# Sudoku Strategy

#### Today's Dragon Tip

**Diagonals**

Need a change from regular Sudoku? Try the diagonal version (sometimes known as X-Sudoku) for a new twist.

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There are only a few strategies that you need to know in order to solve Sudoku puzzles. Please take a look at our Sudoku introduction page for background on terminology and also our theory page. Sudoku Dragon comes with a range of guides that take you through these strategies step by step. You can share your tips and experiences on our strategy message forum. There follows a summary of the techniques you may find useful up to 'advanced' level.

## 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. Sudoku allows squares to be solved in different ways using different strategies.

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## Two out of three rule

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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 that can be solved. 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 the free guides.

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 be allocated.

You can then look at the **2**s 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** through **9**. Whenever you allocate a square this may, in turn, unlock other squares so it is worth restarting the same procedure over for the whole grid.

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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. The way it works is that three rows or columns consist of three regions each of these can only take the symbol once.

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.

## Sub-group exclusion rule

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

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Scaling up to a regular 9x9 Sudoku example, the subgroup exclusion happens in the central region **Dd**. It is the sub-group of the central 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 the central region **Dd**. The subgroup exclusion rule requires that a '5' can not go in the remaining shared squares in that region highlighted in red: **Ed** or **Ef**.

## Hidden Twin exclusion rule

You may find you need to use the twin (or triplet) exclusion rules in order to solve more challenging Sudoku puzzles. It is the strategy to use when simpler strategies have been applied and you are still stuck. 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 Sudoku Dragon puzzle solver).

If you have 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 you have a twin. 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 eliminated as an option. 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 lot of easier squares that could be filled in, but we've ignored them as we are illustrating the hidden twin rule. Look at the green region Aa, none of the squares have yet been allocated. Both **2** and **3** must go somewhere in the region but it turns out that there are only two possible squares. So we have detected a twin {2,3} in squares **Aa** and **Ba**. Because of this twin the possibility of a **1** in
square **Aa** can be knocked out. Also square **Ba** looked like it might be 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 the **2** in **Aa** and **3** in **Ba** or alternatively **3** in **Aa** and **2** in **Ba**. These are the only two ways that {2, 3} can be set in this region. So in both of the two possible ways the squares **Aa** and **Ba** are allocated. 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.] **Note**: 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. And on it goes, the same sort of rule applies to quadruplets, quintuplets etc. but these are very rarely found in real Sudoku puzzles.

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

This rule is named the hidden twin rule as the twins are only found by considering other squares in the group. Discovering the twins is the challenge.

## Naked Twin exclusion rule

Another way to exclude possibilities in a group is with **naked twins**. In this case the twin squares are evident on their own (and so they are termed 'naked' to distinguish them from the previous 'hidden' case) and these are used to exclude possibilities in other squares in the same group. Here's how it works.

This 4x4 Sudoku has the region **Ca** highlighted in green. The 'naked twins' are located in **Ca** and **Cb** with possibilities {2, 3}. Because these two squares have no other possibilities we can deduce 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 for these two squares. 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 number of possibilities (2,3,4...). However there are also **chains**. A chain can take in any number of squares, for example if three squares in a group allow just the possibilities {1,7}; {4,7} and {1,4} there is a closed chain of three symbols {1,4,7} which is neither a twin nor a triplet. Detecting this chain lets you safely exclude a possible 1, 4 and 7 elsewhere in the same group. So the logic applies equally for chains as it does for twins, there are 'naked chains' and 'hidden chains'.

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## 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 two groups are looked at to eliminate possibilities. The **X-Wing** requires looking at multiple groups 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 left corner. No other option is possible for these four squares and this number. Just as with the sub-group rule, this can knock out possibilities somewhere else in the Sudoku grid.

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 blue - this is the vital starting point. Moreover these **4**s form the corners of a rectangular box (highlighted in orange). How is this useful? Well, because there must the **4** in column **a** must either go in **Ca** or **Ga** and nowhere else in that column. Similarly in column **f** (the **4** must be in either **Cf** or **Gf** and so we can exclude all the other **4**s from these two columns. So all the yellow highlighted squares Aa, Ba, Bf and Ha can have the possibility of **4** safely discounted. If you are lucky then eliminating the **4**s will mean you can allocate one of these excluded squares or at least a square in the same group as them. **Note**: The term X-Wing is probably derived from the name of Star Wars fighter which had an X shaped cross-section.

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Believe it or not the complexity does not end at the X-Wing, the **Swordfish** is a further refinement of the X-Wing. Instead of four squares forming a box of possible allocations the Swordfish rule uses six squares. In the example puzzle there are not just two pairs of squares for **9** but three pairs: in columns **b**; **e** and **h**. These squares are highlighted in blue/purple color. They are linked by rows to form a box with an extension 'sword' jutting out on one side : hence the term **Swordfish**. The other squares forming the swordfish are highlighted in orange and yellow. Because all these three columns have coinciding end squares the rule applies again. Any **9**s that we find in rows that link the columns can be safely excluded because we know that a **9** must occur in one of the two highlighted squares in that row; they can't be allocated elsewhere. These excluded squares are highlighted in yellow (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 use it to solve a Sudoku puzzle.

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## More advanced strategies

Further complex strategies are available for fiendishly difficult puzzles. They require a lot more thought and analysis to learn about and use correctly. The techniques include the X-Y Wing or Hook and powerful Alternate Pair , they are explained in full on our separate **Advanced Strategy** page.

## 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 one strategy alone. You just work logically through all the possible alternatives in every square in order until you find the allocations that work out. If you choose a wrong option at some stage later you will find a logical inconsistency and have to go back, undoing all allocations and then trying another option. Because there are so many alternatives (billions) you won't want to use it too often. You start with a square and choose one number from the available possibilities. This is a completely different type of strategy as it uses 'brute force' rather than 'logic'. It is a contentious Sudoku solving technique and so we have a full description of it with examples on our separate ** Guessing page**.

See also

Sudoku Strategy ➚ Some of the more complex puzzle solving strategies explained.

Sudoku Solution Hints ➚ Good introduction to the various strategies for solving puzzles including X-Wing; XY-Wing.

Solving Sudoku ➚ Detailed step by step solution of Sudoku puzzles.

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