## Sudoku Solver
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## Sudoku StrategyThere 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 choice ruleThere 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 (
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## Single possibility ruleWhen you look at individual squares you will often find that there is only one possibility left for the square.
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
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## Only square ruleOften 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
In this case the highlighted column 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.
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## Two out of three ruleThe 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 You can then look at the
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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. 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.
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## Sub-group exclusion ruleMore rarely needed in Sudoku, but exceptionally useful is the A The sub-group exclusion strategy is when you can prove that a number occurs in one of the sub-group squares even though it can't be deduced which of the three sub-group squares it does go in. If you then look beyond that sub-group to the row or column it is in, you can exclude that number from the other intersecting squares. This may not solve a square, but it narrows down the possibilities. A couple of examples follow to explain it. Our Sudoku Dragon has a free tutorial that explains what is going on step by step. Here is a brief example using the simpler 4x4 puzzle size, so there are only four possibilities to think about instead of nine. Sudoku Dragon 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 The other subgroup we could have used in this Sudoku puzzle example is the one shared between column Scaling up to a regular 9x9 Sudoku example, the subgroup exclusion happens in the central region
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## Hidden Twin exclusion ruleYou 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
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 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.
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## Naked Twin exclusion rule
Another way to exclude possibilities in a group is with This 4x4 Sudoku has the region
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## General permutation ruleThe 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. The twin, triplet, quadruplet rules just reflect different number of possibilities (2,3,4...). However there are also ## X-Wing and SwordfishOne 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 Here's an example (and good X-Wings are hard to find). Sudoku Dragon has highlighted all the squares where a
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Believe it or not the complexity does not end at the X-Wing, the 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 strategiesFurther 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 ## Backtracking or Trial and ErrorWhen 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 Sudoku Dragon offers the best range of features for both the newbie and expert. It will solve and generate puzzles of all sorts of sizes. Read more...
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