How to solve difficult Sudoku. Logic puzzles

When solving Sudoku, be consistent in your reasoning. Check your actions periodically, because if you make a mistake at the beginning of the solution, it may ultimately 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 discovered in the solved puzzle.

The following methods for solving Sudoku are presented in order of their difficulty and frequency of use in practice.

Selection of candidates

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

For example, consider cell A2, it is marked gray. “1” – available in the block, “2” – available in the row, “3” – available in the block and row, “4” – available in the row, “5” – available in the column, “7” – available 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, there are several candidates for each cell. Let's fill 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 - A2 and D9, they are called the only candidates. After finding the only candidates, it is also necessary to cross them out from the candidates in other cells (cells of this column, row, block). So, by deleting the number “6” from line 2, column A and block 1, we also get the only candidate in cell B1 – the number “2”. We will continue to do so in the same way.

However, there are also “hidden” single candidates. For example, let's take cell I7. This cell is located 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, and it is also present in line 8. Accordingly, of the three candidates for cell I7, we leave only the number “5”.

Elimination of candidates

The methods described above allow you to unambiguously determine which number needs to be entered in a particular cell, the following will allow you to reduce their number, which will ultimately lead to only one candidate.

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

Let's 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 the number “4” is in, it cannot be in line 5 in other blocks, so it can be safely crossed out from the candidate cells G5.

There is also the opposite option 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 the number “4” is in, it can no longer be in block 2 in other cells, so it can be safely crossed out from the candidate 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 in that block, row, or column.

Cells G9 and H9 contain the candidate pair "6" and "8". Accordingly, no matter which of these two cells contains the numbers “6” and “8” (if “6” is in G9, then “8” is in H9, and vice versa), they cannot be in block 9 in other cells, the same as in line 9. Therefore, they can be safely deleted from the candidate cells H7, G8, B9, C9, F9.

This method can also be used 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, among the candidates there is a pair of candidates that is not found in any other cell of the block, row or column, then no other cells in the block, row or column can contain candidates from this pair. Therefore, all other candidates from these two cells can be crossed out.

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” threes and fours.

There are also more complex methods used to solve 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 at the same time 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, from two rows, excluded candidates will then be in columns.

In column A, the number “2” can only appear 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.

A certain dependence has formed:

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, and it will not be in cell A6 either, i.e. because 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, and it will not be in cell A4 either, i.e. because 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, you can safely cross out the number “2” from other cells on lines 4 and 6. In addition, this method can be applied to blocks. Since in block 4 the number “2” will definitely be in cells A4 or A6, it can also be crossed out from the candidate cells of block 4.

These are the main ways in which you can solve classic Sudoku. If Sudoku is not difficult, then it can be solved using the first methods. When solving more complex puzzles, you cannot do without the latest methods. But these methods are not formulaic; 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 you won’t have to write down all the candidates, and you can easily keep them “in your head.”

An example of solving a classic Sudoku

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

First, let's write down all the candidates.

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

Now let’s find the “hidden” single candidates (gray cells). And cross them out from 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 the candidates in other cells of blocks, rows, columns.

Now take cell H5. In line 5, the number "2" appears 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 in 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 options for solving 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 only one correct solution and it can be found in a logical way, and not by searching through numbers.

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!

It often happens that you need to occupy yourself with something, entertain yourself - while waiting, or on a trip, or simply when there is nothing to do. In such cases, various crossword puzzles and scanword puzzles can come to the rescue, but their disadvantage is that the questions there are often repeated and remembering the correct answers and then entering them “automatically” is not difficult for a person with a good memory. Therefore, there is an alternative version of crossword puzzles - Sudoku. How to solve them and what is it all about?

What is Sudoku?

Magic square, Latin square - Sudoku has a lot of different names. Whatever you call the game, its essence will not change - it is a number puzzle, the same crossword puzzle, only not with words, but with numbers, and compiled according to a certain pattern. IN Lately is a very popular way to brighten up your leisure time.

History of the puzzle

It is generally accepted that Sudoku is a Japanese pleasure. This, however, is not entirely true. Three centuries ago, the Swiss mathematician Leonhard Euler, as a result of his research, developed the game “Latin Square”. It was on its basis that in the seventies of the last century in the USA they came up with number square puzzles. From America they came to Japan, where they received, firstly, their name, and secondly, unexpected wild popularity. This happened in the mid-eighties of the last century.

Already from Japan, the numerical problem went to travel around the world and also reached Russia. Since 2004, British newspapers began to actively distribute Sudoku, and a year later electronic versions of this sensational game appeared.

Terminology

Before talking in detail about how to correctly solve Sudoku, you should devote some time to studying the terminology of this game in order to be confident in the future that you correctly understand what is happening. So, the main element of the puzzle is the cell (there are 81 of them in the game). Each of them is included in one row (consists of 9 cells horizontally), one column (9 cells vertically) and one area (a square of 9 cells). A row can also be called a row, a column can be called a column, and an area can be called a block. Another name for a cell is a cell.

A segment is three horizontal or vertical cells located in the same area. Accordingly, there are six of them in one area (three horizontally and three vertically). All those numbers that can be in a particular cell are called candidates (because they are competing to get into that cell). There can be several candidates in a cell - from one to five. If there are two of them, they are called a pair, if there are three, they are called a trio, if there are four, they are called a quartet.

How to solve Sudoku: rules

So, first, you need to decide what Sudoku is. This is a large square of eighty-one cells (as mentioned earlier), which, in turn, are divided into blocks of nine cells. So there are a total of nine small blocks in this large Sudoku board. The player’s task is to enter numbers from one to nine into all Sudoku cells so that they are not repeated horizontally, vertically, or in a small area. Initially, some numbers are already in place. These are hints given to make solving Sudoku easier. According to experts, a correctly composed puzzle can only be solved in one correct way.

Depending on how many numbers are already in Sudoku, the degrees of difficulty of this game vary. In the simplest ones, accessible even to a child, there are a lot of numbers, in the most complex ones there are practically none, but that makes it all the more interesting to solve.

Varieties of Sudoku

The classic type of puzzle is a large nine by nine square. However, lately, different versions of the game have become increasingly common:

Basic solution algorithms: rules and secrets

How to solve Sudoku? There are two basic principles that can help solve almost any puzzle.

1. We remember that each cell contains a number from one to nine, and these numbers should not be repeated vertically, horizontally or in one small square. Let's try to use the method of elimination to find a cell only in which it is possible to find a number. Let's look at an example - in the figure above, take the ninth block (lower right). Let's try to find a place in it for one. There are four free cells in the block, but you cannot place a unit in the third one in the top row - it is already in this column. It is forbidden to put a unit in both cells of the middle row - it also already has such a number, in the area next door. Thus, for a given block it is permissible for a unit to be in only one cell - the first one in the last row. Thus, using the method of elimination, cutting off unnecessary cells, you can find the only correct cells for certain numbers both in a specific area and in a row or column. The main rule is that this number should not be in the neighborhood. The name of this method is “hidden singles”.
2. Another way to solve Sudoku is to eliminate extra numbers. In the same figure, consider the central block, the cell in the middle. It cannot contain the numbers 1, 8, 7 and 9 - they are already in this column. The numbers 3, 6 and 2 are also not allowed for this cell - they are located in the area we need. And the number 4 is in this row. Therefore, the only possible number for this cell is five. It should be entered into the central cell. This method is called “singles”.

Very often, the two methods described above are enough to quickly solve Sudoku.

How to solve Sudoku: secrets and methods

It is recommended to adopt the following rule: write down in fine detail in the corner of each cell the numbers that could appear there. As new information is received, extra numbers need to be crossed out, and then in the end it will be clear the right decision. In addition, first of all you need to pay attention to those columns, rows or areas where there are already numbers, and in as many numbers as possible - the fewer options left, the easier it is to cope. This method will help you quickly solve Sudoku. As experts recommend, before entering the answer into a cell, you need to double-check it again so as not to make a mistake, because because of one incorrectly entered number, the entire puzzle can “fly” and it will no longer be possible to solve it.

If there is such a situation that in one area, one row or one column in any three cells it is permissible to find the numbers 4, 5; 4, 5 and 4, 6 - this means that the third cell will definitely contain the number six. After all, if there were a four in it, then there could only be five in the first two cells, but this is impossible.

Below are other rules and secrets on how to solve Sudoku.

Locked Candidate Method

When you are working with one specific block, a situation may arise that a certain number in a given area can only be in one row or in one column. This means that in other rows/columns of this block there will absolutely not be such a number. The method is called “locked candidate” because the number is, as it were, “locked” within one row or one column, and later, with the appearance of new information, it becomes clear exactly in which cell of a given row or column this number is located.

In the figure above, consider block number six - central right. The number nine in it can only be in the column in the middle (in cells five or eight). This means that in other cells of this area there will definitely not be a nine.

Open Pairs Method

The next secret of how to solve Sudoku is: if in one column/one row/one area two cells can contain only any two identical numbers (for example, two and three), then they can be found in no other cells of this block/row/column will not. This often makes the task much easier. The same rule applies in a situation with three identical numbers in any three cells of the same row/block/column, and with four - respectively, in four.

Hidden pairs method

It differs from the above in the following way: if in two cells of the same row/area/column, among all possible candidates, there are two identical numbers that do not appear in other cells, then they will be located in these places. However, other numbers can be excluded from these cells. For example, if there are five free cells in one block, but only two of them contain the numbers one and two, then that is where they are located. This method works for three and four numbers/cells.

x-wing method

If a specific number (for example, five) can only be located in two cells of a certain row/column/area, then that is where it is located. Moreover, if in an adjacent row/column/area the placement of a five is allowed in the same cells, then this number is not found in any other cell of the row/column/area.

Difficult Sudoku: solution methods

How to solve difficult Sudoku? The secrets, in general, are still the same, that is, all the methods described above work in these cases. The only thing is that in complex Sudoku there are often situations when you have to abandon logic and act at random. This method even has its own name - “Ariadne’s Thread”. We take a number and insert it into the correct cell, and then, like Ariadne, we unravel a ball of thread, checking whether the puzzle fits together. There are two options here - either it worked or it didn’t. If not, then you need to “wind up the ball”, return to the original one, take another number and try all over again. In order to avoid unnecessary scribbles, it is recommended to do this all on a draft.

Another way to solve complex Sudoku is to analyze three blocks horizontally or vertically. You need to choose a number and see if you can substitute it in all three areas at once. In addition, in cases of solving complex Sudoku, it is not only recommended, but absolutely necessary to recheck all the cells, return to what you missed before - after all, new information, which must be applied to the playing field.

Mathematical rules

Mathematicians do not remain aloof from this problem. Mathematical methods How to solve Sudoku are as follows:

1. The sum of all numbers in one area/column/row is forty-five.
2. If in some area/column/row three cells are not filled, and it is known that two of them must contain certain numbers (for example, three and six), then the desired third number is found using the example 45 - (3+6+ S), where S is the sum of all filled cells in this area/column/row.

How to increase your guessing speed?

The following rule will help you solve Sudoku faster. You need to take a number that is already in its place in most blocks/rows/columns, and by eliminating extra cells, find cells for this number in the remaining blocks/rows/columns.

Game versions

More recently, Sudoku remained only printed game, published in magazines, newspapers and individual books. However, recently all kinds of versions of this game have appeared, for example board Sudoku. In Russia they are produced by the well-known company Astrel.

There are also computer variations of Sudoku - and you can either download this game to your computer or solve the puzzle online. Sudoku is being released for completely different platforms, so it doesn’t matter what exactly is installed on your personal computer.

And just recently they appeared mobile applications with the game Sudoku - both for Android and iPhone, the puzzle is now available for download. And I must say that this application is very popular among cell phone owners.

1. The minimum possible number of clues for a Sudoku puzzle is seventeen.
2. There is an important recommendation on how to solve Sudoku: take your time. This game is considered relaxing.
3. It is recommended to solve the puzzle with a pencil, not a pen, so that you can erase the wrong number.

This puzzle is truly exciting game. And if you know the methods of how to solve Sudoku, then everything becomes even more interesting. Time will fly by for the benefit of the mind and completely unnoticed!

Use numbers from 1 to 9

Sudoku is played on a playing field consisting of 9 by 9 cells, with a total of 81 cells. Inside the playing field there are 9 "squares" (consisting of 3 x 3 cells). Each horizontal row, vertical column and square (9 squares each) must be filled with numbers 1-9, without repeating any numbers in a row, column or square. Does this sound complicated? As you can see from the image below, each Sudoku game board has several cells that are already filled. The more cells are initially filled, the easier the game. The fewer cells are initially filled, the more difficult the game.

Don't repeat any numbers

As you can see, in the top left square (circled in blue) 7 of the 9 cells are already filled. Singular numbers The ones that are missing from that square are the numbers 5 and 6. By seeing which numbers are missing from each square, row, or column, we can use the process of elimination and deductive reasoning to decide which numbers should be in each square.

For example, in the top left square we know that to complete the square we need to add the numbers 5 and 6, but looking at the adjacent rows and squares we cannot yet clearly determine which number to add to which cell. This means that we must now skip the top left square for now and instead try to fill in the gaps in some other places on the playing field.

No need to guess

Sudoku is logic game, so no need to guess. If you don't know what number to put in a certain space, keep scanning other areas of the game board until you see the option to put the number you want. But don't try to "force" anything - Sudoku rewards patience, understanding and solving different combinations, not blind luck or guessing.

Use the elimination method

What do we do when we use the "method of elimination" in Sudoku? Here's an example. In this Sudoku grid (shown below), only a few numbers are missing from the left vertical column (outlined in blue): 1, 5, and 6.

One way to figure out which numbers can be inserted into each square is to use the "method of elimination", checking what other numbers are already in each square, since numbers 1-9 are not allowed to be duplicated in each square, row or column.

In this case, we can quickly notice that there is already a 1 in the top left and center left squares (the 1's are circled in red). This means that in the leftmost column there is only one place where the number 1 can be inserted (circled in green). This is how the elimination method works in Sudoku - you find out which cells are empty, which numbers are missing, and then eliminate the numbers that are already present in the square, columns and rows. Accordingly, fill in the empty cells with the missing numbers.

The rules of Sudoku are relatively simple - but the game is incredibly varied, with millions of possible number combinations and a wide range of difficulty levels. But it's all based on the simple principles of using numbers 1-9, filling in the blanks using deductive reasoning, and never repeating numbers in each square, row, or column.