How to solve a difficult sudoku. Logic puzzles

How to play Sudoku?

Sudoku is a very popular number puzzle. Once you understand how to play Sudoku, you won’t be able to put it down!

The essence of the game:

The cells of the playing field must be filled with numbers from 1 to 9. There should not be repeated numbers in each vertical and horizontal line. Also, they cannot be repeated in small squares (3x3 cells). At the very beginning of the game there are already numbers (depending on the difficulty of the level, the number of initially given numbers may differ).

Rules for playing Sudoku:

• Select a row, column or square with the maximum number of given numbers. Fill in what is missing (it is better to use a pencil). In almost all cases, there is a place where only 1 number fits.
• Next, look through each column in turn, compare which numbers can fit into each cell. You can write down options on a separate piece of paper.
• When also looking at lines and squares, eliminate numbers that are repeated.
• As you fill the puzzle with numbers, it will become easier to solve.

Start playing Sudoku with easy tasks, because the ability to solve the puzzle comes with experience. Or play Sudoku online - incorrect numbers will be highlighted in a different color. This will help you get used to the game. During this lesson, logic develops, so you can gradually complicate the level. Also watch the video attached to the article.

Sudoku is an interesting puzzle for training logic, unlike scanword puzzles, which require erudition and memory. Sudoku has many countries of origin, one way or another, it was played in Ancient China, in Japan, North America... In order for you and me to learn the game, we have made a selection How to solve Sudoku from easy to difficult.

To begin with, let's tell you that Sudoku is a square measuring 9x9, which in turn consists of 9 squares measuring 3x3. Each square must be filled with numbers from one to nine so that each number is used only once along a vertical and horizontal line, and only in a 3x3 square.

When you fill in all the cells, you should have all the numbers from 1 to 9 in each of the 9 squares. So, along the horizontal line all the numbers are from 1 to 9. And along the vertical line the same thing, see the picture:

It would seem that, simple rules, but in order to answer the question of how to solve Sudoku, and even more so, if you want to know how to solve complex Sudoku (especially for those who are just starting their journey), you need to solve at least a couple of easy problems. Then it will be clear what we are talking about. Below are the games. Try printing them out and filling them out so that everything fits together:

How to solve difficult Sudoku

I hope you have read the text above and solved the task that you need in order to understand what will be discussed next. If yes, then let's continue.

This part of the article will answer the questions:

How to solve difficult Sudoku?

How to solve Sudoku: methods?

How to solve Sudoku: methods and methods of cells and fields?

So, you were given two games, by solving which you acquired skills and got a general idea. In order to save your time, I will tell you a couple of life hacks for quickly solving Sudoku.

1. Always start with number 1 and go first along the lines and then along the squares. This way you will definitely not get confused and will prevent yourself from making many mistakes.

2. Always check which number is missing where there are fewer empty cells left. This will save time. And be sure to pay attention to how many and what numbers are missing in the 3 by 3 square (both horizontal and vertical lines).

3. If there are a lot of empty cells in a square and you reach a dead end, try dividing the square along lines in your mind. Think about what numbers might be there, and from this you can understand what numbers will be on the same lines in other squares (and perhaps even understand what numbers will be in other squares on another line).

4. Don’t be afraid of anything, it’s better to make a mistake and understand why than to do nothing!

5. More practice and you will become a master.

And if people who solve Sudoku also have abstract intelligence, which gives powerful potential to its owner, then one can move far forward. Read more about such people.

Below you will find a selection of “How to solve difficult Sudoku”, after which you will be able to do a lot!

Feb 27, 2015 —

Sudoku is a number puzzle. Today it is so popular that most people are familiar with it or have simply seen it in print. In our article we will tell you where this game came from, as well as who invented Sudoku.

Despite the Japanese name, the history of Sudoku does not begin in Japan. The prototype of the puzzle is considered to be the Latin squares of Leonhard Euler, a famous mathematician who lived in the 18th century. However, in the form in which it is known today, it was invented by Howard Garnes. Being an architect by training, Garnes simultaneously invented puzzles for magazines and newspapers. In 1979, an American publication called "Dell Pencil" Puzzles and Word Games" published Sudoku on its pages for the first time. However, then the puzzle did not arouse interest among readers.

It was the Japanese who were the first to appreciate the rebus. In 1984, a Japanese publication published the puzzle for the first time. It immediately became widespread. It was then that the puzzle got its name - Sudoku. In Japanese, “su” means “number” and “doku” means “standing alone.” Some time later, this rebus appeared in many printed publications in Japan. In addition, separate collections of Sudoku were published. In 2004, the puzzle began to be published in UK newspapers, which marked the beginning of the game's spread outside Japan.

The puzzle is a square field with a side of 9 cells, divided in turn into squares measuring 3 by 3. Thus, a large square is divided into 9 small ones, total There are 81 cells. Some cells initially contain clue numbers. The essence of the rebus is to fill empty cells with numbers so that they are not repeated in rows, columns, or squares. Sudoku only uses numbers from 1 to 9. The difficulty of the puzzle depends on the location of the clue numbers. The most difficult, of course, is the one that has only one solution.

The history of Sudoku continues in our time, and successfully. The game is becoming an increasingly common puzzle game, largely due to the fact that it can now be found not only on the pages of the newspaper, but also on your phone or computer. In addition, various variations of this rebus have appeared - letters are used instead of numbers, the number of cells and the shape change.

Select the topic you are interested in:

Sumdoku

Sumdoku is also known as killer sudoku or killer sudoku. In this type of puzzle, numbers are arranged in the same way as in classic Sudoku. But the field additionally contains colored blocks, for each of which the sum of numbers is indicated. Please note that sometimes numbers may be repeated in these blocks!

How to solve sumdoku?

Consider sumdoku (in the picture on the right). To solve it, remember that the sum of the numbers in any row, any column and any small rectangle is the same. For our case, this is 1+2+3+…+9+10 = 55. For sumdoku 9x9 it would be 45.

Let's pay attention to the highlighted gray blocks. They almost completely (except for one number) cover the two lower rectangles. Let's calculate the sum of the numbers in all marked blocks: 13 + 8 + 13 + 15 + 13 + 7 + 14 + 12 + 5 = (13+13+14) + (13+7) + (12+8) + (15+5 ) = 40 + 20 + 20 + 20 = 100. So, the sum of the numbers in the marked blocks is 100. But if we take the two lower rectangles completely, then the sum of the numbers in them should be 55 + 55 = 110. This means that in the only unmarked cell the number is 10.

As you can see, by constantly solving sumdoku, you will become a master of arithmetic. You can, of course, use a calculator, but this dark and slippery path is not for real samurai

Let us now consider the blocks highlighted in the figure on the right. They cover one penultimate horizontal line of the Sudoku and two “extra” cells. Let's calculate the sum of numbers in blocks: 13 + 8 + 15 + 13 + 10 + 14 = (13+13+14) + (10+15) + 8 = 40 + 25 + 8 = 73. But we know that the sum of numbers in horizontal line is 55, which means you can find out the sum of the numbers in two “extra” cells: 73 - 55 = 18.

Let's write down all possible combinations of numbers in these “extra” cells: 10+8, 9+9, 8+10.

History of Sudoku

9+9 - eliminated, since the cells are located on the same horizontal line, leaving 10+8 and 8+10. But if you put 8 in the first “extra” cell, then in the penultimate horizontal line you will get two fives, and the numbers in the horizontal lines should not be repeated. Thus, we find that the first “extra” cell can only contain 10. We immediately arrange the remaining obvious numbers.

06/15/2013 How to solve Sudoku, rules with example.

I would like to say that Sudoku is a really interesting and exciting task, a riddle, a puzzle, a puzzle, a digital crossword, you can call it whatever you like. The solution of which will not only bring real pleasure to thinking people, but will also allow in the process exciting game develop and train logical thinking, memory, perseverance.

For those who are already familiar with the game in any of its manifestations, the rules are known and understandable. And for those who are just thinking about starting, our information may be useful.

The rules for playing Sudoku are not complicated; they are found on the pages of newspapers or can be found quite easily on the Internet.

The main points fit into two lines: the main task The player fills all the cells with numbers from 1 to 9. This must be done in such a way that in the row, column and mini-square 3x3, none of the numbers are repeated twice.

Today we offer you several versions of the Sudoku-4tune electronic game, including more than a million built-in puzzle options in each game player.

For clarity and a better understanding of the process of solving the riddle, consider one of simple options, first difficulty level Sudoku-4tune, 6** series.

And so, a playing field is given, consisting of 81 cells, which in turn make up: 9 rows, 9 columns and 9 mini-squares measuring 3x3 cells. (Fig.1.)

Do not be confused by the further mention of an electronic game. You can find the game on the pages of newspapers or magazines, the basic principle remains the same.

The electronic version of the game provides great opportunities to choose the difficulty level of the puzzle, options for the puzzle itself and their number, at the request of the player, depending on his preparation.

When you turn on the electronic toy, key numbers will be given in the cells of the playing field. Which cannot be transferred or changed. You can choose the option that is more suitable for the solution, in your opinion. Reasoning logically, starting from the given numbers, it is necessary to gradually fill the entire playing field with numbers from 1 to 9.

An example of the initial arrangement of numbers is shown in Fig. 2. Key figures are usually in electronic version games are marked with an underscore or a dot in the cell. In order not to confuse them in the future with the numbers that will be set by you.

Looking at the playing field. It is necessary to decide where to start the solution. Typically, you need to determine the row, column, or mini square that has the minimum number of empty cells. In the version we have presented, we can immediately select two lines, top and bottom. These lines are missing just one digit. Thus, a simple decision is made, having determined the missing numbers -7 for the first line and 4 for the last, we enter them into the free cells of Fig. 3.

The resulting result: two completed lines with numbers from 1 to 9 without repetitions.

Next move. Column number 5 (from left to right) has only two free cells. After some thought, we determine the missing numbers - 5 and 8.

To achieve a successful result in the game, you need to understand that you need to navigate in three main directions: column, row and mini-square.

In this example, it is difficult to navigate only by rows or columns, but if you pay attention to the mini-squares, it becomes clear. It is impossible to enter the number 8 in the second (from the top) cell of the column in question, otherwise there will be two eights in the second mine-square. Likewise with the number 5 for the second cell (bottom) and the second lower mini-square in Fig. 4 (wrong location).

Although the solution seems correct for a column, nine digits, in a column, without repetition, it contradicts the basic rules. In mini-squares, numbers should also not be repeated.

Accordingly, for the correct solution, you need to enter 5 in the second (top) cell, and 8 in the second (bottom) cell. This decision fully complies with the rules.

For the correct option, see Figure 5.

Further solution to a seemingly simple task requires careful consideration of the playing field and the use of logical thinking.

How to solve Sudoku - ways, methods and strategy

You can again use the principle of the minimum number of free cells and pay attention to the third and seventh columns (from left to right). There were three cells left unfilled. Having counted the missing numbers, we determine their values ​​- these are 2,3 and 9 for the third column and 1,3 and 6 for the seventh. Let's leave filling out the third column for now, since there is no certain clarity with it, unlike the seventh. In the seventh column you can immediately determine the location of the number 6 - this is the second free cell from the bottom. What is this conclusion based on?

When examining the mini-square, which includes the second cell, it becomes clear that it already contains the numbers 1 and 3. Of the digital combinations 1,3 and 6 we need, there is no other alternative. Filling the remaining two free cells of the seventh column is also not difficult. Since the third row already contains a filled 1, 3 is entered into the third cell from the top of the seventh column, and 1 is entered into the only remaining free second cell. For an example, see Figure 6.

Let's leave the third column for now for a clearer understanding of the moment. Although, if you wish, you can make a note for yourself and enter the expected version of the numbers required for installation in these cells, which can be corrected if the situation becomes clearer. Electronic games Sudoku-4tune, 6** series allow you to enter more than one number in the cells for a reminder.

Having analyzed the situation, we turn to the ninth (lower right) mini-square, in which, after our decision, there were three free cells left.

Having analyzed the situation, you can notice (an example of filling a mini-square) that the following numbers 2.5 and 8 are missing to completely fill it. Having examined the middle, free cell, you can see that of the necessary numbers only 5 fits here. Since 2 is present in the top cell column, and 8 in a row, which, in addition to the mini-square, includes this cell. Accordingly, in the middle cell of the last mini-square we enter the number 2 (it is not included in either the row or the column), and in the top cell of this square we enter 8. Thus, we have the lower right (9th) mini-square completely filled. a square with numbers from 1 to 9, while the numbers are not repeated in columns or rows, Fig. 7.

As free cells are filled, their number decreases, and we are gradually getting closer to solving our puzzle. But at the same time, solving a problem can be both simplified and complicated. And the first method of filling the minimum number of cells in rows, columns or mini-squares ceases to be effective. Because the number of explicitly defined digits in a particular row, column, or mini-square decreases. (Example: the third column we left). In this case, you need to use the method of searching for individual cells, setting numbers that do not raise any doubts.

IN electronic games Sudoku-4tune, 6** series provides the ability to use hints. Four times per game you can use this function and the computer itself will set the correct number in the cell you have chosen. In the 8** series models there is no such function, and the use of the second method becomes the most relevant.

Let's look at the second method in the example we're using.

For clarity, let's take the fourth column. The empty number of cells in it is quite large, six. Having calculated the missing numbers, we determine them - these are 1,4,6,7,8 and 9. To reduce the number of options, you can take as a basis the average mini-square, which has enough a large number of certain numbers and only two free cells in this column. Comparing them with the numbers we need, we can see that 1,6, and 4 can be excluded. They should not be in this mini-square to avoid repetition. That leaves 7,8 and 9. Please note that in the row (fourth from the top), which includes the cell we need, there are already numbers 7 and 8 from the three remaining ones that we need. Thus, the only option left for this cell is number 9, Fig. 8 Doubts about the correctness this option The fact that all the figures we considered and excluded were originally given in the assignment does not cause a decision. That is, they are not subject to any change or transfer, confirming the uniqueness of the number we have chosen for installation in this particular cell.

Using two methods simultaneously depending on the situation, analyzing and thinking logically, you will fill in all the empty cells and come to the correct solution to any Sudoku puzzle, and this riddle in particular. Try to complete the solution to our example in Fig. 9 yourself and compare it with the final answer shown in Fig. 10.

Perhaps you will determine for yourself any additional key points in solving puzzles, and develop your own system. Or take our advice, and they will be useful for you, and will allow you to join a large number of lovers and fans of this game. Good luck.

Sudoku ("Sudoku") is a number puzzle. Translated from Japanese, “su” means “digit”, and “doku” means “standing alone”. In the traditional Sudoku puzzle, the grid is a square of size 9 x 9, divided into smaller squares with a side of 3 cells ("regions"). Thus, the entire field has 81 cells. Some of them already contain numbers (from 1 to 9). Depending on how many cells have already been filled, the puzzle can be classified as easy or difficult.

The Sudoku puzzle has only one rule. It is necessary to fill in the empty cells so that in each row, in each column and in each small square 3 x 3 each digit from 1 to 9 would appear only once.

Program Cross+A knows how to solve a large number of varieties of Sudoku.

The task can be complicated: the main diagonals of the square must also contain numbers from 1 to 9. This puzzle is called sudoku diagonals ("Sudoku X"). To solve these tasks you need to check the box Diagonals.

Sudoku-argyle (Argyle Sudoku) contains a pattern of lines arranged diagonally.

Sudoku rules

The argyle pattern, consisting of multi-colored diamonds of the same size, was present on the kilts of one of the Scottish clans. Each of the marked diagonals must contain non-repeating numbers.

The puzzle may contain free-form regions; these are called sudoku geometric or curly ("Jigsaw Sudoku", "Geometry Sudoku", "Irregular Sudoku", "Kikagaku Nanpure").

Letters can be used instead of numbers in Sudoku; these types of puzzles are called Godoku ("Wordoku", "Alphabet Sudoku"). After the solution, you can read the keyword in any row or column.

Sudoku-asterisk ("Asterisk") is a variation of Sudoku that contains an additional area of ​​9 squares. These cells must also contain numbers from 1 to 9.

Sudoku girandole ("Girandola") also contains an additional area of ​​9 cells, with numbers from 1 to 9 (a girandole is a fountain of several jets in the form of fireworks, a “fire wheel”).

Sudoku with center points ("Center Dot") is a variant of Sudoku, where the central cells of each region 3 x 3 form an additional area.

The cells in this additional area must contain numbers from 1 to 9.

Sudoku can contain four additional regions 3 x 3. This type of puzzle is called sudoku window ("Windoku", "Four-Box Sudoku", "Hyper Sudoku").

Sudoku puzzle ("Offset Sudoku", "Sudoku-DG") contains additional 9 groups of 9 cells. Cells within a group do not touch each other and are highlighted in the same color. In each group, each number from 1 to 9 should appear only once.

Not a horse's step ("Anti-Knight Sudoku") has an additional condition: identical numbers cannot “beat” each other with a knight’s move.

IN sudoku hermits ("Anti-King Sudoku", "Touchless Sudoku", "Sudoku without touching") identical numbers cannot be in adjacent cells (both diagonally, horizontally and vertically).

IN sudoku-antidiagonal ("Anti Diagonal Sudoku") each diagonal of the square contains no more than three different digits.

Killer Sudoku ("Killer Sudoku", "Sums Sudoku", "Sums Number Place", "Samunamupure", "Kikagaku Nampure"; another name - Sum-do-ku) is a variation of regular Sudoku. The only difference: additional numbers are specified - the sums of values ​​in groups of cells. Numbers contained in a group cannot be repeated.

Sudoku more less ("Greater Than Sudoku") contains comparison signs (">" and "<«), которые показывают, как соотносятся между собой числа в соседних ячейках. Еще одно название — Compdoku.

Sudoku even-odd ("Even-Odd Sudoku") contains information about whether the numbers in the cells are even or odd. Cells containing even numbers are marked in gray, cells containing odd numbers are marked in white.

Sudoku neighbors ("Consecutive Sudoku", "Sudoku with partitions") is a variation of regular Sudoku. It marks the boundaries between adjacent cells that contain consecutive numbers (that is, numbers that differ from each other by one).

IN Non-Consecutive Sudoku numbers in adjacent cells (horizontally and vertically) must differ by more than one. For example, if a cell contains the number 3, adjacent cells should not contain the numbers 2 or 4.

Sudoku points ("Kropki Sudoku", Dots Sudoku, "Sudoku with dots") contains white and black dots at the boundaries between cells. If the numbers in neighboring cells differ by one, then there is a white dot between them. If in neighboring cells one number is twice as large as the other, then the cells are separated by a black dot. Between 1 and 2 there can be a dot of any of these colors.

Sukaku ("Sukaku", "Suuji Kakure", "Pencilmark Sudoku") is a square of size 9 x 9, containing 81 groups of numbers. It is necessary to leave only one number in each cell so that in each row, in each column and in each small square 3 x 3 each number from 1 to 9 would appear only once.

Sudoku chains ("Chain Sudoku", "Strimko", "Sudoku-convolutions") is a square consisting of circles.

It is necessary to arrange the numbers in the circles so that in each horizontal and each vertical all the numbers are different. In the links of one chain, all numbers must also be different.

The program can solve and create puzzles ranging in size from 4 x 4 before 9 x 9.

Sudoku-rama ("Frame Sudoku", "Outside Sum Sudoku", "Sudoku - sums on the side", "Sudoku with sums") is an empty square of size. The numbers outside the playing field indicate the sum of the nearest three digits in a row or column.

Skyscraper Sudoku ("Skyscraper Sudoku") contains key numbers along the sides of the grid. It is necessary to arrange the numbers in a grid; each number indicates the number of floors in the skyscraper. Key numbers outside the grid indicate exactly how many houses are visible in the corresponding row or column when viewed from that number.

Sudoku tripod (Tripod Sudoku) is a type of Sudoku in which the boundaries between regions are not indicated; instead, points are specified at the intersections of the lines. The dots indicate where regional boundaries intersect. Only three lines can extend from each point. It is necessary to restore the boundaries of the regions and fill the grid with numbers so that they are not repeated in each row, each column and each region.

Sudoku mines ("Sudoku Mine") combines the features of Sudoku and “minesweeper” puzzles.

The task is a square in size, divided into smaller squares with a side of 3 cells. You need to place the mines in the grid so that there are three mines in each row, each column and each small square. The numbers show how many mines are in neighboring cells.

Sudoku-half ("Sujiken") was invented by the American George Heineman. The puzzle is a triangular grid containing 45 cells. Some cells contain numbers. It is necessary to fill in all the cells of the grid with numbers from 1 to 9 so that the numbers are not repeated in each row, in each column and on each diagonal. Also, the same number cannot appear twice in each of the regions separated by thick lines.

Sudoku XV ("Sudoku XV") is a variation of regular Sudoku. If the border between adjacent cells is marked with a Roman numeral "X", the sum of the values ​​in these two cells is 10, if the Roman numeral "V" is the sum is 5. If the border between two cells is not marked, the sum of the values ​​in these cells cannot be equal to 5 or 10.

Sudoku Edge ("Outside Sudoku") is a variation of the regular Sudoku puzzle. Outside the grid are numbers that must be present in the first three cells of the corresponding row or column.);

• 16 x 16(size of regions 4 x 4).

Cross+A can solve and create variations of Sudoku consisting of several squares 9 x 9.

Such puzzles are called "Gattai"(translated from Japanese: "connected", "connected"). Depending on the number of squares, the puzzles are designated "Gattai-3", "Gattai-4", "Gattai-5" and so on.

Samurai Sudoku ("Samurai Sudoku", "Gattai-5") is a type of Sudoku puzzle. The playing field consists of five squares of size 9 x 9. The numbers 1 to 9 must be placed correctly in all five squares.

Sudoku flower ("Flower Sudoku", Musketry Sudoku) is similar to Samurai Sudoku. The playing field consists of five squares of size 9 x 9; the central square is entirely covered by four others. The numbers 1 to 9 must be placed correctly in all five squares.

Sudoku-sohei ("Sohei Sudoku") named after warrior monks in medieval Japan. The playing field contains four squares of size 9 x 9

Sudoku mill ("Kazaguruma", "Windmill Sudoku") consists of five squares of size 9 x 9: one in the center, the other four squares almost completely cover the central square. The numbers 1 to 9 must be placed correctly in all five squares.

Butterfly Sudoku ("Butterfly Sudoku") contains four intersecting squares of size 9 x 9, which form a single square of size 12 x 12. The numbers 1 to 9 must be placed correctly in all four squares.

Sudoku cross ("Cross Sudoku") consists of five squares. The numbers 1 to 9 must be placed correctly in all five squares.

Sudoku three ("Gattai-3") consists of three squares of size 9 x 9.

Double Sudoku ("Twodoku", "Sensei Sudoku", "DoubleDoku") consist of two squares of size 9 x 9. The numbers 1 to 9 must be placed correctly in both squares.

The program can solve double sudokus in which the regions have arbitrary shapes:

Triple Sudoku ("Triple Doku") are a puzzle of three squares of size 9 x 9. The numbers 1 to 9 must be placed correctly in all squares.

Sudoku twins ("Twin Corresponding Sudoku") is a pair of regular Sudoku puzzles, each of which contains several starting numbers. Both puzzles must be solved; in this case, each type of numbers in the first grid corresponds to the same type of numbers in the second grid. For example, if the number 9 is in the upper left corner of the first Sudoku puzzle, and the number 4 is in the upper left corner of the second puzzle, then in all cells where there is a 9 in the first grid, there is a 4 in the second grid.

Hoshi ("Hoshi") consists of six large triangles; The numbers 1 to 9 must be placed in the triangular cells of each large triangle. Each line (of any length, even dashed) contains non-repeating numbers.

Unlike Hoshi, in sudoku star ("Star Sudoku") a row on the outer edge of the grid includes a cell located at the nearest sharp end of the figure.

Tridoku ("Tridoku") was invented by Japheth Light from the USA. The puzzle consists of nine large triangles; each one contains nine small triangles. The numbers from 1 to 9 must be placed in the cells of each large triangle. The field contains additional lines, the cells of which must also contain non-repeating numbers. Two touching triangular cells must not contain the same numbers (even if the cells touch each other by only one point).

Online assistant for solving Sudoku.

If you can't solve a difficult Sudoku, try this with a helper. It will highlight possible options for you.

The Sudoku field is a table of 9x9 cells. A number from 1 to 9 is entered in each cell. The goal of the game is to arrange the numbers in such a way that there are no repetitions in each row, in each column and in each 3x3 block. In other words, every column, row, and block must contain all the numbers 1 through 9.

To solve the problem, you can write candidates in the empty cells. For example, consider the cell of the 2nd column of the 4th row: the column in which it is located already has the numbers 7 and 8, the row has the numbers 1, 6, 9 and 4, the block has 1, 2, 8 and 9 Therefore, from the candidates in this cell we cross out 1, 2, 4, 6, 7, 8, 9, and we are left with only two possible candidates - 3 and 5.

Similarly, we consider possible candidates for other cells and get the following table:

It is more interesting to decide with candidates and you can use various logical methods. Next we will look at some of them.

Singles

The method is to find singletons in the table, i.e. cells in which only one digit is possible and no other. We write this number in this cell and exclude it from other cells in this row, column and block. For example: in this table there are three “singles” (they are highlighted in yellow).

Hidden singles

If there are several candidates in a cell, but one of them does not appear in any other cell in a given row (column or block), then such a candidate is called a “hidden singleton”. In the following example, candidate "4" in the green block is found only in the center cell. This means that there will definitely be a “4” in this cell. We enter “4” in this cell and cross it out from other cells of the 2nd column and 5th row. Similarly, in the yellow column, candidate “2” occurs once, therefore, we enter “2” in this cell and exclude “2” from the cells of the 7th row and the corresponding block.

The previous two methods are the only methods that uniquely determine the contents of a cell. The following methods only allow you to reduce the number of candidates in cells, which will sooner or later lead to singletons or hidden singletons.

Locked Candidate

There are times when a candidate within a block is only in one row (or one column). Due to the fact that one of these cells will necessarily contain this candidate, this candidate can be excluded from all other cells in a given row (column).

In the example below, the center block contains candidate "2" only in the center column (yellow cells). This means that one of these two cells must definitely be "2", and no other cells in that row outside of this block can be "2". Therefore, "2" can be excluded as a candidate from other cells in this column (cells in green).

Open pairs

If two cells in a group (row, column, block) contain an identical candidate pair and nothing else, then no other cells in that group can have the value of that pair. These 2 candidates may be excluded from other cells in the group. In the example below, candidates "1" and "5" in columns eight and nine form an Open Pair within the block (yellow cells). Therefore, since one of these cells must be "1" and the other must be "5", candidates "1" and "5" are excluded from all other cells in this block (green cells).

The same can be formulated for 3 and 4 candidates, only 3 and 4 cells are already participating, respectively. Open triples: from green cells we exclude the values ​​of yellow cells.

Open fours: from green cells we exclude the values ​​of yellow cells.

Hidden couples

If two cells in a group (row, column, block) contain candidates that include an identical pair that is not found in any other cell in that block, then no other cells in that group can have the value of that pair. Therefore, all other candidates of these two cells can be eliminated. In the example below, candidates “7” and “5” in the central column are only in the yellow cells, which means that all other candidates from these cells can be excluded.

Similarly, you can look for hidden threes and fours.

x-wing

If a value has only two possible locations in some row (column), then it must be assigned to one of those cells. If there is another row (column) where the same candidate can also be in only two cells and the columns (rows) of these cells coincide, then no other cell of these columns (rows) can contain this digit. Let's look at an example:

In the 4th and 5th lines, the number “2” can only appear in two yellow cells, and these cells are in the same columns. Therefore, the number “2” can be written only in two ways: 1) if “2” is written in the 5th column of the 4th line, then the “2” must be excluded from the yellow cells and then the position “2” in the 5th line is determined uniquely by the 7th column.

2) if “2” is written in the 7th column of the 4th line, then “2” must be excluded from the yellow cells and then in the 5th line the position of “2” is determined uniquely by the 5th column.

Therefore, the 5th and 7th columns will definitely have the number “2” either in the 4th line or in the 5th. Then the number “2” can be excluded from other cells of these columns (green cells).

"Swordfish"

This method is a variation of the .

The rules of the puzzle state that if a candidate is in three rows and only three columns, then in the other rows that candidate in those columns can be eliminated.

Algorithm:

• We are looking for lines in which the candidate appears no more than three times, but at the same time it belongs to exactly three columns.
• We exclude the candidate in these three columns from the other rows.

The same logic applies in the case of three columns, where the candidate is limited to three rows.

Let's look at an example. In three lines (3, 5 and 7th), candidate “5” appears no more than three times (cells are highlighted in yellow). Moreover, they belong to only three columns: 3, 4 and 7th. According to the Swordfish method, candidate “5” can be excluded from other cells in these columns (green cells).

In the example below, the “Swordfish” method is also used, but for the case of three columns. We exclude candidate “1” from the green cells.

"X-wing" and "swordfish" can be generalized to the case of four rows and four columns. This method will be called “Medusa”.

Colors

There are situations where a candidate appears only twice in a group (in a row, column or block). Then the required number will definitely be in one of them. The Colors method strategy is to view this relationship using two colors, such as yellow and green. In this case, the solution can be in cells of only one color.

We select all interconnected chains and make a decision:

• If some unshaded candidate has two differently colored neighbors in a group (row, column or block), then it can be excluded.
• If there are two identical colors in a group (row, column, or block), then that color is false. A candidate from all cells of this color can be eliminated.

The following example applies the Colors method to cells with candidate "9". We start coloring from the cell in the upper left block (2nd row, 2nd column), paint it yellow. In its block it has only one neighbor with “9”, let’s paint it green. It also has only one neighbor in the column, so we paint it green too.

We work in the same way with the remaining cells containing the number “9”. We get:

Candidate "9" can be either only in all yellow cells or in all green cells. In the right middle block there are two cells of the same color, therefore, the green color is incorrect, since in this block there are two “9”, which is unacceptable. We exclude “9” from all green cells.

Another example on the “Colors” method. Let us mark the paired cells for candidate “6”.

The cell with “6” in the upper central block (highlighted in lilac) has two different-colored candidates:

“6” will definitely be in either a yellow or green cell, therefore, “6” can be excluded from this lilac cell.

In this article we will look in detail at how to solve complex Sudoku using the example of diagonal Sudoku.

We get condition number 437, which is shown in Figure 1. And the first square immediately catches your eye, it is the most saturated with open numbers. The numbers 1, 3,4,9 are missing. But since the horizontal line a already contains three, the number three is placed on c1. We cannot accurately place the rest. So let’s look at what else we have. For example, the vertical is 4 and here the number four can only be on b4, due to the presence of a four in the fifth square and on the horizontal c. We will not put the remaining numbers for now.

All the techniques and methods that we will use further apply to solving both simple and complex Sudoku.

What do we have on horizontal b? There is not enough three here and it can only stand on b8. (In the second square it is already there and on vertical 9). And if we carefully examine the horizontal line b further, we will find that we have a hidden single - the number 9 on cell b9. Because the other candidates (these are 1 and 5) cannot stand on this square!

What can we do next? If we consider square five. Here the numbers 3 and 5 can be either on d5 or e6. This means that we do not consider these cells for the remaining numbers. Based on this, there is only one place left for the one - cell d6.

The result of our actions is shown in Figure 2. Thanks to our analysis, row b is filled in completely. One on b5, five on b6. What gives us the right to place 3 and 5 in the fifth square!

Let's continue the analysis of the fifth square. It lacks the number 7, it is not on the main diagonals, and what is most interesting is on the vertical 4. Thanks to this very vertical, we can say for sure that the number seven in the fifth square can be either on f4 or e4. Since the horizontal lines c and d already contain seven. And she cannot stand on e5 because of vertical 4. Next, let’s turn to the main horizontals. And then the sevens are immediately placed! On i9 and f4.

What we got can be seen in Figure 3. Next, we will continue the analysis of the main diagonals. If we look at the one coming from square a1, then it lacks a two, which is placed only on h8. This diagonal also lacks 1, 8 and 9. The 1 can only be placed on a1, put it quickly! But the eight cannot stand on d4, since it is already on the horizontal d. We arrange - d4 -9, e5 -8.

But now we can completely fill the fifth and first squares! What we got is shown in Figure 4.

Pay attention to vertical 3. Here you need to place 1, 6, 7. The unit is placed only on f3, and based on this the rest are placed - e3 -7, h3-6. Next in line we have vertical 9, as its placement is simply fabulous. d9-2, g9-6, h9-8.

What if we check for open singles?! For example, the number three is safely placed on cells d2 and h5. Although further analysis of singletons does not yield anything. Then let's turn to the remaining diagonal. She is missing 6, 2, 4. The number six can only be on c7. The rest is easy to fill out.

Why is vertical 4 not set to the end? Let's fix it. s4 -8.

The result of our research is shown in Figure 5. Now let's fill the horizontal line c. s8-1, s5-9, s6-2. And this is all based on the presence of these numbers in other verticals. Based on the horizontal c, it is easy to fill the horizontal d. d1-6, d7 -4. Then the third square is quite simply filled in. But the second square has not yet been filled, although there are also only two candidates - six and seven. But they do not occur along verticals five and six, and therefore we will put them aside for now.

Having analyzed all the verticals and horizontals, we come to the conclusion that it is impossible to put a single number unambiguously. Therefore, let's move on to considering squares. Let's turn to the sixth square. 5,6,8,9 are missing here. But we can definitely put numbers 6 and 8 on cells f7 and f8. Thanks to our analysis, the entire f horizontal line is marked! f1 -9, f2 -5. And what we see here is that the fourth square is completely filled! e1- 4, e2 -2.

What we got can be seen in Figure 6. Now let’s turn to square nine. Here we have one open single - number one on i7. Thanks to which we can put a one in the seventh square on g2. Eight on i2.