How to solve sudoku secrets lessons. Ways to solve classic Sudoku

When solving Sudoku, be consistent in your reasoning. Periodically check your actions, because if you make a mistake at the beginning of the solution, then it can eventually lead to an incorrect solution to the entire puzzle. It is easier to avoid mistakes at the beginning of a solution than when a contradiction is found in a solved puzzle.

The following ways to solve Sudoku are listed in order of difficulty and frequency of use in practice.

Selection of candidates

With this technique, they begin to solve any Sudoku, regardless of its complexity. In accordance with the proposed task, it is necessary to enter variants of numbers in empty cells, which can be determined by excluding the numbers already present in rows, columns or blocks.

For example, consider cell A2, it is marked in gray. "1" is in the block, "2" is in the row, "3" is in the block and row, "4" is in the row, "5" is in the column, "7" is in the block, "8" is in the row, "9" is in the column. Accordingly, the only option for this cell is the number "6".

But in most cases, for each cell there are several candidates at once. Fill in the grid with all possible candidates for each cell.

As you can see, there are only two cells in which there is only one candidate each - A2 and D9, they are called the only candidates. After finding the only candidates, it is also necessary to cross them out of the candidates for other cells (cells of this column, row, block). So, deleting the number "6" from line 2, column A and block 1, we will also get the only candidate in cell B1 - the number "2". We proceed in the same way.

However, there are also "hidden" single candidates. Let's take cell I7 as an example. This cell is in block 9. In this block, the number 5 can only be in cell I7, since columns G and H already have the number 5, it is also present in row 8. Accordingly, of the three candidates for cell I7, we leave only the number "5".

Exclusion of candidates

The methods described above allow you to unambiguously determine which number to enter in a particular cell, the following will reduce their number, which ultimately will lead to the only candidates.

During the solution process, a situation may arise when a certain number in a block can only be located in one row or column within this block. As a consequence, this number cannot be in other cells of this row or column outside the block.

Consider block 5. In this block, the number "4" can only be in cells D5 and F5, i.e. in line 5. Accordingly, no matter which of these two cells contains the number "4", it can no longer be in line 5 in other blocks, so it can be safely deleted from the candidates of cell G5.

There is also an alternative to the previous method. If a certain number in a row or column can only be located within one block, then the same number cannot be located in other cells of the block in question.

So in line 1, the number "4" can only be in cells D1 and F1, i.e. in block 2. Therefore, no matter which of these two cells contains the number "4", it cannot be in block 2 in other cells, so it can be safely deleted from the candidates of cells D3 and F3.

If two cells in a block, row, or column contain only a pair of identical candidates, then these candidates cannot be in other cells of this block, row, or column.

Cells G9 and H9 contain a pair of candidates "6" and "8". Accordingly, no matter which of these two cells contains the numbers "6" and "8" (if "6" in G9, then "8" in H9, and vice versa), in block 9 in other cells they can no longer be, as well as in line 9. Therefore, they can be safely deleted from the candidate cells H7, G8, B9, C9, F9.

Also, this method can be applied for three and four candidates, only cells in a block, row, column must be taken three and four, respectively.

From the cells highlighted in yellow - B7, E7, H7 and I7 we cross out the candidates contained in the cells highlighted in gray - A7, D7 and F7.

We do the same with fours. From the cells highlighted in yellow - C1 and C6 we cross out the candidates contained in the cells highlighted in gray - C4, C5, C8 and C9.

But there are often "hidden" pairs of candidates. If in two cells in a block, row or column, a pair of candidates occurs among the candidates that does not occur in any other cell of the block, row, or column, then no other cells of the block, row, or column can contain candidates from this pair. Therefore, all other candidates from these two cells can be crossed out.

So, for example, in column G, the pair of numbers "7" and "9" occurs only in cells G1 and G2. Therefore, all other candidates from these cells can be removed.

You can also look for "hidden" triples and fours.

There are more complex methods used in solving Sudoku. They are not so much difficult to understand as when to apply them. So, for example, if in one of the columns a candidate can only be in two cells, and there is a column in which the same candidate can also be in only two cells, and all these four cells form a rectangle, then this candidate can be excluded from other cells of these lines.

By analogy, out of two rows, the excluded candidates would then be in columns.

In column A, the number "2" can only be in two cells A4 and A6, and in column E in E4 and E6. Accordingly, these pairs of cells are in the same rows - 4 and 6, forming a rectangle.

There is a certain dependency:

If the number "2" is in cell A4, then it will also be in cell E6 (it cannot be in cell E4, because the number "2" will already be in line 4, it will not be in cell A6, because j. the number "2" will already be in column A and block 4);

If the number "2" is in cell A6, then it will also be in cell E4 (it cannot be in cell E6, because the number "2" will already be in line 6, it will not be in cell A4, because since the number "2" will already be in column E and block 5).

Therefore, wherever the number "2" is located, in cells A4 and E6 or A6 and E4, from other cells of lines 4 and 6, you can safely cross out the number "2". In addition, this method can be applied to blocks. Since in block 4 the number "2" will necessarily be in cells A4 or A6, it can also be deleted from the candidate cells of block 4.

These are the main ways in which you can solve classic Sudoku. If the Sudoku is not difficult, then it can be solved using the first methods. When solving more complex puzzles, the latter methods are indispensable. But these methods are not stereotyped, in the process of guessing you will develop your own tactics and strategy. The more you solve Sudoku, the better you will get at it. And all the candidates will not need to be written down, and you can easily keep them “in your head”.

An example of a classic Sudoku solution

Now let's try to solve the following Sudoku in its entirety.

To begin with, we will write down all the candidates.

Now let's identify the only candidates (gray cells). And cross them out of the candidates for other cells in blocks, rows, columns (yellow cells).

At the same time, in some cells, we again have the only candidates (for example, in line 1, the number "2" is only in cell B1), we also cross them out of the candidates for other cells of blocks, rows, columns.

Now let's find the "hidden" single candidates (gray cells). And cross them out of the candidates for other cells in blocks, drains, columns (yellow cells).

At the same time, in some cells, we again have “hidden” unique candidates (for example, in line 1, the number “5” is only in cell C1), we also cross them out from candidates for other cells of blocks, rows, columns.

Now we take cell H5. In line 5, the number "2" occurs only in this cell. We continue to solve our Sudoku regarding this cell.

After only the only candidates remain in some cells, we cross them out from other cells of rows, columns and blocks.

As a result, we get the following combination.

Having solved it, we come to the only correct solution:

This is one of the ways to solve this Sudoku. Of course, it was possible to start the solution from other cells and in other ways, but this solution shows that Sudoku has the only correct solution and it can be found in a logical way, and not by enumeration of numbers.

The goal of Sudoku is to arrange all the numbers so that there are no identical numbers in 3x3 squares, rows and columns. Here is an example of a Sudoku already solved:

You can check that there are no repeating numbers in each of the nine squares, as well as in all rows and columns. When solving Sudoku, you need to use this number “uniqueness” rule and, sequentially excluding candidates (small numbers in a cell indicate which numbers, in the player’s opinion, can stand in this cell), find places where only one number can stand.

When we open the Sudoku, we see that each cell contains all the little gray numbers. You can immediately uncheck the already set numbers (marks are removed by right-clicking on a small number):

I'll start with the number that is in this crossword puzzle in one copy - 6, so that it would be more convenient to show the exclusion of candidates.

Numbers are excluded in the square with the number, in the row and column, the candidates to be removed are marked in red - we will right-click on them, noting that there cannot be sixes in these places (otherwise there will be two sixes in the square / column / row, which is against the rules).

Now, if we return to units, then the pattern of exceptions will be as follows:

We remove candidates 1 in each free cell of the square where there is already a 1, in each row where there is a 1 and in each column where there is a 1. In total, for three units there will be 3 squares, 3 columns and 3 rows.

Next, let's go straight to 4, there are more numbers, but the principle is the same. And if you look closely, you can see that in the upper left 3x3 square there is only one free cell (marked in green), where 4 can stand. So, put the number 4 there and erase all the candidates (there can no longer be other numbers). In simple Sudoku, quite a lot of fields can be filled in this way.

After a new number is set, you can double-check the previous ones, because adding a new number narrows the search circle, for example, in this crossword puzzle, thanks to the four set, there is only one cell left in this square (green):

Of the three available cells, only one is not occupied by the unit, and we put the unit there.

Thus, we remove all obvious candidates for all numbers (from 1 to 9) and put down the numbers if possible:

After removing all obviously unsuitable candidates, a cell was obtained where only 1 candidate (green) remained, which means that this number is three, and it is worth it.

The numbers are also put if the candidate is the last in the square, row or column:

These are examples on fives, you can see that there are no fives in the orange cells, and the only candidate in the region remains in the green cells, which means that the fives are there.

These are the most basic ways to put numbers in Sudoku, you can already try them out by solving Sudoku on simple difficulty (one star), for example: Sudoku No. 12433, Sudoku No. 14048, Sudoku No. 526. Sudokus shown are completely solved using the information above. But if you can’t find the next number, you can resort to the selection method - save the Sudoku, and try to put down some number at random, and in case of failure, load the Sudoku.

If you want to learn more complex methods, read on.

Locked Candidates

Locked Candidate in a Square

Consider the following situation:

In the square highlighted in blue, the number 4 candidates (green cells) are located in two cells on the same line. If the number 4 is on this line (orange cells), then there will be nowhere to put 4 in the blue square, which means that we exclude 4 from all orange cells.

A similar example for the number 2:

Locked candidate in a row

This example is similar to the previous one, but here in row (blue) candidates 7 are in the same square. This means that sevens are removed from all the remaining cells of the square (orange).

Locked Candidate in a Column

Similar to the previous example, only in the column 8 candidates are located in the same square. All candidates 8 from other cells of the square are also removed.

Having mastered the locked candidates, you can solve Sudoku of medium difficulty without selection, for example: Sudoku No. 11466, Sudoku No. 13121, Sudoku No. 11528.

Number groups

Groups are harder to see than locked candidates, but they help clear many dead ends in complex crossword puzzles.

naked couples

The simplest subspecies of groups are two identical pairs of numbers in one square, row or column. For example, a bare pair of numbers in a string:

If in any other cell in the orange line there is 7 or 8, then in the green cells there will be 7 and 7, or 8 and 8, but according to the rules it is impossible for the line to have 2 identical numbers, so all 7 and all 8 are removed from the orange cells .

Another example:

A naked couple is in the same column and in the same square at the same time. Extra candidates (red) are removed both from the column and from the square.

An important note - the group must be exactly “naked”, that is, it must not contain other numbers in these cells. That is, and are a naked group, but and are not, since the group is no longer naked, there is an extra number - 6. They are also not a naked group, since the numbers must be the same, but here there are 3 different numbers in the group.

Naked triplets

Naked triples are similar to naked pairs, but they are more difficult to detect - these are 3 naked numbers in three cells.

In the example, the numbers in one line are repeated 3 times. There are only 3 numbers in the group and they are located on 3 cells, which means that the extra numbers 1, 2, 6 from the orange cells are removed.

A naked triple may not contain a number in full, for example, a combination would be suitable:, and - these are all the same 3 types of numbers in three cells, just in an incomplete composition.

Naked Fours

The next extension of bare groups is bare fours.

Numbers , , , form a bare quadruple of four numbers 2, 5, 6 and 7 located in four cells. This four is located in one square, which means that all the numbers 2, 5, 6, 7 from the remaining cells of the square (orange) are removed.

hidden couples

The next variation of groups is hidden groups. Consider an example:

In the topmost line, the numbers 6 and 9 are located only in two cells; there are no such numbers in the other cells of this line. And if you put another number in one of the green cells (for example, 1), then there will be no room left in the line for one of the numbers: 6 or 9, so you need to delete all the numbers in the green cells, except for 6 and 9.

As a result, after removing the excess, only a bare pair of numbers should remain.

Hidden triplets

Similar to hidden pairs - 3 numbers stand in 3 cells of a square, row or column, and only in these three cells. There may be other numbers in the same cells - they are removed

In the example, the numbers 4, 8 and 9 are hidden. There are no these numbers in the other cells of the column, which means we remove unnecessary candidates from the green cells.

hidden fours

Similarly with hidden triples, only 4 numbers in 4 cells.

In the example, four numbers 2, 3, 8, 9 in four cells (green) of one column form a hidden four, since these numbers are not in other cells of the column (orange). Extra candidates from green cells are removed.

This concludes the consideration of groups of numbers. For practice, try to solve the following crossword puzzles (without selection): Sudoku No. 13091, Sudoku No. 10710

X-wing and fish sword

These strange words are the names of two similar ways of eliminating Sudoku candidates.

X-wing

X-wing is considered for candidates of one number, consider 3:

There are only 2 triples in two rows (blue) and these triples lie on only two lines. This combination has only 2 triples solutions, and the other triples in the orange columns contradict this solution (check why), so the red triple candidates should be removed.

Similarly for candidates for 2 and columns.

In fact, the X-wing is quite common, but not so often the encounter with this situation promises the exclusion of extra numbers.

This is an advanced version of X-wing for three rows or columns:

We also consider 1 number, in the example it is 3. 3 columns (blue) contain triplets that belong to the same three rows.

Numbers may not be contained in all cells, but the intersection of three horizontal and three vertical lines is important to us. Either vertically or horizontally, there should be no numbers in all cells except green ones, in the example this is a vertical - columns. Then all the extra numbers in the lines should be removed so that 3 remains only at the intersections of the lines - in green cells.

Additional analytics

The relationship between hidden and naked groups.

And also the answer to the question: why are they not looking for hidden / naked fives, sixes, etc.?

Let's look at the following 2 examples:

This is one Sudoku where one numeric column is considered. 2 numbers 4 (marked in red) are eliminated in 2 different ways - using a hidden pair or using a bare pair.

Next example:

Another Sudoku, where in the same square there is both a bare pair and a hidden three, which remove the same numbers.

If you look at the examples of bare and hidden groups in the previous paragraphs, you will notice that with 4 free cells with a bare group, the remaining 2 cells will necessarily be a bare pair. With 8 free cells and a naked four, the remaining 4 cells will be a hidden four:

If we consider the relationship between bare and hidden groups, then we can find out that if there is a bare group in the remaining cells, there will necessarily be a hidden group and vice versa.

And from this we can conclude that if we have 9 cells free in a row, and among them there is definitely a naked six, then it will be easier to find a hidden triple than to look for a relationship between 6 cells. It is the same with the hidden and naked five - it is easier to find the naked / hidden four, so the fives are not even looked for.

And one more conclusion - it makes sense to look for groups of numbers only if there are at least eight free cells in a square, row or column, with a smaller number of cells, you can limit yourself to hidden and naked triples. And with five free cells or less, you can not look for triples - twos will be enough.

Final word

Here are the most famous methods for solving Sudoku, but when solving complex Sudoku, the use of these methods does not always lead to a complete solution. In any case, the selection method will always come to the rescue - save the Sudoku in a dead end, substitute any available number and try to solve the puzzle. If this substitution leads you to an impossible situation, then you need to boot up and remove the substitution number from the candidates.

Sudoku is a very interesting puzzle game. It is necessary to arrange the numbers from 1 to 9 in the field in such a way that each row, column and block of 3 x 3 cells contains all the numbers, and at the same time they should not be repeated. Consider step-by-step instructions on how to play Sudoku, basic methods and a solution strategy.

Solution algorithm: from simple to complex

The algorithm for solving the Sudoku mind game is quite simple: you need to repeat the following steps until the problem is completely solved. Gradually move from the simplest steps to more complex ones, when the first ones no longer allow you to open a cell or exclude a candidate.

Single Candidates

First of all, for a more visual explanation of how to play Sudoku, let's introduce a numbering system for blocks and cells of the field. Both cells and blocks are numbered from top to bottom and from left to right.

Let's start looking at our field. First you need to find single candidates for a place in the cell. They can be hidden or explicit. Consider possible candidates for the sixth block: we see that only one of the five free cells contains a unique number, therefore, the four can be safely entered in the fourth cell. Considering this block further, we can conclude: the second cell should contain the number 8, since after the exclusion of the four, the eight in the block does not occur anywhere else. With the same justification, we put the number 5.

Carefully review all possible options. Looking at the central cell of the fifth block, we find that there can be no other options besides the number 9 - this is a clear single candidate for this cell. The nine can be crossed out from the rest of the cells of this block, after which the remaining numbers can be easily put down. Using the same method, we pass through the cells of other blocks.

How to discover hidden and explicit "naked couples"

Having entered the necessary numbers in the fourth block, let's return to the empty cells of the sixth block: it is obvious that the number 6 should be in the third cell, and 9 in the ninth.

The concept of "naked pair" is present only in the game of Sudoku. The rules for their detection are as follows: if two cells of the same block, row or column contain an identical pair of candidates (and only this pair!), then the other cells of the group cannot have them. Let's explain this on the example of the eighth block. Putting possible candidates in each cell, we find an obvious "naked pair". The numbers 1 and 3 are present in the second and fifth cells of this block, and there and there there are only 2 candidates each, therefore, they can be safely excluded from the remaining cells.

Completion of the puzzle

If you learned the lesson on how to play Sudoku and followed the above instructions step by step, then you should end up with something like this picture:

Here you can find single candidates: a one in the seventh cell of the ninth block and a two in the fourth cell of the third block. Try to solve the puzzle to the end. Now compare your result with the correct solution.

Happened? Congratulations, this means that you have successfully mastered the lessons on how to play Sudoku and learned how to solve the simplest puzzles. There are many varieties of this game: Sudoku of different sizes, Sudoku with additional areas and additional conditions. The playing field can vary from 4 x 4 to 25 x 25 cells. You may come across a puzzle in which the numbers cannot be repeated in an additional area, for example, diagonally.

Start with simple options and gradually move on to more complex ones, because with training comes experience.

I will not talk about the rules, but immediately move on to the methods.
To solve a puzzle, no matter how complex or simple, cells that are obvious to fill are initially searched for.

1.1 "The Last Hero"

Consider the seventh square. Only four free cells, so something can be quickly filled.
"8 " on the D3 blocks padding H3 and J3; similar " 8 " on the G5 closes G1 and G2
With a clear conscience we put " 8 " on the H1

1.2 "Last Hero" in a row

After viewing the squares for obvious solutions, move on to the columns and rows.
Consider " 4 " on the field. It is clear that it will be somewhere in the line A.
We have " 4 " on the G3 that covers A3, there is " 4 " on the F7, cleaning A7. And one more " 4 " in the second square prohibits its repetition on A4 and A6.
"The Last Hero" for our " 4 " This A2

1.3 "No Choice"

Sometimes there are multiple reasons for a particular location. " 4 " in J8 would be a great example.
Blue the arrows indicate that this is the last possible number squared. Red and blue the arrows give us the last number in the column 8 . Greens the arrows give the last possible number in the line J.
As you can see, we have no choice but to put this " 4 "in place.

1.4 "And who, if not me?"

Filling in numbers is easier to do using the methods described above. However, checking the number as the last possible value also yields results. The method should be used when it seems that all the numbers are there, but something is missing.
"5 " in B1 is set based on the fact that all numbers from " 1 " before " 9 ", Besides " 5 " is in the row, column and square (marked in green).

In jargon it is " naked loner". If you fill in the field with possible values ​​​​(candidates), then in the cell such a number will be the only possible one. Developing this technique, you can search for " hidden loners" - numbers unique for a particular row, column or square.

2. "Naked Mile"

2.1 Naked couples

""Naked" couple" - a set of two candidates located in two cells belonging to one common block: row, column, square.
It is clear that the correct solutions of the puzzle will be only in these cells and only with these values, while all other candidates from the general block can be removed.


In this example, there are several "naked pairs".
red in line BUT cells are highlighted A2 and A3, both containing " 1 " and " 6 ". I don't know exactly how they are located here yet, but I can safely remove all the others " 1 " and " 6 " from string A(marked in yellow). Also A2 and A3 belong to a common square, so we remove " 1 " from C1.

2.2 "Threesome"

"Naked Threes"- a complicated version of "naked couples".
Any group of three cells in one block containing all in all three candidates is "naked trio". When such a group is found, these three candidates can be removed from other cells of the block.

Candidate combinations for "naked trio" may be like this:

// three numbers in three cells.
// any combinations.
// any combinations.

In this example, everything is pretty obvious. In the fifth square of the cell E4, E5, E6 contain [ 5,8,9 ], [5,8 ], [5,9 ] respectively. It turns out that in general these three cells have [ 5,8,9 ], and only these numbers can be there. This allows us to remove them from other block candidates. This trick gives us the solution " 3 " for cell E7.

2.3 "Fab Four"

"Naked Four" a very rare occurrence, especially in its full form, and yet produces results when detected. The solution logic is the same as "naked triplets".

In the above example, in the first square of the cell A1, B1, B2 and C1 generally contain [ 1,5,6,8 ], so these numbers will occupy only those cells and no others. We remove the candidates highlighted in yellow.

3. "Everything hidden becomes clear"

3.1 Hidden pairs

A great way to open the field is to search hidden pairs. This method allows you to remove unnecessary candidates from the cell and give rise to more interesting strategies.

In this puzzle we see that 6 and 7 is in the first and second squares. Besides 6 and 7 is in the column 7 . Combining these conditions, we can assert that in the cells A8 and A9 there will be only these values ​​and we remove all other candidates.

More interesting and complex example hidden pairs. The pair [ 2,4 ] in D3 and E3, cleaning 3 , 5 , 6 , 7 from these cells. Highlighted in red are two hidden pairs consisting of [ 3,7 ]. On the one hand, they are unique for two cells in 7 column, on the other hand - for a row E. Candidates highlighted in yellow are removed.

3.1 Hidden triplets

We can develop hidden couples before hidden triplets or even hidden fours. The Hidden Three consists of three pairs of numbers located in one block. Such as, and. However, as in the case with "naked triplets", each of the three cells does not have to contain three numbers. will work Total three numbers in three cells. For example , , . Hidden triplets will be masked by other candidates in the cells, so first you need to make sure that troika applicable to a specific block.

In this complex example, there are two hidden triplets. The first, marked in red, in the column BUT. Cell A4 contains [ 2,5,6 ], A7 - [2,6 ] and cell A9 -[2,5 ]. These three cells are the only ones where there can be 2 , 5 or 6, so they will be the only ones there. Therefore, we remove unnecessary candidates.

Second, in a column 9 . [4,7,8 ] are unique to cells B9, C9 and F9. Using the same logic, we remove candidates.

3.1 Hidden fours

Perfect example hidden fours. [1,4,6,9 ] in the fifth square can only be in four cells D4, D6, F4, F6. Following our logic, we remove all other candidates (marked in yellow).

4. "Non-rubber"

If any of the numbers appear twice or thrice in the same block (row, column, square), then we can remove that number from the conjugate block. There are four types of pairing:

1. Pair or Three in a square - if they are located in one line, then you can remove all other similar values ​​​​from the corresponding line.
2. Pair or Three in a square - if they are located in one column, then you can remove all other similar values ​​​​from the corresponding column.
3. Pair or Three in a row - if they are located in the same square, then you can remove all other similar values ​​​​from the corresponding square.
4. Pair or Three in a column - if they are located in the same square, then you can remove all other similar values ​​\u200b\u200bfrom the corresponding square.

4.1 Pointing pairs, triplets

Let me show you this puzzle as an example. In the third square 3 "is only in B7 and B9. Following the statement №1 , we remove candidates from B1, B2, B3. Likewise, " 2 " from the eighth square removes a possible value from G2.

Special puzzle. Very difficult to solve, but if you look closely, you can see a few pointing pairs. It is clear that it is not always necessary to find them all in order to advance in the solution, but each such find makes our task easier.

4.2 Reducing the irreducible

This strategy involves carefully parsing and comparing rows and columns with the contents of the squares (rules №3 , №4 ).
Consider the line BUT. "2 "are possible only in A4 and A5. following the rule №3 , remove " 2 " them B5, C4, C5.

Let's continue to solve the puzzle. We have a single location 4 "within one square in 8 column. According to the rule №4 , we remove unnecessary candidates and, in addition, we obtain the solution " 2 " for C7.

Many people like to force themselves to think: for someone - for the development of intelligence, for someone - to keep their brains in good shape (yes, not only the body needs exercise), and various games for logic and puzzles are the best simulator for the mind. One of the options for such educational entertainment can be called Sudoku. However, some have not heard about such a game, let alone knowledge of the rules or other interesting points. Thanks to the article, you will learn all the necessary information, for example, how to solve Sudoku, as well as their rules and types.

General

Sudoku is a puzzle. Sometimes complex, difficult to reveal, but always interesting and addictive for any person who decides to play this game. The name comes from Japanese: "su" means "number", and "doku" is "standing apart".

Not everyone knows how to solve Sudoku. Complex puzzles, for example, are within the power of either smart, well-thinking beginners, or professionals in their field who have been practicing the game for more than one day. Just take it and solve the task in five minutes will not be possible for everyone.

rules

So, how to solve Sudoku. The rules are very simple and clear, easy to remember. However, do not think that simple rules promise a "painless" solution; you will have to think a lot, apply logical and strategic thinking, strive to recreate the picture. You probably need to love numbers to solve Sudoku.

First, a 9 x 9 square is drawn. Then, with thicker lines, it is divided into so-called "regions" of three squares each. The result is 81 cells, which should eventually be completely filled with numbers. This is where the difficulty lies: the numbers from 1 to 9 placed around the entire perimeter should not be repeated either in the “regions” (squares 3 x 3), or in the lines vertically and / or horizontally. In any Sudoku, there are initially some filled cells. Without this, the game is simply impossible, because otherwise it will turn out not to solve, but to invent. The difficulty of the puzzle depends on the number of digits. Complex Sudokus contain few numbers, often arranged in such a way that you have to rack your brains before solving them. In the lungs - about half of the numbers are already in place, making it much easier to unravel.

Completely disassembled example

It is difficult to understand how to solve Sudoku if there is no specific sample showing step by step how, where and what to insert. The provided picture is considered to be uncomplicated, since many of the mini-squares are already filled with the necessary numbers. By the way, it is on them that we will rely for a solution.

For starters, you can look at lines or squares, where there are especially many numbers. For example, the second column from the left fits perfectly, there are only two numbers missing. If you look at those that are already there, it becomes obvious that there are not enough 5 and 9 in the empty cells on the second and eighth lines. With the five, not everything is clear yet, it can be both there and there, but if you look at the nine, everything becomes clear. Since the second line already has the number 9 (in the seventh column), it means that in order to avoid repetitions, the nine must be put down to the 8th line. Using the elimination method, we add 5 to the 2nd row - and now we already have one filled column.

In a similar way, you can solve the entire Sudoku puzzle, however, in more complex cases, when one column, row or square lacks not a couple of numbers, but much more, you will have to use a slightly different method. We will also analyze it now.

This time we will take as a basis the average “region”, which lacks five digits: 3, 5, 6, 7, 8. We fill each cell not with large effective numbers, but with small, “rough” ones. We just write in each box those numbers that are missing and that may be there due to their lack. In the upper cell, these are 5, 6, 7 (3 on this line is already in the “region” on the right, and 8 on the left); in the cell on the left there can be 5, 6, 7; in the very middle - 5, 6, 7; right - 5, 7, 8; bottom - 3, 5, 6.

So, now we look at which mini-digits contain numbers different from others. 3: there is only in one place, in the rest it is not. So, it can be corrected for a large one. 5, 6 and 7 are in at least two cells, so we leave them alone. 8 is only in one, which means that the remaining numbers disappear and you can leave the eight.

Alternating these two ways, we continue to solve Sudoku. In our example, we will use the first method, but it should be recalled that in complex variations the second is necessary. Without it, it will be extremely difficult.

By the way, when the middle seven is found in the upper “region”, it can be removed from the mini-numbers of the middle square. If you do this, you will notice that there is only one 7 left in that region, so you can only leave it.

That's all; finished result:

Kinds

Sudoku puzzles are different. In some, a prerequisite is the absence of identical numbers not only in rows, columns and mini-squares, but also diagonally. Some instead of the usual "regions" contain other figures, which makes it much more difficult to solve the problem. One way or another, how to solve Sudoku is at least the basic rule that applies to any kind, you know. This will always help to cope with a puzzle of any complexity, the main thing is to try your best to achieve your goal.

Conclusion

Now you know how to solve Sudoku, and therefore you can download similar puzzles from various sites, solve them online or buy paper versions at newsstands. In any case, now you will have an occupation for long hours, or even days, because it is unrealistic to drag out Sudoku, especially when you have to actually figure out the principle of their solution. Practice, practice and more practice - and then you will click this puzzle like nuts.