Sudoku Strategy

There are only a few solution Strategies that you need to know in order to solve any Sudoku puzzle. Take a look at our sudoku introduction page for background on terminology and also our theory page. SudokuDragon comes with a range of tutorials that take you through these strategies step by step. We also have a discussion area on our strategy message board.

Only choice rule

There may be only one possible choice for a particular Sudoku square. In the simplest case you have a group (row, column or region) that has eight squares allocated so that leaves only one remaining choice available. So this number must be allocated in that free square.

Looking at the second row (B) all the squares except the first square Ba have been allocated so the missing number 4 has no choice but to go in the square Ba.

Sometimes rather than a Sudoku square having only one choice for it, you find a Sudoku group that has one square in which only one number is possible. For example if a group has seven squares allocated with only two numbers left to allocate it is often the case that an intersecting (or shared) group will mean that a number can not go in one of these two squares and so it must go in the other.

In this case the highlighted column c has seven numbers allocated. The missing numbers are 1 and 3. But you can see that there is already a 3 in row I (square If) so a 3 can not go in square Ic, the 3 is forced to be allocated in the other square Ac it is the only square in column c where a 3 can be allocated.

Note: whenever there are eight allocated in a group with only one remaining you can assign a symbol by applying the 'only choice' or 'only square' rule as both rules result in the same implication.

One of the most useful solution strategies involves a special form of the 'Only Square' rule. Other Sudoku authors refer to it as 'slicing and slotting'. It is a quick way of spotting squares to solve as it can be easily done in your head by scanning the puzzle grid. It almost always finds a square or two you can immediately solve. At the heart of the technique is taking groups of three rows and columns in turn, working methodically through the whole grid. First look for all the 1s then all the 2s,3s etc. all the way through to 9. Here's an example of how it works, for more details look at our Two three strategy page or download our puzzle solver and take the free tutorial for this topic.

Look at the top three rows at where the '1's are located - they are in row A column e (Ae) and Row C column a (Ca) There is no '1' in row B, but there must be one to allocate in one of the blank squares. Because of the 1 in Ae it can not go in any other of the squares in region Ad that is Bd;Be or Bf. By force of elimination then there is only one place '1' can go in row B and that is in the highlighted square Bi. Using the same logic for the following three rows D;E;F there is once more two of them with a 1 in them: squares Eh and Ff. There is therefore a 1 missing from row D and because of the 1 in Eh it can't be in Di, '1' must therefore be assigned to Dc. For the last three rows there are already three 1s Gd; Hb and Ig so there is no '1' left to allocate.

You can then look at the 2s in these three sets of three rows. In rows A;B;C there are 2 in Ai and Cb so there is a 2 missing in row B, but in this case there are 3 unallocated squares Bd;Be and Bf but it can't be quickly decided in which one should go the 2. The same happens in rows D;E;F there are two 2s but both Ed and Ef are possible. Finally in G;H;I there are two 2s Gg and Hc and this means there is a 2 missing in row I. The existing 2s mean there is only one place it can go - square Id. You can then continue this scan through all rows then all columns in groups of three and then through all the numbers 1 to 9 whenever you find a square you can allocate this may unlock other squares so it may well be worth starting again for the whole grid.

The general strategy is to scan rows and columns in groups of three and look to see where if anywhere the number being scanned has been allocated. If you find two out of the three then you know that the missing number can only go in one of three squares in this row (or column), and more often than not only one of these is possible and must therefore be the solution.

More rarely needed in Sudoku, but exceptionally useful is the sub-group exclusion rule. This takes quite a bit more explaining as instead of 'forcing' an allocation of a number to a square, it is an application of logic that 'excludes' possibilities that at first sight look possible. By excluding one possibility for a square may mean another possibility can then be safely assigned there as it was the only other alternative left for the square. Here's an example of how to use the sub-group rule.

A sub-group is a term used here to describe a group of three squares that are the intersection between a Sudoku region and a row or column. Every row and column has three subgroups in the three regions it crosses. In this example the region Aa has been color coded to show the three subgroups it forms with columns a; b and c. The three pink squares are the sub-group intersecting region Aa and column a; the yellow squares with column b and the green ones with column c. The region also has three sub-groups with the Rows A; B and C. In fact every square in the grid belongs to two subgroups - one for the column it is in and one for the row it is in.

The Sudoku rule makes use of the fact that it's often possible to prove that a number must only occur somewhere in a sub-group and not elsewhere in a group but it can't be directly deduced which of the three sub-group squares it must occur in. In effect we have a partial allocation : the number must be placed in only some squares in a sub-group and not in the other squares. This is useful because it excludes it occurring elsewhere, even though it does not lead to a specific square allocation. It's just narrowing down possibilities a bit. Having established allocation within a sub-group this knowledge can be applied to the rest of the region excluding the possibility of the same number occurring in those squares.

Here is a brief example using the simpler 4x4 grid. SudokuDragon has been used with possibilities enabled and exclusions switched on so that the grid directly shows the squares where the exclusion rule comes into play.

First look at column d, you'll see that the '1' must go in the square Cd and that's the only place it can go in the region Cc. Applying the subgroup rule for the subgroup shared between row d and region Cc (highlighted wuth blue) means that '1' can only occur in this subgroup and can not go in any other square in the region, in this case Cc or Dc, so that is why '1' is shown as excluded with a dark background by the SudokuDragon puzzle solver. Moreover because Dc could only take a '1' or '4' it's now certain that '4' must be allocated here.

The other subgroup we could have used in this Sudoku puzzle example is the one shared between column a and region Cb (highlighted in red). Here we can tell that '4' must be allocated in Ca as that is the only place in column a that can take it. So using the subgroup rule '4' can not go in Cb or Db, and so we can safely assign 1 to Db.

Note that these 4x4 Sudoku examples do not really show the full impact of the scheme as it's just not possible to have many options so the simpler rules could have been used to solve these squares more easily.

In this more complex 9x9 puzzle the action happens in the central region Dd. It is the subgroup of this region with the highlighted row F that is of interest. Look at the squares in row F that a 5 can go, it can't go in Fa (because of Aa) nor in Fh (because of Dh) nor in Fi (because of Bi). That only leaves Fd and Ff both of which occur as a shared sub-group with region Dd. The subgroup exclusion rule means that a 5 can not go in the remaining shared squares highlighted in purple : Ed or Ef.

Sudoku Dragon has highlighted quite a few possibilities in other squares that can be safely excluded using this same rule - for example the 8s in Ce and Cf and the 4s in Eg and Eh. This last example is particularly useful since square Eg is left with only one possibility with the 4 excluded, 7 must be allocated there.

Occasionally you may find you need to use the twin (or triplet) exclusion rule in order to solve some of the more difficult Sudoku puzzles. It is the strategy to use when the other strategies have been applied and they don't give any more squares you can solve. In essence it is about spotting matching groups of possibilities in a group (row, column or region). Spotting these groups takes time and it is quite easy to make mistakes, so this is where you need pencil and paper (or the SudokuDragon puzzle solver software).

In its simplest case there are two or more unallocated squares in a region and there are two numbers that can only go in the same two squares and no others. This does not help with allocating the numbers directly as the number could go in either square. However, if there is another possible number in either of the two squares then this number can be discounted as it is excluded because of the presence of a twin elsewhere in the group. It all stems from the presence of a twin. An example is the best way to get your mind around this rule.

Look at this 4x4 grid. There are a number of easier squares that could be filled in, but we'll ignore them as this is only to show the use of the twin rule. Look at the region Aa, none of the squares have yet been allocated. Both '2' and '3' must go somewhere in the region and there are only two squares into which they can go. We have a twin in squares Aa and Ba. In square Aa a 1 was also flagged as possible this can now be safely excluded, also square Ba seems to allow a 4 but this is also excluded because of the twin. How does this work? There are only two ways that the twin of {2,3} can be allocated, either as '2' in Aa and the '3' in Ba or alternatively '3' in Aa and the '2' in Ba. This does not allow the possibility of the '1' being allocated in Aa or the '4' being allocated in Ba - they must be allocated elsewhere in the groups. Whenever there are a restricted number of possibilities restricted to the same number of squares this logic can be applied. See the theory page for futher explanation.

Our Sudoku Dragon software has a free tutorial that explains twins in more detail with an animated guide.

The rule for twins extends to triplets too. If you find that three symbols have only three shared possible squares in a group (row, column or region) then all other possibilities in these three squares can be discounted too. And on it goes, the same sort of rule applies to quadruplets, quintruplets etc. but these are very rarely found in real puzzles.

This rule is sometimes called the hidden twin rule as the twins are not immediately evident and discovering the twin is the challenge.

Similar but slightly different is the use of 'naked' twins to exclude other possibilities in the group. In this case the twin squares are evident on their own (naked) and are used to exclude possibilities in other squares in the same group. Here's how it works...

The 4x4 grid has the region Ca highlighted. The 'naked twins' are located in Ca and Cb. Because these two squares have no other possibilities we know that a '2' must go in Ca and '3' in Cb or else '3' in Ca and '2' in Cb, no other valid possibility exists. So looking at Da the naked twin excludes '2' occurring here, this square must be allocated to '1'.

The 'twin' rules are examples of a general logical property of Sudoku puzzles. To follow this you may need to look at the theory of permutations. Each group is just a permutation of the numbers 1 to 9 (for a 9x9 grid). If you can identify a group within this permutation that is restricted to the same number of squares then you have a sudoku permutation rule. In fact the 'only square' and 'only choice' are just special cases of this general rule - the subset size happens to be of size one. There are more exotic situations for this general rule to apply too.

The twin, triplet, quadruplet rules are stated in terms of the size of the sub-group (2,3,4...) but a chain is also possible. A chain can take in any number of squares, for example if the first three squares in a group allow possibilities {1,7}; {4,7} and {1,4} we have a closed chain group of three symbols {1,4,7} this is not a twin or a triplet but the general permutation rule means that if you are clever enough to spot it then 1, 4 and 7 elsewhere in the group can be safely excluded as possibilities.

By far the most complex method to cover in this introduction to Sudoku strategy is the 'X-Wing' and its variant the 'Swordfish'. These rules are useful for solving the really difficult Sudoku puzzles when all else fails.

In looking for twins and general permutations we have restricted ourselves to look at possibilities within a single group at a time. The shared sub-group rule is an example of a rule where more than one intersecting group is used to determine possibilities. The X-Wing is similar to this in that it requires looking at multiple groups at once. A better name for this strategy might be 'box' as the key feature is four squares forming the corners of a box. These squares must be the only permitted squares for that symbol in that row of one symbol (or column). This box arrangement now forms a two dimensional pairing. If the symbol spotted occurs in the top left of the box it must then also only occur in the bottom right square of the box, the only other alternative is if it occurs in the top right in which case it must occur in the bottom left square. No other option is possible for these squares and this symbol. The usefulness of this rule is that both the possible allocations put the symbol in the two columns (or rows) involved and so this knocks out possibilities anywhere else in the column (or row).

Here's an example (and it takes time to find a good example of a puzzle with an X-Wing in it). Sudoku Dragon has highlighted all the squares where a 4 is allocated or looks like it can be allocated. The rows C and G are crucial. They both have only two squares that can take a 4: Ca; Cf; Ga and Gf - highlighted in green - this is a vital precondition. Moreover the 4s form the corners of a rectangular box (highlighted in yellow). How is this useful? Well, because there must be a 4 in both column a (either in Ca or Ga) and also in column f (either in Cf or Gf), we can exclude all other possible 4s from these columns. So all the purple highlighted squares Aa; Ba; Bf; and Ha can have the possibility of a '4' discounted. If you are lucky then eliminating the '4' will mean you can allocate one of these squares or at least a square in the same group as these squares.

Note: The term X-Wing is probably derived from the name of Star Wars fighter which had an X shaped cross-section.

Believe it or not the Swordfish is a further complication on top of the X-Wing, instead of four squares forming a linked cycle of possible allocations the Swordfish rule allows more linked pairs to form the group. In this case we have not just two pairs of two possibilities for 9 but three : in columns b; e and h. These squares are highlighted in orange. It's a box with an extension 'sword' jutting out on one side : hence the term Swordfish. Because these three columns have these coinciding end squares the same rule applies. Any '9's that we find in rows that are part of the grid can be safely excluded because we know that a 9 must occur in one of the two orange squares in the row and so can't be located elsewhere. These excluded squares are highlighted in purple (Fc; Gf and Fg).

Of course, the Swordfish is not the end of the matter we can extend the logic to four interlinking pairs of possibilities and then five etc.. You'll feel a real sense of achievement if you locate a Swordfish and can use it to help solve a Sudoku puzzle!

When all else fails, there is one technique that is guaranteed to always work, indeed you can solve any Sudoku puzzle just using this strategy and nothing else. It's simply a matter of working through all the alternatives until it is solved. Because there are so many alternatives (billions) you won't want to use it too often. You start with a square and choose arbitrarily one number from the available possibilities. You need to make a note of the choice you make though. Now continue solving squares until the puzzle is solved or it is un-solvable. To see this approach in action take a look at this external web site that has an animated example : Animated solver A Sudoku puzzle becomes un-solvable because an incorrect allocation has been made. This shows up as two squares in the same region having the same number (breaking the rules) or as a square in a region unable to take any number at all (breaking the rules). If this happens you have to back-track all the allocations until you come back to the last square you had to guess, and then choose one of the other alternatives. Sudoku Dragon makes this easy because it supports 'Undo' - you don't need to remember the squares you allocated. You can also add a note to a square to note the guess that you made.

The reason that the Labyrinth is mentioned is that this is the strategy to get your way out of any Maze, you can always escape from a maze as long as you run through all the alternatives in a consistent, logical fashion. If turning right leads to a dead end, retrace steps and try straight ahead, if that fails try left. To keep track of where you have been you need plenty of string, some chalk or a bag of rice to make a mark of where you have been already. In this way all of the possible routes are tried out in turn - until you find a path that leads you out.

Give our SudokuDragon puzzle solver a free 22 day trial by visiting our download page. All the main Sudoku strategies have clear, easy to follow guide tutorials. It can selectively help you quickly identify possibilities and exclude the ones that actually aren't possible.

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References

Sudoku Strategy

Sudoku Solution Hints

Solving Sudoku

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