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Sudoku StrategyThere are very few strategies that you need to know in order to solve Sudoku puzzles. 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 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 this number must go in that empty square.
Single possibility ruleWhen you look at individual squares you will often find that there is only one possiblity left for the square. If there eight squares solved in the group then this is just the same as the only choice rule. However 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 no choice, the number must go in that square.
Only square ruleOften you will find a group of Sudoku squares where only one of the squares 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 will mean that a number can not go in one of these two squares and so it must go in the other one. You are left with an 'only square' for a number to go in.
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 choose, it depends which one you find easiest. 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 imply the same thing. It is a key feature of Sudoku that it can be solved in several ways using different strategies. Two out of three ruleOne of the most useful solution strategies involves a special form of the Only Square rule. Some 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 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 9. 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 tutorial for this topic.
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, but in this case there are three unallocated squares Bd; Be and Bf so it can't be quickly decided in which one 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 may unlock other squares so it is usually worth starting again for the whole grid.
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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 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 some squares that you could also have found using the only choice, only square and single possibility rules. 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 ruleMore rarely needed in Sudoku, but exceptionally useful is the sub-group exclusion rule. This takes more explanation as instead of 'forcing' an allocation of a number to a particular 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. Here's an example of the use of the sub-group rule.
A sub-group is a term used here 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. This strategy comes into play when you can 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 go in. The number must be placed in only within a sub-group but not elsewhere even though it may not lead to solving a specific square. It's all about narrowing down the possibilities a little. Having established an allocation can only be made within a sub-group then all other squares in the rest of that region can have this number safely excluded. Our Sudoku Dragon has a free tutorial that explains what is going on step by step.
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 column d and region Cc (highlighted in blue) means that '1' can only occur in this subgroup and can not go in any other square in the region, in this case squares 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 impossible to have many solution strategies so the simpler rules could have been used to solve these squares much more easily.
Sudoku Dragon has highlighted quite a few possibilities in other squares that can be safely excluded using the same rule - for example the '8's in Ce and Cf and the '4's 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 go there. Hidden Twin exclusion ruleOccasionally 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 simpler strategies have been applied and they don't solve any more squares. In essence it is about spotting matching patterns 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.
Our Sudoku Dragon software has a free tutorial that explains twins in more detail with an animated guide. 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 too. And on it goes, the same sort of rule applies to quadruplets, quintuplets etc. but these are very rarely found in actual puzzles. This rule is sometimes called the hidden twin rule as the twins are not immediately evident and discovering the twin is the challenge. Naked Twin exclusion rule
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 square Da the naked twin rule excludes '2' from occurring here, '1' must be allocated there. For the same reason the naked twin also eliminates '2' from square Cc. General permutation ruleThe '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. [Note: In fact the 'only square'; 'single possibility' and 'only choice' are just special cases of this general rule - the subset size is one in this case.] There are more exotic situations for application of this general rule. 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 can spot it then 1, 4 and 7 elsewhere in the same group can be safely excluded as possibilities. X-Wing and SwordfishOne of the more complicated Sudoku strategies is the 'X-Wing' and its variant the 'Swordfish'. These rules are useful for solving the really difficult Sudoku puzzles when everything else has been tried. In looking for twins and permutations we have restricted ourselves to look at possibilities within a single group. 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 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 for 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 occur in the bottom right square of the box. The only other alternative is if it occurs in the top right square in which case it must then 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 same two columns (or rows) involved and so knocks out possibilities anywhere else in the column (or row).
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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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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! Further strategies are also available for fiendishly difficult puzzles. These require a lot more thought and analysis to use in 'real' rather than 'theoretical' puzzles. These are the X-Y Wing and Alternate Pair techniques, they are explained in full on our separate Advanced Strategy page. 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 this strategy and nothing else. It's simply a matter of working logically through all the 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 alternative 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. 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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