Sudoku

Sudoku is not a numerical puzzle, as you might suppose. Sudoku consists of a square grid to be filled in with numbers 1 to 9. Actually though, these need not be numbers and could just as well be letters or colours or shapes or whatever - as long as they are distinct things. The puzzle is simply to place these symbols in the correct place to complete the grid.

There is just one simple rule controlling where you can place numbers in the grid. A symbol must occur once and only once in each group of nine grid squares. The groups of nine squares include the rows, columns and regions within the grid. Such a simple rule leads to all the amazing variability of the Sudoku puzzles.

Sudoku Terminology

First let's introduce the terms we will use on this web site, as not everyone uses the same convention.

The whole puzzle area is called the grid, it is divided into rows (horizontal lines) and columns (vertical lines) made up of individual Sudoku squares.

Rows

To save confusion we use letters rather than numbers to refer to rows and columns. The names are shown in the heading of the grid. In this example Sudoku grid row C (capital C) is highlighted. If we used numbers we would end up having to say things like in row 3 there is only one place for a 5 while in column 2 there are 3. Confusing isn't it? Using letters for grid references makes it easier to follow.

Columns

Sudoku columns are given a lower case letter.
Column e (small e) is highlighted.
Using row and column letters lets us unambiguously refer to squares. For example He is row H column e, there is a 2 allocated in this Sudoku square.

Regions

A region is a set of nine adjacent Sudoku squares, in this example the top left region Aa is highlighted.

The whole grid has nine regions. Some Sudoku sites use the term 'mini-grid'; 'box' or 'subgrid' for 'region' .

but we think region is simpler. A region is referenced by the top-left square, so Dd is the central region. A symbol must occur once and once only in each of the regions within the grid as well as each row and column. This was one of the innovations that made Sudoku such an interesting puzzle to solve.

Groups

Rules

How to Play Sudoku

The process of solving a Sudoku puzzle is to fill in all the empty squares. However each square has only one solution as it must obey the Sudoku rule.

Sometimes it is obvious what must go in a square while for others a great deal of mental torture is involved in working through the possibilities (it can take half an hour to solve one square!). Much like placing a single piece in a jigsaw, there must be a place to fit it in somewhere but finding the place is sometimes easy to spot and sometimes takes an age.

There is no correct sequence of square allocations to make, different people have their own techniques for solving the Sudoku puzzle and will solve the squares in a different order. However, the end result is always the same, there is only one unique solution - but many ways of getting there.

Making mistakes

If you make a mistaken allocation of a number for a Sudoku square this will make the puzzle unsolvable. At some later stage you will find an insurmountable contradiction, a symbol would have to be placed in two squares in the same row, column or region violating the Sudoku rule or else you'll find a square that can take no number at all according to the rule. To correct the mistake you need to backtrack through the allocations that you have made until you find the one in error. Often it's because you overlooked another possibility for a square and thought it was the only choice. Sudoku Dragon helps with this as it immediately alerts you when a puzzle becomes unsolvable or you make an allocation that breaks the Sudoku rule. The program lets you backtrack easily to a solvable state again.

Creating Puzzles

Skill is required to create a challenging Sudoku puzzle. It is not just a matter of randomly allocating numbers to squares. Firstly, to ensure that there is only one unique solution requires that there are quite a number of initial 'exposed' squares to begin with. If there were only a handful there would be many ways to allocate all the squares - but all Sudoku puzzles can have only one, unique solution. The challenge is to reveal just enough Sudoku squares to make the solution unique and an adequate challenge. The pattern of squares can make a pleasing arrangement too, and this is taken into account when devising a puzzle. In general the more squares initially revealed the easier the puzzle will be but it is crucial to reveal just the right ones. If the revealed squares are distributed evenly the puzzle will be generally easier than if there are regions with very few filled squares. Some of the toughest puzzles have a couple of regions with no squares revealed at all, or when a number does not occur at all in the whole Sudoku grid. Solution strategies are discussed in our online forums and strategy page.

When Sudoku was taken on by the Nikoli magazine in Japan they decided to add some extra spice to the original puzzle to form true Sudoku puzzles. They stipulated that the pattern of revealed squares should be symmetric. Most puzzles that you come across will have be symmetric. If you turn the puzzle on its side or upside down the pattern of initial squares is repeated (but not the numbers). Sudoku Dragon supports both symmetric and random patterns of initial squares. The random pattern can often make the games more challenging to solve though it is aesthetically appealing to look at.

Sudoku Puzzle Difficulty

Someone setting a Sudoku puzzle has to judge how easy it is to solve. This decision is tricky because there are so many solution strategies and different people will find puzzles more challenging thatn others. Difficulty is in the eye of the beholder - at least to some extent. The vital measure in establishing the level of puzzle difficulty is working out which Sudoku strategies need to be employed in order to solve it. The easier puzzles tend to require only the more basic only square; single possibility and only choice rules. Moderate puzzles require some application of the twin and excluded choice rules. Truly challenging puzzles require the discovery of X-Wings, X-Y Wings, alternate pairs or may be some degree of trial and error - backtracking after following a blind alley or two before the correct solution is attained.

Sudoku and Jigsaws

The closest puzzle to compare to Sudoku is perhaps the humble jigsaw. There are similarities both in the way it works and the pleasure gained by solving it. In a jigsaw there are lots of pieces to fit in a particular grid, there is only one solution and each piece can only go in one place. Sudoku is similar in that is a matter of putting things in the right place. If you like doing jigsaws you'll probably enjoy Sudoku too.

To solve a jigsaw everyone has their own personal strategy. Most people will hunt and separate the edge pieces and then join these up before tackling pieces with distinct markings and then join these up. When nearing the completion of a jigsaw, particularly with problem areas like large expanses of clear blue sky, you may look out for pieces of a particular shape and size. There are different strategies to apply depending on the completeness of the picture and that makes jigsaws interesting.

In Sudoku there are also strategies that you use at different stages of solving the puzzle. Some of these can become a tough trial and error process just like a jigsaw.

The joy of successfully completing a jigsaw is akin to that of solving a Sudoku puzzle, when the final square has been filled in, the satisfaction of correct completion is like stepping back to enjoy the whole picture when the final piece has been placed in a jigsaw. Everything is in its proper place.

Varieties of Sudoku

With such a simple rule you can apply the same idea in lots of different ways. First of all you can change the size of the grid. Using the standard 9x9 grid is only one option. The simpler 4x4 grid is useful for learning the basics of Sudoku and we use it in our tutorials within Sudoku Dragon. There are only four symbols and four regions to consider, but 4x4 never makes a hard puzzle.

Stepping up the other way 16x16 makes a big challenge, because there are 16 squares and 16 possibilities for each square. It is not possible to use just digits but the letters 'A' through 'I' or hexadecimal digits will do admirably. Sudoku Dragon supports puzzles of this size. It's of course possible to increase the sudoku grid size further to 25x25 and then 36x36 and so on, but 16x16 with a total of 256 squares to complete is challenging enough. After that the level of complexity the puzzle has too many possibilities to carry around in the average sized head.

You can also make the regions making up the grid rectangular rather than square. The Sudoku Dragon supports seven rectangular grids including: 2x3 grid (about the most common rectangular size you will find) and the 4x5 monster sized grid.

Our Theme and Variations describes a number of different forms of Sudokus with example grids. Some newspapers print super-sudokus with overlapping 3x3 puzzles so that one region of the grid is shared by a central 3x3 grid. These are time consuming but rewarding when solved.

How many possible puzzles?

As there are so many Sudokus printed these days, surely all the possible grids have now been solved? Well you may think so.

After a little thought it is clear there are quite a few new puzzles left and we are unlikely to run out in the near future. For each row in isolation there are 9! (shorthand for nine factorial) possible permutations of numbers to Sudoku squares which gives 362,880 possible orderings just for one row. Each of these rows can be combined with 8 other rows, and temporarily ignoring the Sudoku rule for columns there would be 9! to the power 9 which works out to be about 10 to the power 50 possible grids (that's 10 with 50 noughts after it).

However applying the Sudoku rule for columns reduces this substantially. Just considering unique solutions for rows and columns and not regions knocks this down as there are now having assigned the first row only 8 options to choose from for each square in the 2nd row and 7 for the second etc. so this gives a much smaller number.

Taking into account regions will also knock out possibilities in a more complicated way. Fortunately some clever people have used super sized calculators to do the maths and claim there are 6,670,903,752,021,072,936,960 unique Sudoku grids of size 9x9. That's plenty to be getting on with.

But if you then start determining symmetries including rotations and swaps then the number of 'effectively different' puzzles reduces to 5,472,730,538. These puzzles would show up as requiring different strategies to be used for their solution.
For more mathematical analysis

For information about the historical development of Sudoku, please see our Origins page. For tips on solving Sudoku puzzles visit our Strategy page or for more mathematical analysis visit our theory page.
Give our SudokuDragon puzzle solver a free 23 day trial by visiting our download page.

What is Sudoku?