# Su Doku

Su Doku is a Japanese phrase meaning Number Place.

One possible reason it is so popular in Japan is probably because the Japanese language does not lend itself easily to crossword puzzles, so they do Su Doku instead. Su Doku does not require you to speak a particular language, least of all Japanese, nor does it require you to be good at mathematics, as there is no adding up or other arithmetic involved. While it is traditional to place numbers, you could be placing nine different kinds of fruit, letters, or even colours.

### The Puzzle

A Su Doku puzzle is a grid of eighty one small squares (cells) arranged in a larger square. The grid is also divided into nine 3x3 smaller blocks.

The objective is to place the numbers 1 to 9 in the grid so that each horizontal line of nine cells, each vertical line of nine cells, and each 3x3 small block contains one, and only one of each number.

To start you off, the puzzle creator has helpfully placed some numbers for you. Using these numbers, it is possible, using only logic, to deduce the values of all of the other numbers in the grid. The more numbers there are to start with, and the skill with which they are placed determine the difficulty of the puzzle.

Puzzles vary in complexity from easy to fiendish. Sometimes the puzzle creator will describe a puzzle as fiendish, only for the solver to devour it in five minutes. A little luck sometimes helps in spotting the secrets in the puzzle. The experienced Su Doku puzzle solver should be able to complete an easy puzzle in a few minutes. 'Mild' puzzles taking only slightly longer, and 'difficult' puzzles about half an hour. Fiendish puzzles take longer. Sometimes much longer. (The author notes that he only solved the December 24th 2004 fiendish puzzle in The Times on March 19th 2005, and was the first in the office (of six addicted puzzlers) to do so.)

### A simple sample puzzle

Here is a simple starting grid to get you going. As Su Doku puzzles go, this one is yawningly simple.

 8 7 2 9 9 3 5 6 4 1 5 1 5 2 6 3 6 9 7 4 7 6 9 5 6 3 4 7 1 3 4 7 8 5 2

So, where do you start?

The answer is Anywhere you can.

The simplest way of placing a number is to find a cell where only one number can possibly fit. Consider the cell below the 5 in the top right box. The row that it is in has the numbers 1,4,5 and 6. The column that it is in has the numbers 2,3,5,7 and 9. This leaves only the number 8 as a possible solution for that cell.

Another way of finding solutions is by 'slicing'. If you look at the top three rows of the puzzle, particularly at the 9's, you can see that there are 9's in both of the top two rows, and in both the second and third blocks. Therefore, the 9 in the third row must go in the first block, and there is only one free square on that row in that block, and so it can be confidently placed there.

You can repeat the same trick with the 9's by looking at the middle three columns. There is a 9 in the first and second columns, and in the top and middle blocks, so the 9 in the bottom block must be in the third column, and again there is only one place it can go.

Time for a new trick. Now that the 9 is in place in the bottom middle group, there are only three numbers missing from that 3x3 block, which are 1, 2, and 7. As there is already a 7 in the middle column (right at the top of the puzzle), the 7 must go in the vacant square to the left. Of the two remaining squares in that block, the 1 cannot go in the central square, so it must go in the top, which leaves only one place for the 2.

By combining information from both columns and rows, the position of the 7 and 5 in the central block can be determined, and this leaves only one number missing from that block, which must be the 1.

The puzzle slowly starts to reveal its secrets.

 8 7 2 9 9 3 5 6 4 9 1 5 8 1 5 2 7 6 3 6 5 9 7 4 7 6 9 5 6 1 3 4 7 1 3 7 2 4 7 8 5 9 2

Using those two techniques, known as Slicing and Dicing the easier puzzles can be quickly solved.

The rest of the puzzle is left as an exercise for the reader, although if you are deperate, the answer to this puzzle is here.

Apart from simply working out each cell by a process of elimination, there are more advanced techniques which the avid Su Doku solver needs to develop in order to solve some of the more fiendishly difficult puzzles.

One useful technique comes in to play when, in one of the 3x3 boxes, you have a complete row or column filled out. Once this happens, it is possible to determine the column and 3x3 box in which some numbers lie, even if it is not immediately apparent which of the small squares it lies in. Armed with this information, it becomes possible to place numbers in other boxes elsewhere on the grid.

Consider this set of three horizontal boxes:

 9 5 4 3 2 6 6 2 8 1 5

Look in particular at the number 8 in the centre group. There has to be an 8 in the left hand group, but it is not possible to determine (yet) which of the cells it should go in. However, because it cannot go in either of the bottom two cells in that left hand group, it must go in the centre row, which leaves only one possible cell on the top row (in the right hand group) where the 8 can go.

This then leads on to the mystery surrounding the 1 and 7, which are the only two numbers yet to be placed in the top row of the centre group. While it is not possible (yet) to determine which way around these two numbers go, depending on other numbers in the rows not shown, it may be possible to determine where in the centre row in the left hand group the 1 should go.

Harder puzzles require even more advanced strategies, forcing the solver to look one or two moves ahead and solve a number of What If.. situations. If, by elimination, there are two or three possible numbers that can go in a particular square, the solver needs to look ahead to see what the consequences of placing one of the numbers will be. If this leads to an impossibility (having to place two of the same numbers in a row, column, or box) then that number is the wrong choice, so it must be one of the other possible numbers.

### Taking the UK by storm

Since making its UK debut in various national papers in November 2004, Su Doku has spread across the country, slowly driving the polulation insane as they struggle to solve puzzles ranging from simple to fiendishly difficult. The phenomenon has spawned a book, a website, several Su Doku puzzle solving programs, and even a Su Doku puzzle generation program.

Why not discuss your techniques in the conversation forums, or see what other Games and Puzzles the guide has to offer?