Sudoku Strategy

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 leaving only one remaining choice available; so the remaining number must go in that empty square.

Looking at the second row (B) all the squares except the first one Ba have been allocated so the missing number 4 has no choice but to go in the square Ba. You can use this technique by scanning for 8 allocated squares in all rows, columns or regions.

For an extensive run-through example of this strategy  click here

Single possibility rule

When you look at individual squares you will often find that there is only one possibility left for the square. [Note: If there eight squares solved in the group then this is just the same as the only choice rule.] Because of the way that groups intersect you may have a group with more than one unallocated square and yet only one possibility exists for one of the squares. So there is only one possibility for that square, and the number must go there.

In this partially solved Sudoku there are quite a few readily solvable squares. Looking at the purple square Da and running through possibilities: 1;2;3;4;5 and 8 that are allocated in column a leaves only 6; 7 and 9 as possibilities. But in row D there is already a 6 and a 9 so that leaves 7 as the single possibility for square Da. The single possibility rule can be used to solve all the puzzle squares highlighted in green, so that makes it a very useful technique to have up your sleeve. To use this technique you choose a promising square and mentally run through each number in turn that might go in it, if there is only one number left then that number must go in the square.

For an extensive run-through example of this strategy  click here

Only square rule

Often you will find within a group of Sudoku squares that there is only one place that can take a particular number. 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 forces a number to go in one of the squares and not the other one. You are left with an 'only square' within a group for a number to go in. This is different to the 'single possibility' rule where we looked at squares on their own rather than as a group.

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 cannot go in square Ic, the 3 must go in the other square Ac it is the only square in column c where a 3 can be allocated.

You will often find that the same square can be solved by the 'single possibility' rule as well as the 'only square' rule. It doesn't matter which rule you use, as long as the square is solved.

Note: Whenever there are eight allocated in a group with only one remaining empty you can assign a symbol by applying either the 'only choice', 'single possibility' or 'only square' rules as all of them come down to the same thing. It is a feature of Sudoku that squares can be solved in several ways using different strategies.

For an extensive run-through example of this strategy  click here

Two out of three rule

The next useful solution strategy builds on the Only Square rule. Some Sudoku authors refer to it as 'slicing and slotting'. It is a quick way of solving squares as it can be done in your head by scanning the puzzle grid. It almost always finds a square or two to immediately solve. At the heart of the technique is to take 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 the 9s. Here's an example of how it works, for more details look at our 2 out of 3 strategy page or download our puzzle solver and take the free guides for this topic.

Look at the top three rows 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, it must go 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 elimination 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 again two of them with a '1' in them: squares Eh and Ff. There is a '1' missing from row D and because of the '1' in Eh it can't be in Di, '1' must be assigned to Dc. For the last three rows there are already three '1's 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's in Ai and Cb so there is a '2' missing in row B, however in this case there are three unallocated squares Bd; Be and Bf so it can't be quickly decided in which one of these the '2' should go. The same happens in rows D; E; F there are two '2's but both Ed and Ef are possible. Finally in G; H; I there are two '2's Gg and Hc and so there is a '2' missing in row I. The existing '2's 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 in turn may now unlock other squares so it is usually worth restarting the same procedure over for the whole grid.

 To download this puzzle and see it in Sudoku Dragon  click here...

The procedure 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 only one of three squares in this row (or column), and more often than not only one of these is possible and must be allocated there. It will find squares that you could also have found using the only choice, only square and single possibility strategies.

When using the Sudoku Dragon software you can use the automatic allocation feature to automatically find and solve squares that can be solved with the 'only choice', 'single possibility' and 'only square' rules, leaving you free to concentrate on solving the harder squares.

For an extensive run-through example of this strategy  click here

Sub-group exclusion rule

More rarely needed in Sudoku, but exceptionally useful is the sub-group exclusion rule. This takes more explanation as instead of 'forcing' a number in a square, it is an application of logic that 'excludes' possibilities that at first sight looked possible. By excluding one possibility for a square may mean there is only one remaining possibility left, so the square can be safely set to the alternative. Here's an example of the sub-group rule.

A sub-group is a term used to describe three squares in a row or column that intersect a Sudoku region. Every row and column has three sub-groups 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 the sub-group with column b and the green ones the sub-group with column c. The region also has three sub-groups with the rows A; B and C. Every square in the grid belongs to two sub-groups - one for the column it is in and one for the row it is in.

Hidden Twin exclusion rule

You may find you need to use the twin (or triplet) exclusion rule in order to solve more challenging Sudoku puzzles. It is the strategy to use when simpler strategies have been applied and they don't solve any more squares. In essence it is all about spotting matching patterns of possibilities in a group (row, column or region). Spotting these groups takes time and it is difficult to keep track of these in your head, so this is where you need pencil and paper (or the SudokuDragon puzzle solver).

In its simplest form 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 in that group. This does not directly help to allocate squares as the number could go in either of them. However, if the two squares have another possible number then this number can be safely discounted. It is excluded because of the presence of the 'hidden twin' in the group. Studying 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 we are illustrating the hidden twin rule. Look at the region in green Aa, none of the squares have yet been allocated. Both '2' and '3' must go somewhere in the region but there are only two squares that can take them. We have detected a twin {2, 3} in squares Aa and Ba. In square Aa a the '1' that was flagged as possible can now be safely dismissed. In addition square Ba looked like it could a '4' but this too can be excluded due to the same {2, 3} twin. How does this work? There are only two ways that the twin of {2, 3} can be allocated, either '2' in Aa and '3' in Ba or alternatively '3' in Aa and '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 somewhere elsewhere in the group. Whenever there are the same number of possibilities restricted to the same number of squares this logic can be applied. See the theory page for further explanation.

Naked Twin exclusion rule

Another way to exclude possibilities in a group is with the somewhat similar 'naked twins' . In this case the twin squares are evident on their own (and so they are termed 'naked') and these 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 with possibilities 2 and 3. 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, there are no other alternatives. So looking at square Da the naked twin rule excludes '2' from occurring here (because we have just shown that region Ca must have a 2 in either Ca or Cb). As Da is now left with one possibility a '1' can be safely allocated there. For the same reason the naked twin also eliminates '2' from square Cc and a '4' must go there.

General permutation rule

The two 'twin' rules are particular examples of the general Sudoku logic. It is all down to permutations. Each Sudoku group is 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. [Note: In fact the 'only square'; 'single possibility' and 'only choice' are just special cases of this general rule - only one square is involved in this case.] This general rule has more exotic applications.

The twin, triplet, quadruplet rules just reflect different sub-group sizes (2,3,4...). However there are also 'chains'. 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 of three symbols {1,4,7} which is neither a twin nor a triplet. The general permutation rule still applies, if you spot a 1, 4 and 7 elsewhere in the same group can be safely excluded as possibilities. So the logic applies equally for chains as it does for twins, there are 'naked chains' and 'hidden chains'.

For an extensive run-through example of this strategy  click here

X-Wing and Swordfish

One of the more complex Sudoku strategies is the 'X-Wing' and its cousin the 'Swordfish'. These rules are useful for solving the really difficult Sudoku puzzles when all else has been tried and failed.

In looking for twins and permutations we restricted ourselves to looking at possibilities within a single group. The shared sub-group rule is the simplest example of a rule where more than one group is used to eliminate possibilities. The X-Wing requires looking at multiple groups at once as well. A better name for this strategy might be 'Box' as you are looking for four squares forming the corners of a box. These squares must be the only permitted squares for that number in that row (or column) for one particular symbol. This box arrangement forms a two dimensional link. If the symbol spotted occurs in the top left corner of the box it must then also occur in the bottom right corner of the box. The only other alternative is that it occurs in the top right corner in which case it must then occur in the bottom corner. No other option is possible for these four squares and this number. Just as in sub-group rule, both the possible allocations can knock out possibilities somewhere else in the puzzle.

Here's an example (and good X-Wings are hard to find). 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 the vital starting point. Moreover the '4's form the corners of a rectangular box (highlighted in orange). 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 the other '4's from these two columns. So all the purple highlighted squares Aa, Ba, Bf and Ha can have the possibility of a '4' safely discounted. If you are lucky then eliminating the '4' will mean you can allocate one of these 'excluded' squares or at least a square in the same group as them.

 To download this puzzle and see it in Sudoku Dragon  click here...

For an extensive run-through example of this strategy  click here

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

Backtracking or Trial and Error

When all else fails, there is one technique that is guaranteed to always work, indeed you can solve any Sudoku puzzle just using just this strategy alone. It's not really a strategy at all, you just work logically through all the possible alternatives in each square until the puzzle 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.

This is a completely different type of strategy as it uses 'brute force' rather than 'logic'. It is the most contentious Sudoku solving technique and so we have a full description of it with examples on our separate Guessing page.