Solving strategies used by SUDOKIS

Now comes the difficult part, the solution strategies that SUDOKIS uses to solve Sudokus. If you ask SUDOKIS for help, you should have understood some basics of the solution strategies and the presentation in SUDOKIS. However, if you are reading the rule explanations for the first time, it makes sense to skip the difficult rules and come back later. You do not need the difficult rules to solve simple Sudokus.

You should understand how rows, columns and blocks are numbered and how the coloring supports the explanation of the rules.

Naked and Hidden Singles

Naked Single is the simplest rule in Sudoku. It is simply an application of the basic rule that a number may only occur once in a column, row and block. Accordingly, you can delete all the markings of the known numbers in the columns, rows and blocks.

To apply the rule, take a known number of a field and cross out all the corresponding marks in the row, column and block.

In this example, there are a number of fields where it is obvious that the 9 is not possible. You can delete the markings here.

Just by applying this simple rule, you can solve all Sudoku puzzles that are classified as Easy.

With the Direct Rule, known numerical values are simply entered directly into the field. Strictly speaking, the Direct Rule is only a sensible simultaneous application of several Naked Single rules. To apply this rule, select a field that you want to solve and check the surrounding row, column and block to see if you can delete all numbers except one. If you find such a field, you can enter the solution number directly and save yourself the trouble of marking individual numbers.

The disadvantage of this rule is that you have to search a little more than with applying the original Naked Single rule. In addition, with medium-rated Sudokus at the latest, you reach a point where you can no longer find a Direct Rule and have to fall back on the basic Naked Single rule.

In the situation of this example you can enter the 3 directly. It is not necessary to mark many fields with a lot of effort.

Easy rated Sudokus can be solved entirely with the Direct Rule. With Easy 10 rated Sudokus, however, you have to search for the fields a little, with Easy 1 Sudokus every field can be solved immediately with the Direct Rule.

A little preparation is required to apply the Hidden Single rule. Try to delete all possible selections in a row, column or block by applying the Naked Single rule. If you now have a situation in which a number is only possible in a single field, then this number must also be entered there.

In this situation, we know a field in which we can enter a 5, as the 5 has already been excluded from all other fields in the block by applying previous rules.

Medium rated Sudokus can be solved completely by knowing the Hidden Single and Naked Single rules.

Are you wondering why we make even medium-rated Sudokus so simple that we can solve them with just two rules? The answer is simple, Sudokus should also be solvable by small children. We want them to have a sense of achievement, not just to be able to solve Easy-rated Sudokus. For everyone else, the fun begins with advanced Sudokus, which are rated Difficult 1 and above.

Naked and Hidden Subsets

The search for the Naked Pair rule usually begins when the Naked Single and Hidden Single rules fail. To do this, you look for a row, a column or a block in which two numbers are only possible in two fields. In this situation, one field must have one value and the other field must have the other value. We do not know which of the fields gets which value. However, we do know that the two values cannot occur in any other field.

In the situation of this example, we know from a number of fields that neither a 2 nor a 8 can occur there.

Together with the Single rules, knowledge of the Naked Pair rule will enable you to solve Sudoku puzzles with a Difficult 1 rating.

The name of the Hidden Pair rule could just as easily be Naked Seven. The best way to understand this is to look at this example. Here there are seven fields marked in red in which neither the number 1 nor the number 2 appears. It is obvious that all seven fields must have different numbers. Since the set of numbers that must be distributed over the seven fields is limited to the values 3, 4, 5, 6, 7, 8 and 9, we know that we must distribute these values completely over the fields with the red marking. None of the numbers 3, 4, 5, 6, 7, 8 and 9 can therefore appear in the fields marked in green.

You can read the identical, shorter argumentation given by SUDOKIS under the Sudoku box: In the cells [4,6] of block 8, only one value from [1, 2] is possible, as all these values cannot occur in the 7 other cells of block 8.

Together with the previous rules, you can solve Sudoku puzzles with a Difficult 2 rating by knowing the Hidden Pair rule.

Understanding the Naked Triple rule is quite simple if you have already internalized the previous rules. In this example, the values 2, 5 and 8 can only appear in the fields marked in red and therefore not in any of the other 6 fields. In three fields we can already find the solution, in the three other fields marked in green we can exclude 2, 5 and 8.

However, just because it's relatively easy to understand this rule this doesn't mean it's easy to find it in a large Sudoku field. In our opinion, it is easier to find the rules in the next section “Interaction between different houses”. Accordingly, we rate this rule as Difficult 7. If you can solve this rule, all the previous rules and the rules in section “Interaction between different houses”, you can solve all the Sudokus rated Difficult 7.

The Hidden Triple rule, which could also be called Naked 6, works in a similar way to the other Hidden rules. The values 1, 2, 4, 6, 7 and 8 only appear in the fields marked in red, so they can be deleted from the fields marked in green.

Together with the previous rules and the rules from the section “Interaction between different houses”, you can solve Sudoku with a Difficult 8 rating.

With the Naked Foursome rule, you search for 4 values that can only be found in four cells. Here, these are the values 2, 4, 7 and 8, which can only be found in the cells with the red marking in the third column. Accordingly, these values can be excluded from all other fields, marked in green here. The top field is not marked green as the values have obviously already been excluded in advance.

Together with the previous rules and the rules from the section “Interaction between different houses”, you can solve Sudoku with a Difficult 9 rating.

The Hidden Foursome rule, which could also be called Naked 5, works in a similar way to the other Hidden rules. The values 2, 5, 7, 8 and 9 only appear in the fields marked in red, so they can be deleted from the fields marked in green.

Together with the previous rules and the rules from the section “Interaction between different houses”, you can solve Sudoku with a Difficult 10 rating.

Interaction between Different Houses

The Row Block Interaction is a rule for which you have to search within a row for a value that only occurs in cells that belong to a single block.

In this example, let's look at the value 9 in row 1. This value can only occur in the fields marked in red. As one of the two fields outlined in red contains the value 9, it cannot occur in any other field in block 1. The value 9 can therefore be deleted in the fields with the green marking.

With Block Row Interaction, you proceed the other way round. Within a block, you search a value that only occur in cells that belong to a single row.

In this example, let's look at the value 7 in block 2. This value can only occur in the fields marked in red. As one of the two fields outlined in red contains the value 7, it cannot occur in any other field in row 6. The value 7 can therefore be deleted in the fields with the green marking.

The Column Block Interaction is the same rule as the Row Block Interaction, only with a field rotated by 90°. Here you have to search within a column for a value that only occurs in cells that belong to a single block.

In this example, let's look at the value 2 in column 7. This value can only occur in the fields marked in red. As one of the two fields outlined in red contains the value 7, it cannot occur in any other field in block 6. The value 7 can therefore be deleted in the fields with the green marking.

Last but not least, the Block Column Interaction is the same rule as the Block Row Interaction, only with the field rotated by 90°. Here you have to search within a block for a value that only occurs in cells that belong to a single column.

In this example, let's look at the value 8 in block 5. This value can only occur in the fields marked in red. As one of the two fields outlined in red contains the value 8, it cannot occur in any other field in column 4. The value 8 can therefore be deleted in the fields with the green marking.

Sudokus of difficulty levels Difficult 3 to Difficult 6 can be solved with the knowledge of the rules “Row Block / Block Row / Column Block and Block Column Interaction” together with the rules Hidden Pair or easier.

The difficulty level of a Sudoku between Difficult 3 and Difficult 6 depends on how many of these rules have to be found to solve the Sudoku.

Fishes

There are three advanced solution strategies in Sudoku, which are collectively referred to as Fishes. These are the X Wing, the Swordfish and the Jellyfish. Contrary to what the now common name X Wing suggests, this strategy does not belong to the Wings and Chains, but to the Fishes.

The principle of this group of solution strategies is always the same, they only differ in size. While the X Wing is size two, the Swordfish is size three and the Jellyfish is size four. Theoretically, there are also larger fish, but these are unimportant as they do not provide any further insights that the smaller fishes have not already provided.

In the X-Wing, you are looking for two rows in which a value only occurs twice, exactly one above the other. In this example, we are looking at rows 2 and 7. The value 6 only occurs in cells 3 and 7 in each of these rows. Let us now assume that the value 6 occurs in row 2 in the third cell, then it must occur in row 7 in the seventh cell. If the value 6 appears in row 2 in the seventh cell, then it must appear in row 7 in the third cell. Both cases are shown below.

However, if we now look at columns 3 and 7, we know that the value 6 cannot occur in any other row than rows 2 and 7. We can delete the number 6 in the fields outlined in green.

Analogously, the X-Wing can also be found with the rows and columns reversed.

If you understood this rule and all rules above, you can solve Sudokus with rating Difficult 10.

The Swordfish works in the same way as the X-Wing, except that we search for three rows in which a value occurs only three times, exactly one above the other. In this example, we find the value 3 in rows 3, 4 and 7 only three times, distributed over only three columns 2, 7 and 8.

We have exactly three ways in which we can distribute the value 3 across the rows (see below). This means that the value 3 cannot appear anywhere else in the corresponding columns 2, 7 and 8.

In exactly the same situation, we can alternatively find another Swordfish. In this Swordfish, however, we search for columns with the same values and can then delete values, in this case the value 7, in the corresponding rows.

The three combinations below show how the value 7 can be distributed.

The Swordfish is much more difficult to find than the X-Wing. We therefore classify the Swordfish as Very Difficult 2 in our ranking of ratings. In between with a Difficult 1 rating is the easiest chain, the XY Wing.

The Jellyfish works in the same way as the X-Wing and the Swordfish, except that we search for four rows in which a value occurs only four times, exactly one above the other. In this example, we find the value 2 in rows 3, 4, 6 and 7 only four times, distributed over only four columns 2, 3, 7 and 8.

We have exactly 12 ways in which we can distribute the value 2 across the rows (see below). This means that the value 2 cannot appear anywhere else in the corresponding columns 2, 3, 7 and 8.

The Jellyfish is much more difficult to find than the Swordfish. We therefore classify the Jellyfish as Very Difficult 6 in our ranking of ratings. In between with a Difficult 4, a Difficult 5 and a Difficult 6 rating are the XYZ Wing, the X Chain and the XY Chain.

Wings and chains

Wings and Chains are the same category of solution strategies, the principles are identical. There is a starting point from which several logical chains begin in the form of a case differentiation. At this starting point, there are two or more possible values that the starting point can take. A chain must start from each of the possible values. In our example, this means that three possible values of the starting point mean that we have to consider three chains.

Now we build up the logical chains over several intermediate fields if necessary. The intermediate fields always have exactly two open possibilities, otherwise they cannot be used as intermediate fields. Fields can only be connected to each other if they are in the same row, the same column or the same block in Sudoku. Only in this illustration do we initially refrain from showing the exact location in the Sudoku so that the principle can be recognized more clearly. The intermediate fields are displayed by SUDOKIS with a purple color. The chains all end in the target field at the same numerical value.

Now let's look at the first chain. Let's assume that the start field has the value 1. Then the intermediate field A must have the value 7, as both fields are connected via the same row, the same column or the same block. The intermediate field B must have the value 5 and the end field must not have the value 5.

Let's look at the second chain. Let's assume that the start field has the value 2, then the intermediate field C must have the value 4, the intermediate field D must have the value 6, the intermediate field E must have the value 5 and the end field must not have the value 5.

In the third case, if the start field has the value 7, then the intermediate field F must have the value 5 and the end field must not have the value 5.

We do not know the value of the start field, it will have one of three possible values. But no matter which value it is, the end field must not have the value 5.

All chains and wings have the same principle, but are given different names depending on their properties. If chains have a maximum length of 2, then we call them wings. If there are 2 possible values in the start field, so that we have to build two chains, we call them an XY Wing or an XY Chain. If there are three chains, we speak of an XYZ Wing / XYZ Chain. Four chains are called WXYZ Wing / WXYZ Chain, five or more chains are called VWXYZ Wing / VWXYZ Chain.

The following examples show that all the variants listed can be found in real Sudokus.

An XY Wing is the simplest variant of the chains. It has two starting points (XY) and a maximum chain length of 2 (Wing). In this example, start in row 2 cell 9 with the values 2 and 7. As a result, the value 9 in row 4 cell 7 can be deleted.

The XYZ Wing has three starting points (XYZ) and a maximum chain length of 2 (Wing). In this example, start in row 3 cell 5 with the values 2, 5 and 8. As a result, the value 5 in row 3 cell 6 can be deleted.

The XY Chain has two starting points (XY) and no chain length limit (Chain). In this example, start in row 1 cell 7 with the values 5 and 8. As a result, the value 4 in row 1 cell 6 can be deleted.

The XYZ Chain has three starting points (XYZ) and no chain length limit (Chain). In this example, start in row 2 cell 3 with the values 1, 2 and 3. As a result, the value 1 in row 9 cell 3 can be deleted.

The WXYZ Wing has four starting points (WXYZ) and a maximum chain length of 2 (Wing). In this example, start in row 5 cell 4 with the values 1, 2, 6 and 9. As a result, the value 1 in row 5 cell 6 can be deleted.

The WXYZ Chain has four starting points (WXYZ) and no chain length limit (Chain). In this example, start in row 8 cell 1 with the values 5, 6, 7 and 9. As a result, the value 9 in row 9 cell 1 can be deleted.

The VWXYZ Wing has five starting points (VWXYZ) and a maximum chain length of 2 (Wing). In this example, start in row 1 cell 1 with the values 2, 3, 5, 7 and 8. As a result, the value 2 in row 3 cell 1 can be deleted.

The VWXYZ Chain has five starting points (VWXYZ) and no chain length limit (Chain). In this example, start in row 4 cell 4 with the values 2, 4, 6, 7 and 8. As a result, the value 8 in row 4 cell 5 can be deleted.

This example of an TUVWXYZ Chain shows that even more starting points are possible. For the sake of simplicity, we no longer assign letters, but also call this chain VWXYZ Chain.

Just like the normal VWXYZ Chain and the VWXYZ Wing, we rate this chain as Very Difficult 10, which means that to solve Sudokos with this level of difficulty, it may also be necessary to look at chains with an extremely large number of starting points.

X Chains

To create an X Chain, you search for a value that is only possible twice in a row, column or block. Therefore, you know that the value occurs either in one cell or in the other. To indicate this connection, you connect these two cells with a line.

If several of these pairs are found, a long chain, the so-called X Chain, can be created. Within this chain, the value is then either set or not set alternately.

In this example, an X chain is created for the value 7. The alternating occurrence is represented by SUDOKIS using the colors red and purple.

If you find fields in other places that are in the area of influence of both a red and a violet field, the selected number cannot appear there. This is because the number must either be deleted by the red or by the violet field. You do not know which of the two fields is responsible for the deletion, but you do know that it is one of the two.

SUDOKIS marks these findings with a green border and removes the corresponding value. In this example, the value 7 can be deleted in two fields.

When an X Chain is created, it is also possible to delete values in fields of the X Chain itself. These fields then have a function within the chain and also have values that can be deleted at the end.

SUDOKIS marks these fields with dashed lines in both colors.

Chaining Background

There are many more chaining strategies than the Wings/Chains and X-Chains explained above. Let's take a closer look at the individual chain links to get a better understanding of the variety of possible chains.

A simple element of a chain is a weak link. A weak link itself does not have much significance. It simply says that a value can appear in different places in a house. In this case, the 7 can appear both in column 2 and in column 5 (and in other columns). But in this example we are interested in columns 2 and 5.

The statement that can be derived from this is initially rather weak. If you know that the number 7, for example, is placed in column 2 in the 2nd row, then you know that the number cannot be in column 5. If, on the other hand, it is not in the 2nd row, then no further conclusions can be drawn. The number 7 could then be in a number of other columns.

Weak links initially have no direction. The statements apply alternately from column 2 in the direction of column 5, or from column 5 in the direction of column 2.

A first chaining and thus stronger statements arise when the field on one side of a weak link only has two possibilities.

In this case, the statement applies that if the field in column 2 of row 2 contains a 7, then the field in column 5 of row 2 cannot contain a 7. However, as only two values are possible there, it must contain the number 5, and the statement continues with a new weak link to column 5 of row 3, where there must not be a 5.

If we now look at our chains and wings, they are made up entirely of weak links. We have a starting point with several possibilities, intermediate points that are limited to 2 values so that the chain can continue, and a common end point for which we want to obtain a statement.

There is a simple solution strategy behind this: Just look for fields with only two possibilities and try to connect them with weak links. A chain or wing can then be created by selecting a suitable start and end point.

Strong links make a stronger statement than weak links. If a number can only occur twice in a house, it must occur either in one place or in the other.

In this example, the number 7 in the second row can only occur in columns 2 and 5. This means that if the number 7 occurs in column 2, it cannot occur in column 5 (a statement identical to the weak link). However, if the number does not occur in column 2, then we know that it must occur in column 5 (the weak link does not offer this inverse statement).

Similar to the weak link, a strong link initially has no direction; identical statements apply in both directions.

In combination with a weak link, however, statements can be constructed directly that point in one direction. In this example, if the number 7 is not set in column 2 of row 2, then it is set in column 5 of row 2 and a follow-up weak link ensures that it cannot be set in column 3 of row 3.

Alternating statements result when strong links are connected directly with each other. There are two possibilities here, either the 7 is set in row 2 column 2 and row 3 column 5 and not in row 2 column 5, or it is set in row 3 column 5 and not in row 2 column 2 and row 3 column 5.

If we now look at the X chain above, it is a long alternating chain of strong links and then several weak links at the end.

The solution strategy behind this is also pretty simple, just look for pairs of fields with strong links that you can connect and look for closures through weak links.

Now you can come up with many more possible combinations between strong link and weak link concatenations. These are then often presented as new solution strategies. It becomes even more difficult when there are branches of chains that are merged again. Some of these strategies make sense, others do not.

For further chaining technologies to be included in the simpler Very Difficult category, they must fulfill several criteria. Firstly, it must be significantly easier to find them than the General Chains or Brute Force, and then it must show solutions for a large number of Sudokus that could not be solved using simpler rules. We will make this decision carefully after discussions with Sudoku players and automatic evaluations of Sudokus.

While there are certainly still some rules that should be added to SUDOKIS in the future, this definitely does not apply to all the rules that exist. Logic is not always better than guesswork. Extremely complicated logical relationships are often no problem for computers, but SUDOKIS should always recommend the simplest possible next steps for humans. And these are sometimes guesswork. This example of a logical concatenation of weak links and strong links clearly shows that the limit of manageable complexity has been far exceeded here. It simply makes no sense to present this logically consistent connection as a possible solution step.

In the following, we present an even more general chaining technology that is implemented in SUDOKIS. It is no longer intended for use by humans, but merely serves to catch all possible chain combinations. Since this general chaining technology is not considered to be a strategy that can be meaningfully implemented by humans, it is referred to as Extremely Difficult evaluations. But with general chains, there is still the option of having them carried out by people, unlike the chaotic chaining presented above.

General Chains

General Chains are a generalization of many different chain types, some of which are not defined here. In the General Chains you can also find chains that do not only consist of simple 2-value fields, there are short-term branches that quickly merge again, etc.

With many different chains, it is debatable whether they are still rules or whether they are very close to guesswork. General chains are not a simple solution strategy, they are a form of guessing. In contrast to Brute Force, many guessing branches are avoided here. This means that the strategy can still be used if there is no other option. And sometimes you can find easy-to-find chains hidden in the General Chains.

Naked Single Chains are the simplest form of General Chains. The idea behind it is to guess the value of a field and then continue solving the Sudoku using only the Naked Single Rule or the Direct Rule. The same procedure is repeated with all other possible values of the field. At the end, you see which values you can cross out in each of these solutions.

In this example, we have tried out the two values for the field in row 1 and column 1. In the end, it turns out that all fields marked in green have been deleted in both paths.

It often happens that at the end of a chain it turns out that the Sudoku can no longer be solved. In this case, there are many more options for crossing out values. In particular, a value can also be deleted from the starting point. The corresponding field is therefore shown as a dashed line, with the red color for the start value and the green color for a deleted marker.

The Hidden Single Chain and the General Chain follow exactly the same principle. The only difference here is that the hidden single rule, or all possible rules, are also permitted for the execution of the individual branches. Of course, this increases the effort considerably, but provides even more possible solutions.

We rate all types of General Chains as Extremely Difficult, which means “You shouldn't even try this!”

We rate Naked Single Chains as Extremely Difficult 1 to Extremely Difficult 7, depending on how often this rule has to be applied. We rate Hidden Single Chains with Extremely Difficult 8 to Extremely Difficult 10.

General Chains fall out of our normal rating scheme, an Extremely Difficult 11 is appropriate.

Trial and Error - Brute Force

If no rule helps at all, Brute Force is guaranteed to solve any Sudoku. You simply guess values and run through all the possibilities in the form of a decision tree.

The rating for this category is Extremely Difficult 12. However, the use of General Chains is so powerful that we no longer know of many Sudokus that fall into this category. The famous “Most difficult Sudoku in the world” by Arto Inkala belongs to this top category.

50 levels of difficulty in detail

The following overview shows which solution strategies are required to solve a Sudoku of a certain level of difficulty. Getting started is fairly easy, pure common sense is enough to solve all Sudokus in the Easy and Medium categories. This is the right category for children and for solving Sudoku puzzles during a cozy evening in front of the TV. If you are looking for a real challenge, start with the higher categories. However, the fun stops at the Extremely Difficult category at the latest. Often, nothing is possible here without a lot of trial and error.

The question is whether this rating system will always remain constant, is it a kind of law of nature? We will definitely develop the scoring system further. As can be seen from this table, the system is based on the fact that certain solution strategies are implemented in SUDOKIS. Thease are evaluated for the scoring system. However, the list of existing solution strategies in the world is much longer. we consider many of these strategies to be superfluous, either because they do not produce better results than simpler strategies, or because they are simply a combination of several simple strategies. It is usually better to combine two simple strategies than to use one difficult strategy you won't find.

But wo do have a few candidates in the Chains and Fishes area that definitely have the potential to be included. We will examine these candidates carefully and test their success on hundreds of thousands of Sudokus. It makes no sense to learn difficult solution strategies for which there are hardly any Sudokos in the world. The strategies currently included are definitely all helpful in finding solutions.

If other rating systems are presented on other websites, this does not necessarily mean that they are better or worse, they are simply different. At the end of the day, how a solution strategy is rated is also a question of taste. But there are rating systems that simply count the number of pre-assigned fields. This is definitely complete nonsense. The number of pre-assigned fields has nothing to do with the level of difficulty.