- This is a popular form of leisure, which is a puzzle with numbers, which is also called a magic square. Its solution allows developing logical thinking, attention, analytical approach. The benefits of Sudoku lie not only in the benefits for the brain, but also in the ability to distract from problems, to fully concentrate on the task.
This puzzle takes up little space, unlike scanwords, crosswords and so on. The playing field, consisting of 81 squares, the cells are divided into small blocks, 3 * 3 in size. It can easily fit on a piece of paper. The task looks like selectively filled cells, which must be supplemented with values and fill the entire table. In Sudoku, the rules of the game are very simple and allow you to eliminate multiple solutions. Each row or column contains numbers from 1 to 9. Also, the values are not repeated within one small block.
Sudokus differ in the level of difficulty, which depends on the number of cells filled with numbers and the methods of solving. Usually there are about 5 levels, where only real masters can solve the most difficult one.
The game of Sudoku has its own rules and secrets. The simplest puzzles can be solved in a few minutes with the help of deduction, as there is always at least one cell for which only one number fits. Complex Sudoku can be solved for hours. A correctly composed puzzle has only one way to solve it.
To obtain the right decision, you need to consider a few simple rules:
If both points are taken into account, then you can be sure that the cell is filled out correctly.
Consider on specific example how to solve sudoku. The playing field in the picture is a relatively simple version of the game. The rules of the Sudoku game for simple ones come down to identifying dependencies in the horizontal and vertical planes and in individual squares.
For example, the numbers 3, 4, 5 are missing in the central vertical. The four cannot be in the lower square, since it is already present in it. It is also possible to exclude the empty center cell, since we see 4 in the horizontal line. From this we conclude that it is located in the upper square. Similarly, we can put down 3 and 5 and get the following result.
By drawing lines in the upper middle small square 3 * 3, you can exclude cells in which the number 3 cannot be located.
Solve Continuing in this way, it is necessary to fill in the remaining cells. The result is the only correct solution.
This method is called by some The last Hero" or "Single". It is also used as one of several at master levels. The average time spent on the easy difficulty level fluctuates around 20 minutes.
Many people wonder how to solve Sudoku, if there are standard methods and strategy. As in any logic puzzle there is. We have considered the simplest of them. To go to more high level, you need to have more time, perseverance, patience. To solve the puzzle, you will have to make assumptions and, possibly, get the wrong result, returning to the place of choice. In essence, Sudoku is difficult - it's like solving a problem using an algorithm. Let's consider several popular techniques used by professional "Sudokuveds" in the following example.
First of all, it is necessary to fill in the empty cells with possible options in order to make the decision as easy as possible and have the full picture before your eyes.
The answer, how to solve Sudoku is difficult for everyone. Who is more comfortable using different colors for coloring cells or numbers, someone prefers black and white version. The figure shows that there is not a single cell in which there would be a single digit, however, this does not mean that there are no singles in this task. Armed with Sudoku rules and a careful look, you can see that the top line of the middle small block is the number 5, which occurs once in its line. In this regard, you can safely put it down and exclude it from the cells colored in green color. This action will entail the opportunity to put down the number 3 in the orange cell and boldly cross it out of the corresponding purple vertically and in a small block 3 * 3.
In the same way, we check the remaining cells and put down units in the circled cells, since they are also the only ones in their lines.
To figure out how to solve complex Sudokus, you need to arm yourself with a few simple methods.
To clear the field further, you need to find open pairs that allow you to exclude the numbers in them from other cells in the block and rows. In the example, these pairs are 4 and 9 from the third row. They clearly show how to solve complex Sudoku. Their combination suggests that only 4 or 9 can be entered in these cells. This conclusion is made on the basis of Sudoku rules.
You can remove the blue values from the cells highlighted in green and thereby reduce the number of options. At the same time, the combination 1249 located in the first line is called by analogy an “open four”. You can also find "open triplets". Such actions entail the appearance of other open pairs, such as 1 and 2 in the top line, which also provide an opportunity to narrow the circle of combinations. In parallel, we put 7 in the circled cell of the first square, since the five in this line will in any case be located in the lower block.
This method is opposite to open combinations. Its essence lies in the fact that it is necessary to find cells in which numbers are repeated within a square / line that are not found in other cells. How does this help solve Sudoku? The technique allows you to cross out the rest of the numbers, since they serve as a background and cannot be entered in the selected cells. This strategy has several other names, for example, "The cell is not rubber", "The secret becomes clear." The names themselves explain the essence of the method and compliance with the rule, which speaks of the possibility of putting down a single digit.
An example is blue-stained cells. The numbers 4 and 7 are found exclusively in these cells, so the rest can be safely deleted.
The conjugation system works in a similar way when it is possible to exclude from the cells of a block / row / column values that occur several times in an adjacent or conjugated one.
The principle of how to solve Sudoku is the ability to analyze and compare. Another way to exclude options is to have a number in two columns or lines that intersect. In our example similar situation not met, so consider another. The picture shows that the "two" occurs in the second and third middle block once, with a combination of which they are connected and mutually exclude each other. Based on this data, the number 2 can be removed from other cells in the specified columns.
Can also be used for three and four lines. The complexity of the method lies in the difficulties of visualization and identification of relationships.
As a result of each action, the number of options in the cells is reduced and the solution is reduced to the "Singleman" method. This process can be called reduction and is distinguished in separate method, since it involves a thorough analysis of all rows, columns, and small squares with successive exclusion of options. As a result, we come to a single solution.
This strategy differs little from the one described, and consists in the color indication of cells or numbers. The method helps to visualize the entire course of the solution, however, it is not suitable for everyone. Some coloring knocks down and makes it difficult to concentrate. To correctly use the gamut, you need to choose two or three colors and paint the same options in different blocks / lines, as well as controversial cells.
To figure out how to solve Sudoku, it is better to arm yourself with a pen and paper. This approach will allow you to train your head, in contrast to the use of electronic algorithms with hints. The BrainApps team has reviewed some of the most popular, clear and effective techniques, however, there are many other algorithms. For example, the trial and error method, when choosing trial version out of two or three possible and the whole chain is checked. The disadvantage of this technique is the need to use a computer, since it is not so easy to return to the original version on a piece of paper.
Consider the seventh square. Only four free cells, so something can be quickly filled.
"8
" on the D3 blocks padding H3 and J3; similar " 8
" on the G5 closes G1 and G2
With a clear conscience we put " 8
" on the H1
After viewing the squares for obvious solutions, move on to the columns and rows.
Consider " 4
" on the field. It is clear that it will be somewhere in the line A
.
We have " 4
" on the G3 that covers A3, there is " 4
" on the F7, cleaning A7. And another one " 4
" in the second square prohibits its repetition on A4 and A6.
"The Last Hero" for our " 4
" this is A2
In jargon it is " naked loner". If you fill in the field with possible values (candidates), then in the cell such a number will be the only possible one. Developing this technique, you can search for " hidden loners" - numbers unique for a particular row, column or square.
Candidate combinations for "naked trio" may be like this:
// three numbers in three cells.
// any combinations.
// any combinations. 
In this example, everything is pretty obvious. In the fifth square of the cell E4, E5, E6 contain [ 5,8,9
], [5,8
], [5,9
] respectively. It turns out that in general these three cells have [ 5,8,9
], and only these numbers can be there. This allows us to remove them from other block candidates. This trick gives us the solution " 3
" for cell E7.
In the above example, in the first square of the cell A1, B1, B2 and C1 generally contain [ 1,5,6,8
], so these numbers will occupy only those cells and no others. We remove the candidates highlighted in yellow.
Second, in a column 9
. [4,7,8
] are unique to cells B9, C9 and F9. Using the same logic, we remove candidates.
If any of the numbers appear twice or thrice in the same block (row, column, square), then we can remove that number from the conjugate block. There are four types of pairing:
Let me show you this puzzle as an example. In the third square 3
"is only in B7 and B9. Following the statement №1
, we remove candidates from B1, B2, B3. Likewise, " 2
" from the eighth square removes a possible value from G2.
Special puzzle. Very difficult to solve, but if you look closely, you can see a few pointing pairs. It is clear that it is not always necessary to find them all in order to advance in the solution, but each such find makes our task easier.
A mathematical puzzle called "" comes from Japan. It has become widespread throughout the world due to its fascination. To solve it, you will need to concentrate attention, memory, and use logical thinking.
The puzzle is printed in newspapers and magazines, there are computer versions of the game and mobile applications. The essence and rules in any of them are the same.
The puzzle is based on the Latin square. The field for the game is made in the form of this particular geometric figure, each side of which consists of 9 cells. The large square is filled with small square blocks, sub-squares, three squares on a side. At the beginning of the game, some of them are already filled with "hint" numbers.
It is necessary to fill in all the remaining empty cells with natural numbers from 1 to 9.
You need to do this so that the numbers do not repeat:
Thus, in each row and each column of the large square there will be numbers from one to ten, any small square will also contain these numbers without repetition.
The game has only one the right decision. There are different levels of difficulty: a simple puzzle, with large quantity filled cells can be solved in a few minutes. On a complex one, where a small number of numbers are placed, you can spend several hours.
Various approaches to problem solving are used. Consider the most common.
Exclusion method
This is a deductive method, it involves the search for unambiguous options - when only one digit is suitable for writing to a cell.
First of all, we take the square most filled with numbers - the lower left. It lacks one, seven, eight and nine. To find out where to put the one, let's look at the columns and rows where this number is: it is in the second column, so our empty cell (the lowest in the second column) cannot contain it. There are three possible options left. But the bottom line and the second line from the very bottom also contain one - therefore, by the elimination method, we are left with the upper right empty cell in the subsquare under consideration.
Similarly, fill in all empty cells.
Writing Candidate Numbers to a Cell
For the solution, options are written in the upper left corner of the cell - candidate numbers. Then “candidates” that are not suitable according to the rules of the game are crossed out. Thus, all free space is gradually filled.
Experienced players compete with each other in skill, in the speed of filling empty cells, although this puzzle is best solved slowly - and then the successful completion of Sudoku will bring great satisfaction.
Sudoku is a very interesting puzzle game. It is necessary to arrange the numbers from 1 to 9 in the field in such a way that each row, column and block of 3 x 3 cells contains all the numbers, and at the same time they should not be repeated. Consider step by step instructions how to play sudoku, basic methods and solution strategy.
The algorithm for solving the Sudoku mind game is quite simple: you need to repeat the following steps until the problem is completely solved. Gradually move from the simplest steps to more complex ones, when the first ones no longer allow you to open a cell or exclude a candidate.
First of all, for a more visual explanation of how to play Sudoku, let's introduce a numbering system for blocks and cells of the field. Both cells and blocks are numbered from top to bottom and from left to right.
Let's start looking at our field. First you need to find single candidates for a place in the cell. They can be hidden or explicit. Consider possible candidates for the sixth block: we see that only one of the five free cells contains a unique number, therefore, the four can be safely entered in the fourth cell. Considering this block further, we can conclude: the second cell should contain the number 8, since after the exclusion of the four, the eight in the block does not occur anywhere else. With the same justification, we put the number 5.
Look carefully at everything possible options. Looking at the central cell of the fifth block, we find that there can be no other options besides the number 9 - this is a clear single candidate for this cell. The nine can be crossed out from the rest of the cells of this block, after which the remaining numbers can be easily put down. Using the same method, we pass through the cells of other blocks.
Having entered the necessary numbers in the fourth block, let's return to the empty cells of the sixth block: it is obvious that the number 6 should be in the third cell, and 9 in the ninth.
The concept of "naked pair" is present only in the game of Sudoku. The rules for their detection are as follows: if two cells of the same block, row or column contain an identical pair of candidates (and only this pair!), then the other cells of the group cannot have them. Let's explain this on the example of the eighth block. Putting possible candidates in each cell, we find an obvious "naked pair". The numbers 1 and 3 are present in the second and fifth cells of this block, and there and there there are only 2 candidates each, therefore, they can be safely excluded from the remaining cells.
If you learned the lesson on how to play Sudoku and followed the above instructions step by step, then you should end up with something like this picture:
Here you can find single candidates: a one in the seventh cell of the ninth block and a two in the fourth cell of the third block. Try to solve the puzzle to the end. Now compare your result with the correct solution.
Happened? Congratulations, this means that you have successfully mastered the lessons on how to play Sudoku and learned how to solve the simplest puzzles. There are many variations of this game: Sudoku different sizes, Sudoku with additional areas and additional conditions. The playing field can vary from 4 x 4 to 25 x 25 cells. You may come across a puzzle in which the numbers cannot be repeated in an additional area, for example, diagonally.
Start with simple options and gradually move on to more difficult ones, because with training comes experience.
The goal of Sudoku is to arrange all the numbers so that there are no numbers in 3x3 squares, rows and columns same digits. 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 under 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.
Consider the following situation:
In the square highlighted in blue, the number 4 candidates (green cells) are located in two cells on the same line. If the number 4 is on this line (orange cells), then there will be nowhere to put 4 in the blue square, which means that we exclude 4 from all orange cells.
A similar example for the number 2:
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).
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.
Groups are harder to see than locked candidates, but they help clear many dead ends in complex crossword puzzles.
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 a line to have 2 the same number, 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 should be the same, but here 3 different numbers in a group.
Naked triples are similar to naked pairs, but they are more difficult to detect - these are 3 naked numbers in three cells.
In the example, the numbers in one line are repeated 3 times. There are only 3 numbers in the group and they are located on 3 cells, which means that the extra numbers 1, 2, 6 from the orange cells are removed.
A naked triple may not contain a number in full, for example, a combination would be suitable:, and - these are all the same 3 types of numbers in three cells, just in an incomplete composition.
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 quadruple 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.
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.
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.
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
These strange words are the names of two similar ways of eliminating Sudoku candidates.
X-wing is considered for candidates of one number, consider 3:
There are only 2 triples in two rows (blue) and these triples lie on only two lines. This combination has only 2 triples solutions, and the other triples in the orange columns contradict this solution (check why), so the red triple candidates should be removed.
Similarly for candidates for 2 and columns.
In fact, the X-wing is quite common, but not so often the encounter with this situation promises the exclusion of extra numbers.
This is an advanced version of X-wing for three rows or columns:
We also consider 1 number, in the example it is 3. 3 columns (blue) contain triplets that belong to the same three rows.
Numbers may not be contained in all cells, but the intersection of three horizontal and three vertical lines is important to us. Either vertically or horizontally, there should be no numbers in all cells except green ones, in the example this is a vertical - columns. Then all the extra numbers in the lines should be removed so that 3 remains only at the intersections of the lines - in green cells.
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) excluded 2 different ways- with the help of a hidden pair or with the help of a naked 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.
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.
