Difficult sudoku how to solve. How to solve complex sudoku using the example of diagonal sudoku

Carefully look at the rows or columns to fill in the large squares with the missing numbers. Look at the row of three large squares. Check it for two duplicate digits in different large squares. Swipe your finger over the rows that contain these numbers. This number must also be present in the third large square, but it cannot be located in the same two rows that you traced with your finger. It should be in the third row. Sometimes two of the three cells in this row of the square will already be filled with numbers and it will be easy for you to enter the number that you checked in its place.

• If there is an eight in two large squares of the row, it must be checked in the third square. Run your finger along the rows with two eights present, since in these rows the eight cannot stand in the third large square.

Additionally, view the puzzle field in the other direction. Once you understand the principle of looking at the rows or columns of a puzzle, add a look in the other direction to it. Use the above view principle with a little addition. Perhaps when you get to the third large square, in the row in question there will be only one finished number and two empty cells.

• In this case, it will be necessary to check the columns of numbers above and below the empty cells. See if one of the columns contains the same number that you are going to put. If you find this number, you cannot put it in the column where it already exists, so you need to enter it in another empty cell.

Work immediately with groups of numbers. In other words, if you notice a lot of the same numbers on the field, they can help you fill in the rest of the squares with the same numbers. For example, there may be many fives on the puzzle board. Use the above field scan technique to fill it with as many remaining fives as possible.

It often happens that you need something to occupy yourself, entertain yourself - while waiting, or on a trip, or simply when there is nothing to do. In such cases, a variety of crosswords and scanwords can come to the rescue, but their minus is that the questions are often repeated there and remembering the correct answers, and then entering them “on the machine” is not difficult for a person with a good memory. Therefore, there is an alternative version of crossword puzzles - this is 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 from this - this is a numerical puzzle, the same crossword puzzle, only not with words, but with numbers, and compiled according to a certain pattern. Recently, it has become a very popular way to brighten up your leisure time.

The 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 developed the Latin Square game as a result of his research. It was on its basis that in the seventies of the last century in the United States they came up with numerical puzzle squares. From America, they came to Japan, where they received, firstly, their name, and secondly, unexpected wild popularity. It happened in the mid-eighties of the last century.

Already from Japan, the numerical problem went to travel the world and reached, among other things, 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 solve Sudoku correctly, you should devote some time to studying the terminology of this game in order to be sure of the correct understanding of what is happening in the future. So, the main element of the puzzle is the cage (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 (square of 9 cells). A row may otherwise be called a row, a column a column, and an area 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 claim to be in this cell). There can be several candidates in the cell - from one to five. If there are two of them, they are called a pair, if there are three - a trio, if four - 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. Thus, there are nine small blocks in total in this large Sudoku field. The player's task is to enter numbers from one to nine in all Sudoku cells so that they do not repeat either horizontally or vertically, or in a small area. Initially, some numbers are already in place. These are hints given to make it easier to solve Sudoku. According to experts, a correctly composed puzzle can only be solved in the only correct way.

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

Varieties of Sudoku

The classic type of puzzle is a large nine-by-nine square. However, in recent years, various versions of the game have become more and more common:

Basic solution algorithms: rules and secrets

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

1. Remember that each cell contains a number from one to nine, and these numbers should not be repeated vertically, horizontally and in one small square. Let's try by elimination to find a cell, only in which it is possible to find any number. Consider an example - in the figure above, take the ninth block (lower right). Let's try to find a place for the unit in it. There are four free cells in the block, but one cannot be placed in the third 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 figure, in the area next door. Thus, for this block, it is permissible to find a unit in only one cell - the first in the last row. So, acting by the method of exclusion, cutting off extra 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 loners".
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 in the central cell. This method is called "loners".

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

How to solve Sudoku: secrets and methods

It is recommended to adopt the following rule: write small in the corner of each cell those numbers that could be there. As new information is obtained, the extra numbers must be crossed out, and then in the end the correct solution will be seen. In addition, first of all, you need to pay attention to those columns, rows or areas where there are already numbers, and as many as possible - the fewer options left, the easier it is to handle. This method will help you quickly solve Sudoku. As experts recommend, before entering the answer into the cell, you need to double-check it again so as not to make a mistake, because because of one incorrectly entered number, the whole puzzle can “fly”, 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 in the third cell there will definitely be the number six. After all, if there were a four in it, then in the first two cells there could only be five, and this is impossible.

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

Locked Candidate Method

When you work with any one particular block, it may happen 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 be absolutely no such 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 advent of new information, it becomes clear exactly in which cell of this row or this column this number is located.

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

Method "open pairs"

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

Hidden Pair Method

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

x-wing method

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

Difficult Sudoku: Solving Methods

How to solve difficult sudoku? The secrets, in general, are the same, that is, all the methods described above work in these cases. The only thing is that in complex sudoku situations are not uncommon when you have to leave logic and act by the “poke method”. This method even has its own name - "Ariadne's Thread". We take some number and substitute it in the right cell, and then, like Ariadne, we unravel the ball of threads, checking whether the puzzle fits. 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 scribbling, it is recommended to do all this on a draft.

Another way to solve complex sudoku is to analyze three blocks horizontally or vertically. You need to choose some number and see if you can substitute it in all three areas at once. In addition, in cases with solving complex Sudokus, it is not only recommended, but it is necessary to double-check all the cells, return to what you missed before - after all, new information appears that needs to be applied to the playing field.

Math Rules

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

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

How to increase guessing speed?

The following rule will help you solve Sudoku faster. You need to take a number that is already in 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 a printed game, published in magazines, newspapers and individual books. Recently, however, all sorts of versions of this game have appeared, such as 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 comes out for completely different platforms, so it doesn't matter what exactly is on your personal computer.

And more recently, mobile applications with the Sudoku game have appeared - both for Android and for iPhones, 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 advised to solve the puzzle with a pencil, not a pen, so that you can erase the wrong number.

This puzzle is a truly addictive 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!

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 of putting 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 there is a number 4 on this line (orange cells), then there will be nowhere to put 4 in the blue square, which means 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 bare three 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 triples 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.