5 |
|   : 2 :   | 7 : 3 : 1 |   : 6 :   |
| 8 :   :   |   : 4 :   |   : 2 : 3 |
-------------------------------------
            | 2 :   :   |
            | 1 :   : 5 |
            |   :   : 4 |
            -------------

In this middle square, we need to place a 5, a 6, an 8, and a 9. We can place the 5:

            -------------
            |   :   : X |
            | 4 :   : 2 |
            |   :   : X |
-------------------------------------
| 3 : X : X | X : 2 : X | X : X : 5 | <- only one 5
|   : 2 :   | 7 : 3 : 1 |   : 6 :   |     in this row
| 8 :   :   | 5 : 4 : X |   : 2 : 3 |
-------------------------------------
            | 2 :   : X |
            | 1 :   : 5 |
            |   :   : 4 |
            -------------
                      ^
                      only one 5 in this column

But we can't place any more. Onwards:

                        -------------
                        | 4 : 8 :   |
                        | 3 :   :   |
                        | 2 :   : 6 |
-------------------------------------
| 3 :   :   |   : 2 :   |   :   : 5 |
|   : 2 :   | 7 : 3 : 1 |   : 6 :   |
| 8 :   :   | 5 : 4 :   |   : 2 : 3 |
-------------------------------------
                        | 6 :   : 8 |
                        |   :   : 2 |
                        | 5 : 3 :   |
                        -------------

We need to place a 1, a 4, a 7, an 8, and a 9 in this left center square. We can actually place the 8. Can you see how?

                        -------------
                        | 4 : 8 : X |
                        | 3 : X : X |
                        | 2 : X : 6 |
-------------------------------------
| 3 :   :   |   : 2 :   |   : X : 5 |
|   : 2 :   | 7 : 3 : 1 |   : 6 : X |
| 8 : X : X | 5 : 4 : X | X : 2 : 3 | <- one 8 in this row
-------------------------------------
                        | 6 : X : 8 |
                        |   : X : 2 |
                        | 5 : 3 : X |
                        -------------
                              ^   ^
                  one 8 in this column
                              |
                              |
             one 8 in this column

At this point, we still have two open squares. But there is one key fact here: we KNOW the eight HAS to go in the top row of the middle square in the middle row. Why? The middle row in the middle square of the middle row is completely full, and the bottom row already has an eight in it. So...

                        -------------
       8 must go in one | 4 : 8 : X |
   of these two squares | 3 : X : X |
               /     \  | 2 : X : 6 |
--------------|-------|--------------
| 3 :   :   | 0 : 2 : 0 | X : X : 5 | <- only one 8
|   : 2 :   | 7 : 3 : 1 | 8 : 6 : X |         in this row
| 8 : X : X | 5 : 4 : X | X : 2 : 3 |
-------------------------------------
                        | 6 : X : 8 |
                        |   : X : 2 |
                        | 5 : 3 : X |
                        -------------

Let's tackle another square:

-------------
| 2 : 1 : 3 |
|   :   : 8 |
| 9 : 4 : 7 |
-------------
| 3 :   :   |
|   : 2 :   |
| 8 :   :   |
-------------------------------------
| 4 :   : 5 | 2 :   :   | 6 :   : 8 |
|   :   :   | 1 :   : 5 |   :   : 2 |
|   : 8 : 2 |   :   : 4 | 5 : 3 :   |
-------------------------------------

We need to put a 1, a 3, a 6, a 7, and a 9 in that lower left square. We can place the 1:

-------------
| 2 : 1 : 3 |
|   : X : 8 |
| 9 : 4 : 7 |
-------------
| 3 : X :   |
|   : 2 :   |
| 8 : X :   |
-------------------------------------
| 4 : X : 5 | 2 :   :   | 6 :   : 8 |
| X : X : X | 1 : X : 5 | X : X : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 :   |
-------------------------------------

In the bottom center square:

            -------------
            |   :   :   |
            | 4 :   : 2 |
            |   :   :   |
            -------------
            |   : 2 :   |
            | 7 : 3 : 1 |
            | 5 : 4 :   |
-------------------------------------
| 4 :   : 5 | 2 :   :   | 6 :   : 8 |
|   :   :   | 1 :   : 5 |   :   : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 :   |
-------------------------------------

We need to put a 3, a 6, a 7, an 8, and a 9. We can place the 3:

            -------------
            |   : X :   |
            | 4 : X : 2 |
            |   : X :   |
            -------------
            |   : 2 :   |
            | 7 : 3 : 1 |
            | 5 : 4 :   |
-------------------------------------
| 4 :   : 5 | 2 : X : 3 | 6 :   : 8 |
|   :   :   | 1 : X : 5 | X : X : 2 |
| 1 : 8 : 2 | X : X : 4 | 5 : 3 : X |
-------------------------------------

We can also place the 8:

            -------------
            |   :   :   |
            | 4 :   : 2 |
            |   :   :   |
            -------------
            |   : 2 :   |
            | 7 : 3 : 1 |
            | 5 : 4 :   |
-------------------------------------
| 4 : X : 5 | 2 : X : 3 | 6 : X : 8 |
|   :   :   | 1 : 8 : 5 |   :   : 2 |
| 1 : 8 : 2 | X : X : 4 | 5 : 3 : X |
-------------------------------------

In the lower right square:

                        -------------
                        | 4 : 8 :   |
                        | 3 :   :   |
                        | 2 :   : 6 |
                        -------------
                        |   :   : 5 |
                        | 8 : 6 :   |
                        |   : 2 : 3 |
-------------------------------------
| 4 :   : 5 | 2 :   : 3 | 6 :   : 8 |
|   :   :   | 1 : 8 : 5 |   :   : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 :   |
-------------------------------------

We need to put a 1, a 4, a 7, and a 9 here. We can put in the 1:

                        -------------
                        | 4 : 8 :   |
                        | 3 :   :   |
                        | 2 :   : 6 |
                        -------------
                        |   :   : 5 |
                        | 8 : 6 :   |
                        |   : 2 : 3 |
-------------------------------------
| 4 :   : 5 | 2 :   : 3 | 6 : 1 : 8 |
| X : X : X | 1 : 8 : 5 | X : X : 2 |
| 1 : 8 : 2 | X : X : 4 | 5 : 3 : X |
-------------------------------------

We can put in the 4:

                        -------------
                        | 4 : 8 :   |
                        | 3 :   :   |
                        | 2 :   : 6 |
                        -------------
                        | X :   : 5 |
                        | 8 : 6 :   |
                        | X : 2 : 3 |
-------------------------------------
| 4 :   : 5 | 2 :   : 3 | 6 : 1 : 8 |
|   :   :   | 1 : 8 : 5 | X : 4 : 2 |
| 1 : 8 : 2 | X : X : 4 | 5 : 3 : X |
-------------------------------------

So, where are we?

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 :   |
|   :   : 8 | 4 :   : 2 | 3 :   :   |
| 9 : 4 : 7 |   :   :   | 2 :   : 6 |
-------------------------------------
| 3 :   :   |   : 2 :   |   :   : 5 |
|   : 2 :   | 7 : 3 : 1 | 8 : 6 :   |
| 8 :   :   | 5 : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 :   : 5 | 2 :   : 3 | 6 : 1 : 8 |
|   :   :   | 1 : 8 : 5 |   : 4 : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 :   |
-------------------------------------

At this point we just keep moving from square to square. The center right one needs a 1, a 4, a 7, and a 9. We can place the 4:

                        -------------
                        | 4 : 8 :   |
                        | 3 : X :   |
                        | 2 : X : 6 |
-------------------------------------
| 3 :   :   |   : 2 :   | X : X : 5 |
|   : 2 :   | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 :   :   | 5 : 4 :   | X : 2 : 3 |
-------------------------------------
                        | 6 : 1 : 8 |
                        | X : 4 : 2 |
                        | 5 : 3 :   |
                        -------------

In the center left one, we need to place a 1, a 4, a 5, a 6, a 7, and a 9. We can place the 4:

-------------
| 2 : 1 : 3 |
| X : X : 8 |
| 9 : 4 : 7 |
-------------------------------------
| 3 : X : 4 |   : 2 :   |   :   : 5 |
| X : 2 : X | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 : X : X | 5 : 4 : X | X : 2 : 3 |
-------------------------------------
| 4 : X : 5 |
| X : X :   |
| 1 : 8 : 2 |
-------------

And the 5:

-------------
| 2 : 1 : 3 |
|   :   : 8 |
| 9 : 4 : 7 |
-------------------------------------
| 3 : X : 4 | X : 2 : X | X : X : 5 |
| 5 : 2 : X | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 : X : X | 5 : 4 : X | X : 2 : 3 |
-------------------------------------
| 4 :   : 5 |
|   :   : X |
| 1 : 8 : 2 |
-------------

And the 9. How can we place the 9? We've placed eight of the nine numbers in that middle row now, and the only one missing is the 9.

-------------
| 2 : 1 : 3 |
|   :   : 8 |
| 9 : 4 : 7 |
-------------------------------------
| 3 :   : 4 |   : 2 :   |   :   : 5 |
| 5 : 2 : 9 | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 :   :   | 5 : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 :   : 5 |
|   :   :   |
| 1 : 8 : 2 |
-------------

And the 1:

-------------
| 2 : 1 : 3 |
| X : X : 8 |
| 9 : 4 : 7 |
-------------------------------------
| 3 : X : 4 |   : 2 :   |   :   : 5 |
| 5 : 2 : 9 | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 : X : 1 | 5 : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 : X : 5 |
| X : X :   |
| 1 : 8 : 2 |
-------------

We can finish up the upper left square now, as we only have to put a 5 and a 6 in it:

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 :   |
| 6 : 5 : 8 | 4 :   : 2 | 3 :   :   |
| 9 : 4 : 7 |   :   :   | 2 :   : 6 |
-------------------------------------
| 3 :   : 4 |
| 5 : 2 : 9 |
| 8 :   : 1 |
-------------
| 4 :   : 5 |
|   :   :   |
| 1 : 8 : 2 |
-------------
  ^
  there can only be one 5 in this column,
   so this column can't have another 5 added
   thus we add a 6; after that, we can put 5
   next to it, because it's the only missing
   number in that square

Now we can finish the lower left square now. We have to place a 3, a 6, a 7, and a 9 in it. Let's place the 6 and the 7; we can do this because the first and third columns each have all but one number filled in, meaning we can easily see which number needs to be entered in each of the columns:

-------------
| 2 : 1 : 3 |
| 6 : 5 : 8 |
| 9 : 4 : 7 |
-------------
| 3 :   : 4 |
| 5 : 2 : 9 |
| 8 :   : 1 |
-------------------------------------
| 4 :   : 5 | 2 :   : 3 | 6 : 1 : 8 |
| 7 :   : 6 | 1 : 8 : 5 |   : 4 : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 :   |
-------------------------------------

We can then fit in the 3:

-------------
| 2 : 1 : 3 |
| 6 : 5 : 8 |
| 9 : 4 : 7 |
-------------
| 3 :   : 4 |
| 5 : 2 : 9 |
| 8 :   : 1 |
-------------------------------------
| 4 : X : 5 | 2 : X : 3 | 6 : 1 : 8 |
| 7 : 3 : 6 | 1 : 8 : 5 |   : 4 : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 :   |
-------------------------------------

And the 9 goes in right above the 3, because it's the only number missing in that square. Let's see our whole grid as it stands right now:

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 :   |
| 6 : 5 : 8 | 4 :   : 2 | 3 :   :   |
| 9 : 4 : 7 |   :   :   | 2 :   : 6 |
-------------------------------------
| 3 :   : 4 |   : 2 :   |   :   : 5 |
| 5 : 2 : 9 | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 :   : 1 | 5 : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 : 9 : 5 | 2 :   : 3 | 6 : 1 : 8 |
| 7 : 3 : 6 | 1 : 8 : 5 |   : 4 : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 :   |
-------------------------------------

On to the home stretch! That upper middle square is pretty bare; let's shoot to fill that in as best we can. To prepare, let's fill in the upper right square; it still needs a 1, a 5, a 7, and a 9, and we can easily place the 1:

-------------------------------------
| 2 : 1 : 3 | X : X : X | 4 : 8 : X |
| 6 : 5 : 8 | 4 :   : 2 | 3 : X : 1 |
| 9 : 4 : 7 |   :   :   | 2 : X : 6 |
-------------------------------------
                        |   : X : 5 |
                        | 8 : 6 : 4 |
                        |   : 2 : 3 |
                        -------------
                        | 6 : 1 : 8 |
                        |   : 4 : 2 |
                        | 5 : 3 :   |
                        -------------

And the 5:

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 : X |
| 6 : 5 : 8 | 4 : X : 2 | 3 : X : 1 |
| 9 : 4 : 7 |   :   :   | 2 : 5 : 6 |
-------------------------------------
                        |   :   : 5 |
                        | 8 : 6 : 4 |
                        |   : 2 : 3 |
                        -------------
                        | 6 : 1 : 8 |
                        |   : 4 : 2 |
                        | 5 : 3 : X |
                        -------------

Now the upper middle square. It needs a 1, a 3, a 5, a 6, a 7, an 8, and a 9.

-------------------------------------
| 2 : 1 : 3 |   :   :   | 4 : 8 :   |
| 6 : 5 : 8 | 4 :   : 2 | 3 :   : 1 |
| 9 : 4 : 7 |   :   :   | 2 : 5 : 6 |
-------------------------------------
            |   : 2 :   |
            | 7 : 3 : 1 |
            | 5 : 4 :   |
            -------------
            | 2 :   : 3 |
            | 1 : 8 : 5 |
            |   :   : 4 |
            -------------

The 1:

-------------------------------------
| 2 : 1 : 3 | X : X : X | 4 : 8 : X |
| 6 : 5 : 8 | 4 : X : 2 | 3 : X : 1 |
| 9 : 4 : 7 | X : 1 : X | 2 : 5 : 6 |
-------------------------------------
            | X : 2 : X |
            | 7 : 3 : 1 |
            | 5 : 4 : X |
            -------------
            | 2 :   : 3 |
            | 1 : 8 : 5 |
            | X :   : 4 |
            -------------

The 3:

-------------------------------------
| 2 : 1 : 3 | X : X : X | 4 : 8 : X |
| 6 : 5 : 8 | 4 : X : 2 | 3 : X : 1 |
| 9 : 4 : 7 | 3 : 1 : X | 2 : 5 : 6 |
-------------------------------------
            |   : 2 : X |
            | 7 : 3 : 1 |
            | 5 : 4 : X |
            -------------
            | 2 :   : 3 |
            | 1 : 8 : 5 |
            |   :   : 4 |
            -------------

The 5:

-------------------------------------
| 2 : 1 : 3 | X : 5 : X | 4 : 8 :   |
| 6 : 5 : 8 | 4 : X : 2 | 3 : X : 1 |
| 9 : 4 : 7 | 3 : 1 : X | 2 : 5 : 6 |
-------------------------------------
            | X : 2 : X |
            | 7 : 3 : 1 |
            | 5 : 4 : X |
            -------------
            | 2 :   : 3 |
            | 1 : 8 : 5 |
            | X :   : 4 |
            -------------

The 8:

-------------------------------------
| 2 : 1 : 3 | X : 5 : X | 4 : 8 : X |
| 6 : 5 : 8 | 4 : X : 2 | 3 :   : 1 |
| 9 : 4 : 7 | 3 : 1 : 8 | 2 : 5 : 6 |
-------------------------------------
            |   : 2 :   |
            | 7 : 3 : 1 |
            | 5 : 4 :   |
            -------------
            | 2 : X : 3 |
            | 1 : 8 : 5 |
            |   : X : 4 |
            -------------

Now let's see where we are, and whether we can quickly fill in any columns or rows immediately because they're only missing one number.

-------------------------------------
| 2 : 1 : 3 |   : 5 :   | 4 : 8 :   |
| 6 : 5 : 8 | 4 :   : 2 | 3 :   : 1 |
| 9 : 4 : 7 | 3 : 1 : 8 | 2 : 5 : 6 |
-------------------------------------
| 3 :   : 4 |   : 2 :   |   :   : 5 |
| 5 : 2 : 9 | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 :   : 1 | 5 : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 : 9 : 5 | 2 :   : 3 | 6 : 1 : 8 |
| 7 : 3 : 6 | 1 : 8 : 5 |   : 4 : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 :   |
-------------------------------------

Row 7 needs only a 7 added to it.
Row 8 needs only a 9 added to it.
Now we have:

-------------------------------------
| 2 : 1 : 3 |   : 5 :   | 4 : 8 :   |
| 6 : 5 : 8 | 4 :   : 2 | 3 :   : 1 |
| 9 : 4 : 7 | 3 : 1 : 8 | 2 : 5 : 6 |
-------------------------------------
| 3 :   : 4 |   : 2 :   |   :   : 5 |
| 5 : 2 : 9 | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 :   : 1 | 5 : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 : 9 : 5 | 2 : 7 : 3 | 6 : 1 : 8 |
| 7 : 3 : 6 | 1 : 8 : 5 | 9 : 4 : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 :   |
-------------------------------------

Now, the lower right square can be completed with only a 7 (it's the only one missing), and the rightmost column can then be completed with a 9 (it's the only one missing).

-------------------------------------
| 2 : 1 : 3 |   : 5 :   | 4 : 8 : 9 |
| 6 : 5 : 8 | 4 :   : 2 | 3 :   : 1 |
| 9 : 4 : 7 | 3 : 1 : 8 | 2 : 5 : 6 |
-------------------------------------
| 3 :   : 4 |   : 2 :   |   :   : 5 |
| 5 : 2 : 9 | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 :   : 1 | 5 : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 : 9 : 5 | 2 : 7 : 3 | 6 : 1 : 8 |
| 7 : 3 : 6 | 1 : 8 : 5 | 9 : 4 : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 : 7 |
-------------------------------------

The upper right square needs a 7 in the middle (it's the only one missing now from that square), and then we can put a 9 in the middle square of the second row (it's the only one missing from that row) and a 9 in the fourth square of the eighth column (it's the only one missing from that column).

-------------------------------------
| 2 : 1 : 3 |   : 5 :   | 4 : 8 : 9 |
| 6 : 5 : 8 | 4 : 9 : 2 | 3 : 7 : 1 |
| 9 : 4 : 7 | 3 : 1 : 8 | 2 : 5 : 6 |
-------------------------------------
| 3 :   : 4 |   : 2 :   |   : 9 : 5 |
| 5 : 2 : 9 | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 :   : 1 | 5 : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 : 9 : 5 | 2 : 7 : 3 | 6 : 1 : 8 |
| 7 : 3 : 6 | 1 : 8 : 5 | 9 : 4 : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 : 7 |
-------------------------------------

Let's try to solve it now, row by row, and see how we do. The first row needs only a 6 and a 7, and these go in the fourth and sixth columns. However, the fourth column already has a 7 in it (in the fifth row), so we put the 6 in the fourth column and the 7 in the sixth column, and the first row is finished.

-------------------------------------
| 2 : 1 : 3 | 6 : 5 : 7 | 4 : 8 : 9 |
| 6 : 5 : 8 | 4 : 9 : 2 | 3 : 7 : 1 |
| 9 : 4 : 7 | 3 : 1 : 8 | 2 : 5 : 6 |
-------------------------------------
| 3 :   : 4 |   : 2 :   |   : 9 : 5 |
| 5 : 2 : 9 | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 :   : 1 | 5 : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 : 9 : 5 | 2 : 7 : 3 | 6 : 1 : 8 |
| 7 : 3 : 6 | 1 : 8 : 5 | 9 : 4 : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 : 7 |
-------------------------------------

The fourth row needs a 1, a 6, a 7, and an 8, which go in the open second, fourth, sixth, and seventh columns. There is already a 1 in the second column (in the first row), the fourth column (in the eighth row), and the sixth column (in the fifth row), leaving only the seventh column of the fourth row to put a 1 into.

-------------------------------------
| 2 : 1 : 3 | 6 : 5 : 7 | 4 : 8 : 9 |
| 6 : 5 : 8 | 4 : 9 : 2 | 3 : 7 : 1 |
| 9 : 4 : 7 | 3 : 1 : 8 | 2 : 5 : 6 |
-------------------------------------
| 3 :   : 4 |   : 2 :   | 1 : 9 : 5 |
| 5 : 2 : 9 | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 :   : 1 | 5 : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 : 9 : 5 | 2 : 7 : 3 | 6 : 1 : 8 |
| 7 : 3 : 6 | 1 : 8 : 5 | 9 : 4 : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 : 7 |
-------------------------------------

There is already an 8 in column two (row nine) and column six (row three), meaning an eight goes into the fourth row, fourth column.

-------------------------------------
| 2 : 1 : 3 | 6 : 5 : 7 | 4 : 8 : 9 |
| 6 : 5 : 8 | 4 : 9 : 2 | 3 : 7 : 1 |
| 9 : 4 : 7 | 3 : 1 : 8 | 2 : 5 : 6 |
-------------------------------------
| 3 :   : 4 | 8 : 2 :   | 1 : 9 : 5 |
| 5 : 2 : 9 | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 :   : 1 | 5 : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 : 9 : 5 | 2 : 7 : 3 | 6 : 1 : 8 |
| 7 : 3 : 6 | 1 : 8 : 5 | 9 : 4 : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 : 7 |
-------------------------------------

Only the 6 and 7 remain now, in columns two and six of row four. However, there is already a 7 in column six (in row one), so we have to put the 7 in column two of row four and thus the 6 into column six of row four, finishing the row.

-------------------------------------
| 2 : 1 : 3 | 6 : 5 : 7 | 4 : 8 : 9 |
| 6 : 5 : 8 | 4 : 9 : 2 | 3 : 7 : 1 |
| 9 : 4 : 7 | 3 : 1 : 8 | 2 : 5 : 6 |
-------------------------------------
| 3 : 7 : 4 | 8 : 2 : 6 | 1 : 9 : 5 |
| 5 : 2 : 9 | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 :   : 1 | 5 : 4 :   |   : 2 : 3 |
-------------------------------------
| 4 : 9 : 5 | 2 : 7 : 3 | 6 : 1 : 8 |
| 7 : 3 : 6 | 1 : 8 : 5 | 9 : 4 : 2 |
| 1 : 8 : 2 |   :   : 4 | 5 : 3 : 7 |
-------------------------------------

Five numbers remain:
The second column is missing its 6, so we put that into the sixth row.
The fourth column is missing its 9, so we put that into the last row.
The fifth column is missing its 6, so we put that into the last row.
The sixth column is missing its 9, so we put that into the sixth row.
And, finally, the seventh column is missing its 7, so we put that into the sixth row.

Here's the completed puzzle!

-------------------------------------
| 2 : 1 : 3 | 6 : 5 : 7 | 4 : 8 : 9 |
| 6 : 5 : 8 | 4 : 9 : 2 | 3 : 7 : 1 |
| 9 : 4 : 7 | 3 : 1 : 8 | 2 : 5 : 6 |
-------------------------------------
| 3 : 7 : 4 | 8 : 2 : 6 | 1 : 9 : 5 |
| 5 : 2 : 9 | 7 : 3 : 1 | 8 : 6 : 4 |
| 8 : 6 : 1 | 5 : 4 : 9 | 7 : 2 : 3 |
-------------------------------------
| 4 : 9 : 5 | 2 : 7 : 3 | 6 : 1 : 8 |
| 7 : 3 : 6 | 1 : 8 : 5 | 9 : 4 : 2 |
| 1 : 8 : 2 | 9 : 6 : 4 | 5 : 3 : 7 |
-------------------------------------

Solving More SuDoku Puzzles
Every sudoku puzzle can be solved using the techniques demonstrated above. Note, however, that many of the most complex puzzles require multiple techniques used simultaneously. The best way to build this skill?

Practice.

Sudoku puzzles are truly sublime fun if you enjoy logic puzzles.

Solving Sudoku

The only problem with 18thCandidate's otherwise beautiful writeup is, with just one exception, the entire board can be solved using only the "naked single" technique. The exception is at one point the "hidden single" technique is required. These are the two most elementary methods.

Since you aspire to be a Sudoku grandmaster (and really, who doesn't?), you'll have to learn how to solve difficult puzzles. But before I teach you the secret ways of Sudoku, let's be sure we have our terminology correct.

By the (very few) rules of the game, there are nine rows, nine columns, and nine boxes, each of which must contain the numbers one through nine. For consistency, we'll use 1-based indexing. This means the top row is number one, the bottom row is number nine, and so on.

Here's a little diagram that lists the column, row, and box of each cell. Find the hardlinked cell that says "846" in the right-middle box. "8" means it is in the eighth column, "4" means it is in the fourth row, and "6" means it is in the sixth box.

            col 4
             vvv
*-----------------------------------*
|111 211 311|412 512 612|713 813 913|
|121 221 321|422 522 622|723 823 923| <-- row 2
|131 231 331|432 532 632|733 833 933|
|-----------+-----------+-----------|
|144 244 344|445 545 645|746 846 946|
|154 254 354|455 555 655|756 856 956|
|164 264 364|465 565 665|766 866 966|
|-----------+-----------+-----------|
|177 277 377|478 578 678|779 879 979| <
|187 287 387|488 588 688|789 889 989| < box 9
|197 297 397|498 598 698|799 899 999| <
*-----------------------------------*
                         ^^^^^^^^^^^
                            box 9

Now let's get down to business. If you're at all serious about Sudoku, you'll want to get some software. You could do this all on paper, of course, but I prefer to worry about the logic, not the representation. Some people like to use dots to mark the impossible candidates for a cell (which allows you to solve without needing to erase). I invariably make a mess of it, so I heartily recommend Simple Sudoku, available at http://angusj.com/sudoku/. My favorite feature of the program is the filtering. You can filter so that all the cells that include four as a candidate show up in a pale green. It also includes a hint feature which is very handy for when you play puzzles a difficulty level higher than usual. Naturally it can also generate puzzles of a certain difficulty level (the program has five levels). Its only shortcoming is that it doesn't yet know the highly advanced techniques, but that's okay. When you get to the point where you're learning the highly advanced techniques, you'll know what to do.

A naked single is a cell that has only one candidate. This is by far the most fundamental technique; without it you cannot solve any boards. The Boston Globe, my local newspaper, often prints Sudoku boards that can be solved with nothing more than naked singles, sadly. Here's one such board from April 4th, 2006:

 *-----------*
 |..2|.9.|8..|
 |.1.|736|.5.|
 |4..|2.1|..7|
 |---+---+---|
 |1.9|...|3.6|
 |...|...|...|
 |3.7|...|4.5|
 |---+---+---|
 |8..|3.4|..2|
 |.6.|578|.9.|
 |..1|.2.|5..|
 *-----------*

A hidden single is the only cell in a group that has a certain candidate, and so, must be that candidate. For example, if you have the following setup,

 *-----------*
 |...|...|...|
 |5..|...|...|
 |...|...|...|
 |---+---+---|
 |...|...|5..|
 |...|...|...|
 |...|.5.|...|
 |---+---+---|
 |..5|...|...|
 |...|...|...|
 |...|...|...|
 *-----------*

Then the hardlinked cell at R5C2 must be five, because no other cell in box four can have a five. If you try to put a five anywhere else in the box, there would be a conflict (two fives in a single row or column), scary men with large guns would appear, and you will miss your first (and now last) date with that hip girl from the delicatessen. Here's a puzzle generated by Simple Sudoku that can be solved by naked and hidden singles.

 *-----------*
 |..1|..8|.3.|
 |3.4|.7.|.6.|
 |..7|...|..4|
 |---+---+---|
 |...|8..|..7|
 |.3.|.4.|.8.|
 |1..|..9|...|
 |---+---+---|
 |4..|...|8..|
 |.9.|.2.|1.5|
 |.5.|9..|6..|
 *-----------*

These two techniques are really in a class of their own. With one or two advanced exceptions, these are the only that say, "This cell must be a <certain number>." All of the other techniques narrow down the candidates of cells so that these two techniques may be used.

Now we get to the very interesting techniques. Behind each technique name there should be a writeup describing the method in full, because you certainly aren't expected to learn the technique from the executive summaries of each method given below. If a technique doesn't include a summary here, then that just means it is too complicated to summarize concisely -- it does not necessarily mean the technique is difficult to understand!

These techniques are given in the rough order that I learned them. You may find some later in the list easier to grok than some of the earlier ones. Feel free to learn them out of order; this is merely what I recommend from experience. Techniques in bold are known to have a writeup behind them.

  • naked pair: If two numbers are the only candidates of two cells in a group, then the two numbers can be excluded from the rest of the group.
  • hidden pair: If two numbers are the candidates of only two cells in a group, then the rest of the candidates in those two cells can be excluded.
  • locked candidates
    • locked candidates 1: If a candidate in a box is confined to one line, then the candidate can be excluded from cells on the line outside of the box.
    • locked candidates 2: If a candidate in a line is confined to one box, then the candidate can be excluded from cells in the box not on the line.
  • naked subset: If N numbers are the only candidates of N cells in a group, then the N numbers can be excluded from the rest of the group.
  • hidden subset: If N numbers are the candidates of only N cells in a group, then the rest of the candidates in those N cells can be excluded.
  • X-Wing
  • coloring
  • swordfish
  • XY-Wing
  • XYZ-Wing
  • forcing chains: If all the possibilities of a cell lead to some conclusion, that conclusion must be true.
  • uniqueness test: If a certain action would cause the board to have multiple solutions, then we can assume that action is incorrect.

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