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 |
-------------
```
```-------------------------------------
| 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.