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 |
-------------------------------------```