Sudoku is the meaning of the game. About problem solving methods - Sudoku complete course

A mathematical puzzle called "" comes from Japan. It has become widespread all over the world due to its fascination. To solve it you will need to concentrate attention, memory, use logical thinking.

The puzzle is published 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.

How to play

The puzzle is based on a Latin square. The playing field is made in the form of exactly this geometric figure, each side of which consists of 9 cells. The large square is filled with small square blocks, sub-squares, with a side of three squares. At the beginning of the game, some of them already contain “hint” numbers.

All remaining empty cells must be filled in natural numbers from 1 to 9.

This must be done so that the numbers are not repeated:

• in each column,
• in every line,
• in any of the small squares.

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.

Difficulty levels

The game has only one correct solution. There are different levels of difficulty: a simple puzzle, with big amount filled cells can be solved in a few minutes. A complex one, where a small number of numbers are placed, can take several hours.

Solution techniques

Various approaches to solving problems are used. Let's look at the most common ones.

Elimination method

This is a deductive method, it involves searching for unambiguous options - when only one digit is suitable for writing in a cell.

First of all, we take on the square most filled with numbers - the bottom left one. It is missing 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 one in the second column) cannot contain it. Three left possible options. But the bottom line and the second line from the very bottom also contain a 1 - therefore, by the method of elimination, we are left with the upper right empty cell in the subsquare in question.

Similarly, fill all empty cells.

Writing candidate numbers to a cell

To solve the problem, options - candidate numbers - are written in the upper left corner of the cell. Then “candidates” that do not meet the rules of the game are eliminated. In this way, all free space is gradually filled.

Experienced players compete with each other in skill and in the speed of filling empty cells, although this puzzle is best solved slowly - and then successfully completing Sudoku will bring great satisfaction.

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 an already solved Sudoku:

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 rule of “uniqueness” of a number and, sequentially eliminating 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.

Having opened Sudoku, we see that each cell contains all the small gray numbers. You can immediately remove marks from already set numbers (marks can be removed by right-clicking on a small number):

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

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

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

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

Next, let's move 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 left (marked in green), where there can be a 4. So, we put the number 4 there and erase all the candidates (there can be no other numbers there anymore). In simple Sudoku, you can fill in quite a lot of fields in this way.

After a new number has been 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 set of four, there is only one cell (green) left for one in this square:

Of the three available cells for a unit, only one is not occupied, so we put the unit there.

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

After removing all obviously unsuitable candidates, we ended up with a cell where only 1 candidate remained (green), which means that this number is three, and it stands there.

Numbers are also placed if the candidate is the last one left in the square, row or column:

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

These are the most basic ways of putting numbers in Sudoku, you can already try them by solving Sudoku on simple difficulty (one star), for example: Sudoku No. 12433, Sudoku No. 14048, Sudoku No. 526. The above sudoku puzzles can be 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 enter some number at random, and if that fails, load the Sudoku.

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

Locked Candidates

Locked Candidate Squared

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 number 2:

Locked candidate in line

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

Locked candidate in column

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

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

Groups of numbers

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

Naked couples

The simplest subtype of groups is 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 remain 7 and 7, or 8 and 8, but according to the rules it is impossible for a line to have 2 identical numbers, which means that all 7 and all 8 are removed from the orange cells .

Another example:

Naked couple in one column and one square at the same time. Extra candidates (red) are removed from both the column and the square.

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

Naked threesomes

Naked threes are similar to naked pairs, but they are harder to spot - they are 3 naked numbers in three squares.

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 are removed from the orange cells.

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

Naked fours

The next extension of bare groups is bare quadruples.

The numbers , , , form a naked quadruple of four numbers 2, 5, 6 and 7, located in four cells. This four is located in one square, which means that all 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. Let's look at an example:

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

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

Hidden Threes

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. Other cells in the column do not contain these numbers, which means we remove unnecessary candidates from the green cells.

Hidden fours

Same with hidden threes, 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 there are no these numbers in other cells of the column (orange). Excess candidates from green cells are removed.

This concludes our consideration of groups of numbers. To practice, try solving the following crossword puzzles (without matching): Sudoku No. 13091, Sudoku No. 10710

X-wing and swordfish

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

X-wing

X-wing is being considered for candidates of the same number, let's consider 3:

There are only 2 triples in two lines (blue) and these triples lie on only two lines. This combination has only 2 solutions for triplets, and the other triplets in the orange columns contradict this solution (check why), which means the red candidates for triplets must be removed.

Likewise for 2 and column candidates.

In fact, X-wing occurs quite often, but not so often meeting this situation promises the elimination of unnecessary numbers.

This is a complicated variation 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.

The numbers may not be contained in all cells, but what is important to us is the intersection of three horizontal and three vertical lines. Either vertically or horizontally there should be no numbers in all cells except green ones, in the example this is vertical - columns. Then all extra numbers in the lines must 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 aren’t they looking for hidden/naked fives, sixes, etc.?

Let's look at the following 2 examples:

This is one Sudoku where one number column is considered. 2 numbers 4 (marked in red) excluded 2 different ways– using a hidden pair or using a naked pair.

Next example:

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

If you look closely 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 definitely 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, we can find out that if there is a bare group in the remaining cells, there will definitely be a hidden group and vice versa.

And from this we can conclude that if we have 9 free cells in a row, and among them there is definitely a naked six, then it will be easier to find a hidden three than to look for the relationship between 6 cells. It’s the same with a hidden and naked five – it’s easier to find a naked/hidden four, so fives aren’t 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 triplets. And with five free cells or less, you don’t have to look for threes - twos will be enough.

Final word

Here are the most well-known 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 place, 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 substituted number from the candidates.

ALGORITHM FOR SOLVING SUDOKU (SUDOKU) Contents Introduction 1. Techniques for solving Sudoku.* 1.1. Method of small squares.* 1.2. Method of rows and columns.* 1.3. Joint analysis of a row (column) with a small square.* 1.4. Joint analysis of the square of a row and column.* 1.5.Local tables. Couples. Triads..* 1.6.Logical approach.* 1.7.Reliance on undisclosed pairs.* 1.8.An example of solving a complex Sudoku 1.9.Volitional disclosure of pairs and Sudoku with ambiguous solutions 1.10.Non-pairs 1.11.joint use of two techniques 1.12.Half-pairs.* 1.13. Solving Sudoku with a small initial number of digits. Nontriads. 1.14.Quadro 1.15.Recommendations 2.Tabular algorithm for solving Sudoku 3.Practical instructions 4.An example of solving Sudoku using a tabular method 5.Test your strength Note: items not marked with an asterisk (*) can be omitted during the first reading. Introduction Sudoku is a number puzzle. The playing field is a large square consisting of nine rows (9 cells in a row, cells in a row are counted from left to right) and nine columns (9 cells in a column, cells in a column are counted from top to bottom) in total: (9x9 = 81 cells), divided into 9 small squares (each square consists of 3x3 = 9 cells, counting squares - from left to right, top to bottom, counting cells in a small square - left to right, top to bottom). Each cell of the working field belongs simultaneously to one row and one column and has coordinates consisting of two numbers: its column number (X-axis) and row number (Y-axis). The cell in the upper left corner of the playing field has coordinates (1,1), the next cell in the first line is (2,1), the number 7 in this cell will be written in the text as follows: 7(2,1), the number 8 in the third cell in the second line is 8(3,2), etc., and the cell in the lower right corner of the playing field has coordinates (9,9). To solve Sudoku - fill all the empty cells of the playing field with numbers from 1 to 9 so that the numbers are not repeated in any row, in any column, or in any small square. The numbers in the filled cells are the result numbers (RR). The numbers we need to find are the missing numbers - CN. If three numbers are written in some small square, for example, 158 is the CR (commas are omitted, we read: one, two, three), then the SC in this square is 234679. In other words, solve Sudoku - find and correctly arrange all the missing numbers, each CN, the place of which is uniquely determined, becomes a CN. In the figures, the CRs are drawn with indices, index 1 determines the CR found first, 2 - the second, etc. The text indicates either the coordinates of the CR: CR5(6,3) or 5(6,3); or coordinates and index: 5(6,3) ind. 12: or index only: 5-12. Indexing the CR in the pictures makes it easier to understand the process of solving Sudoku. In "diagonal" Sudoku, one more condition is imposed, namely: in both diagonals of the large square, the numbers must not be repeated either. Usually Sudoku has one solution, but there are exceptions - 2, 3 or more solutions. Solving Sudoku requires attention and good lighting. Use ballpoint pens. 1. TECHNIQUES FOR SOLVING SUDOKU* 1.1.Method of small squares - MK.* This is the simplest method for solving Sudoku, it is based on the fact that in each small square, each number out of nine possible can appear only once. You can start solving the puzzle with it. The search for CR can be started with any number, usually we start with one (if they are present in the problem). We find a small square in which this figure is missing. We search for the cell in which the number we have chosen should be located in a given square in the following way. We look through all the rows and columns passing through our small square to see if they contain the number we have chosen. If somewhere (in neighboring small squares), a row or column passing through our square contains our number, then parts of them (rows or columns) in our square will be forbidden ("broken") for setting the number we have chosen. If, after analyzing all the rows and columns (3 and 3) passing through our square, we see that all the cells of our square, except for ONE “bit”, are either occupied by other numbers, then we must enter our number in this ONE cell! 1.1.1.Example. Fig. 11 In Quarter 5 there are five empty cells. All of them, except for the cell with coordinates (5,5), are “bits” in triplets (broken cells are indicated by red crosses), and in this “unbeaten” cell we will enter the result number - CR3 (5,5). 1.1.2.Example with an empty square. Analysis: Fig. 11A. Square 4 is empty, but all its cells, except one, are “bits” with the numbers 7 (broken cells are indicated by red crosses). In this one “unbeaten” cell with coordinates (3.5) we will enter the result number - CR7 (3.5). 1.1.3. Let us analyze the following small squares in the same way. Having worked with one number (successfully or unsuccessfully) on all the squares that do not contain it, we move on to another number. If some number is found in all small squares, we make a note about it. Having finished working with nine, we go back to one and work through all the numbers again. If the next pass does not produce results, then move on to other methods outlined below. The MK method is the simplest, with its help you can solve only the simplest sudoku puzzles (Fig. 11B). Black color - ref. comp., green color- first circle, red - second, third circle - empty cells for CR2. For a better understanding of the matter, I recommend drawing the initial state (black numbers) and going through the entire solution path. 1.1.4. To solve complex Sudoku, it is good to use this method in conjunction with technique 1.12. (half-pairs), marking with small numbers absolutely ALL half-pairs that occur, be it straight, diagonal, corner. 1.2.Method of rows and columns - SiS.* St - column; Page - line. When we see that there is one empty cell left in a particular column, small square or row, we easily fill it. If it doesn’t come to this, and the only thing we managed to achieve was two free cells, then we enter the two missing numbers into each of them - this will be a “pair”. If three empty cells are in the same row or column, then enter the three missing numbers in each of them. If all three empty cells were in one small square, then it is considered that they are now filled and do not participate in further searches in this small square. If there are more empty cells in any row or column, then we use the following techniques. 1.2.1.SiSa. For each missing digit, we check all free cells. If there is only ONE “unbeaten” cell for a given missing digit, then we set this digit in it, this will be the result digit. Fig. 12a: An example of solving a simple Sudoku using the SiSa method.
The red color shows the CRs found as a result of the analysis of columns, and the green color - as a result of the analysis of rows. Solution. Art. 5 there are three empty cells, two of them are bits of twos, and one is not a bit, we write 2-1 in it. Next we find 6-2 and 8-3. Page 3 has five empty cells, four cells are filled with fives, and one is not, so we write 5-4 into it. Art. 1 has two empty cells, one bit is one, and the other is not, we write 1-5 in it, and 3-6 in the other. This Sudoku can be solved to the end using only one SiS technique. 1.2.2.SiSb. If using the CC criterion does not allow you to find more than a single digit of the result (all rows and columns have been checked and everywhere for each missing digit there are several “unbeaten” cells), then you can search among these “unbeaten” cells for one that is “bit” by all the others with the missing digits, except one, and put this missing digit in it. We do it as follows. We write down the missing numbers of any row and check all the columns intersecting this row in empty cells for compliance with criterion 1.2.2. Example. Fig. 12. Line 1: 056497000 (zeros indicate empty cells). The missing numbers in row 1 are: 1238. In row 1, the empty cells are the intersections with columns 1,7,8,9, respectively. Column 1: 000820400. Column 7: 090481052. Column 8: 000069041. Column 9: 004073000.
Analysis: Column 1 “hits” only the two missing digits of the line: 28. Column 7 “hits” three digits: 128, this is what we need, the missing digit 3 remained unbeaten, we will write it in the seventh empty cell of row 1, this is will be the result number CR3(7,1). Now NC Page 1 -128. St. 1 “beats” the two missing numbers (as mentioned earlier) -28, the number 1 remains unbeaten, we write it in the first square cell of St. 1, we get CR1(1,1) (it is not shown in Fig. 12) . With some skill, we perform SiSa and SiSb checks simultaneously. If you analyzed all the rows in this way and did not get a result, then you need to carry out a similar analysis with all the columns (now writing out the missing numbers of the columns). 1.2.3.Fig. 12B: An example of solving a more complex Sudoku using the techniques MK - green, SiSa - red and SiSb - blue. Let's consider the use of the SiSb technique. Search 1-8: Page 7, there are three empty cells in it, cell (8,7) is a two and a nine, but not a one, one will be the CR in this cell: 1-8. Search 7-11: Page 8, there are four empty cells in it, cell (8,8) is bit by one, two and nine, but not by seven, it will be the CR in this cell: 7-11. Using the same technique we find 1-12. 1.3. Joint analysis of a row (column) with a small square.* Example. Fig. 13. Square 1: 013062045. Missing numbers of square 1: 789 Line 2: 062089500. Analysis: Line 2 “beats” an empty cell in the square with coordinates (1,2) with its numbers 89, the missing number 7 in this cell is “unbeaten” and it will be result in this cell is CR7(1,2). 1.3.1.Empty cells are also capable of “beating”. If in a small square only one small row (three numbers), or one small column is empty, then it is easy to calculate the numbers that are latently present in this small row or small column and use their “beat” property for your own purposes. 1.4. Joint analysis of a square, row and column.* Example. Fig. 14. Square 1: 004109060. Missing digits of Square 1: 23578. Row 2: 109346002. Column 2: 006548900. Analysis: Row 2 and column 2 intersect in an empty cell of square 1 with coordinates (2,2). The row “beats” this cell with the numbers 23, and the column with the numbers 58. The missing number 7 remains unbeaten in this cell, and it will be the result: CR7(2,2). 1.5.Local tables. Couples. Triads.* The technique consists of constructing a table similar to that described in Chapter 2, with the difference that the table is not built for the entire working field, but for one structure - a row, column or small square, and in applying the techniques described in the above chapter . 1.5.1.Local table for the column. Couples. We will demonstrate this technique using the example of solving a Sudoku of medium complexity (for a better understanding, you must first read Chapter 2. This is the situation that arose when solving it, black and green numbers. The initial state is black numbers. Fig. 15.
Column 5: 070000005 Missing digits of column 5: 1234689 Square 8: 406901758 Missing digits of square 8: 23 Two empty cells in square 8 belong to column 5 and they will contain a pair: 23 (for pairs, see 1.7, 1.9 and 2.P7. a)), this pair made us pay attention to column 5. Now let’s create a table for column 5, for which we write all its missing numbers in all the empty cells of the column, table 1 will take the form: Let's cross out in each cell the numbers identical to the numbers in the line to which it belongs and in the square, we get table 2: We cross out in other cells the numbers identical to the digits of the pair (23), we get table 3: In its fourth line there is the result digit CR9 (5,4). Taking this into account, column 5 will now look like: Column 5: 070900005 Row 4: 710090468 Further solving this sudoku will not present any difficulties. The next digit of the result is 9(6.3). 1.5.2.Local table for a small square. Triads. Example in Fig. 1.5.1.
Ref. comp. - 28 black numbers. Using the MK technique we find CR 2-1 - 7-14. Local table for Quarter 5. NC - 1345789; Fill out the table, cross out ( green) and we get a triad (triad - when in three cells of any one structure there are three identical CNs) 139 in cells (4,5), (6,5) and in cell (6,6) after purification from the five (purification , if there are options, you need to do it very carefully!). We cross out (in red) the numbers that make up the triad from other cells, we get CR5(6,4)-15; cross out the five in the cell (4,6) - we get CR7(4,6)-16; cross out the sevens - we get a pair of 48. We continue the solution. Small example for cleansing. Let's assume that lok. table for Kv.2 it looks like: 4, 6, 3, 189, 2, 189, 1789, 5, 1789; You can get a triad by clearing one of the two cells containing NC 1789 from the seven. Let’s do this, in the other cell we will get CP7 and continue working. If, as a result of our choice, we come to a contradiction, then we will return to the point of choice, take another cell for cleansing and continue the solution. In practice, if the number of missing numbers in a small square is small, then we don’t draw a table, we perform the necessary actions in our minds, or we simply write the NC on a line to make the work easier. When performing this technique, you can fit up to three digits. Although my drawings have no more than two numbers, I did this for better legibility of the drawing! 1.6.Logical approach* 1.6.1.A simple example. When making a decision, a situation arose. Fig. 161, without the red six.
Analysis.Q.6: QR6 should be either in the upper right cell or in the lower right. Square 4: there are three empty cells in it, the lower right one contains a six, and one of the upper ones may contain a six. This six will hit the top cells in Square 6. This means that the six will be in the lower right cell Kv6.: CR6 (9,6). 1.6.2. A beautiful example. Situation.
In Kv2, CR1 will be located in cells (4,2) or (5,2). In Kv7, CR1 will be located in one of the cells: (1,7); (1.8); (1.9). As a result, all cells in Kv1 will be beaten with the exception of cell (3,3), which will contain CR1(3,3). Next, we continue the solution to the end using the techniques outlined in 1.1 and 1.2. Track. CR: CR9(3.5); CR4(3.2); CR4(1.5); Tsr4(2.8), etc. 1.7. Reliance on undisclosed pairs.* An undisclosed pair (or simply a pair) is two cells in a row, column or small square, which contain two identical missing numbers, unique for each of the above-described structures. A pair can appear naturally (there are two empty cells left in the structure), or as a result of a targeted search for it (this can happen even in an empty structure). After opening, the pair contains one result digit in each cell. An undisclosed pair can: 1.7.1. By its very presence, occupying two cells simplifies the situation by reducing by two the number of missing numbers in the structure. When analyzing rows and columns, unexpanded pairs are perceived as expanded if they are entirely within the body of the analyzed page. (Art.) (in Fig. 1.7.1 - pairs E and D, which are entirely located in the body of the analyzed page 4), or are entirely located in one of the small squares through which the anal passes. Page (Art.) not being part of her (him) (in the figure - pairs B, C). OR the pair is partially or completely outside such squares, but is located perpendicular to the anal. Page (Art.) (in the Fig. - pair A) and can even cross it (him), again without being part of it (him) (in the Fig. - pairs G, F). IF ONE cell of an undisclosed pair belongs to anal, Page. (St.), then during the analysis it is considered that in this cell there can only be digits of this pair, and for the rest NC. Page (v.) this cell is occupied (in the Fig. - pairs K, M). A diagonal unopened pair is perceived as open if it is entirely located in one of the squares through which the anal passes. (art.) (in Fig. - pair B). If such a pair is located outside these squares, then it is not taken into account at all in the analysis (pair H in Fig.). A similar approach is used in small squares analysis. 1.7.2.Participate in the creation of a new pair. 1.7.3. Reveal another pair if the pairs are located perpendicular to each other, or the pair to be revealed is diagonal (the cells of the pair are not on the same horizontal or vertical). The technique is good for use in empty squares, and when solving minimal sudokus. Example, Fig. A1.
The original numbers are black, without indexes. Apartment 5 is empty. We find the first CRs with indices 1-6. Analyzing Square 8 and Page 9, we see that in the upper two cells there will be a pair of 79, and in the bottom line of the square there will be the numbers 158. The lower right cell of the bit is filled with numbers 15 from Article 6 and in it there will be CR8 (6,9 )-7, and in two adjacent cells there is a pair of 15. In Page 9 the numbers 234 remain undefined. Looking at Article 7, we see that tsr2(7,9)-8 has places to be. Now empty Apt. 5. Sevens hit the two left columns and the middle row, and sixes do the same. The result is pair 76. The eights hit the top and bottom rows and the right column - pair 48. We find CR3(5,6), index 9 and CR1(4,6), index 10. This unit reveals the pair 15 - CR5(4,9 ) and CR1(5,9) indices 11 and 12. (Figure A2).
Next, we find the CR with indices 13-17. Page 4 contains a cell with the numbers 76 and an empty cell, broken by a seven, put CR6(1,4) index 18 into it and open the pair 76 CR7(6,4) index 19 and CR6( 6,6) index 20. Next we find the CR with indices 21 - 34. CR9(2,7) index 34 reveals the pair 79 - CR7(5,7) and CR9(5,8) indexes 35 and 36. Next we find the CR with indices 37 - 52. Four with ind.52 and eight with ind.53 reveals the pair 48 - CR4(4.5) ind.54 and TsR8(5,5) ind.55. The above techniques can be used in any order. 1.8.An example of solving a complex Sudoku. Fig.1.8. To better perceive the text and benefit from its reading, the reader must draw the playing field in its original state and, guided by the text, consciously fill in the empty cells. The initial state is 25 black digits. Using the techniques of Mk and SiSa we find the CR: (red) 3(4.5)-1; 9(6.5); 8(5.4) and 5(5.6); further: 8(1.5); 8(6.2); 4(6.9); 8(9.8); 8(8.3); 8(2,9)-10; pairs: 57, 15, 47; 7(3.5)-12; 2-13; 3-14; 4-15; 4-16 reveals pair 47; pair 36(Q4); To find 5(8,7)-17 we use a logical approach. In Q2 the five will be in the top line, in Q3. the five will be in one of the two empty cells of the bottom row, in Square 6 the five will appear after the pair 15 is revealed in one of the two cells of the pair, based on the above, the five in Square 9 will be in the middle cell of the top row: 5(8,7)- 17(green). Para 19 (Art. 8); Page 9 two empty cells in Kv.8 are bits with three and six, we get a chain of pairs 36 We build a local table for art. 4: we cross out, in the bottom cell we get - 19 (4,9). The result is a chain of pairs 19. 7(5,9)-18 reveals pair 57; 4-19; 3-20; pair 26; 6-21 reveals the chain of pairs 36 and pair 26; pair 12(Page 2); 3-22; 4-23; 5-24; 6-25; 6-26; pair 79 (St. 2) and pair 79 (Sq. 7; pair 12 (St. 1) and pair 12 (St. 5); 5-27; 9-28 reveals pair 79 (Sq. 1), chain of pairs 19, chain pars 12; 9-29 reveals pair 79 (Square 7); 7-30; 1-31 reveals pair 15. End. 1.9. Willful disclosure of pairs and Sudoku with an ambiguous solution. 1.9.1. This paragraph and paragraph 1.9.2 . You don’t have to read it during initial reading. These points can be used to solve sudokus that are not quite correctly composed, which is now a rare occurrence. Volitional opening of pairs is used when the use of other techniques does not produce results. The decision you make may turn out to be wrong, you will determine this , when you notice that you have two identical numbers in any structure, or you are trying to do this. In this case, you need to change your choice when revealing the pair to the opposite one and continue the solution from the point of revealing the pair.
Example Fig. 190. Solution. Ref. comp. 28 black numbers, we use the techniques - MK, SiSa and once - SiSb - 5-7; after 1-22 - para37; after 1-24 - pair 89; 3-25; 6-26; pair 17; two pairs of 27 - red and green. dead end. We open the voluntarist pair 37, which causes the opening of pair 17; further - 1-27; 3-28; dead end. We open the chain of pairs 27; 7-29 - 4-39; 8-40 reveals a pair of 89. That's it. We were lucky that during the solution all the pairs were revealed correctly, otherwise we would have had to go back and reveal the pairs alternatively. To simplify the process, the volitional disclosure of pairs and further decisions must be made in pencil, so that in case of failure, new numbers can be written in ink. 1.9.2. Sudokus with ambiguous solutions have not one, but several correct solutions.
Example. Fig. 191. Solution. Ref. comp. 33 numbers in black. We find green CRs up to 7(9.5)-21; four pairs of green - 37,48,45,25. Dead end. A chain of pairs 45 is opened at random; we find new pairs of red color59,24; open pair 25; new pair 28. Open pairs 37,48 and find 7-1 red, new. pair 35, open it and find 3-2, also red: new pairs 45,49 - open them, taking into account the fact that their parts are in the same Square 2, where there are fives; then pairs24,28 are revealed; 9-3; 5-4;8-5. In Fig. 192 I show the second solution, two more options are shown in Fig. 193, 194 (see illustration). 1.10.Non-pairs. A non-pair is a cell with two different numbers, the combination of which is unique to a given structure. if the structure contains two cells with a given combination of numbers, then this is a pair. Non-pairs appear as a result of using local tables or as a result of their targeted search. They are revealed as a result of prevailing conditions, or by a volitional decision. Example. Fig. 1.101. Solution. Ref. comp. - 26 black numbers. Find the CR (green): 4-1 - 2-7; pairs 58,23,89,17; 6-8; 2-9; Square 3 bits in pairs 58 and 89 - find 8-10; 5-11 - 7-15; pair 17 opens; pair 46 is revealed by the six from Art. 1; 6-16; 8-17; pair 34; 5-18 - 4-20; Lock. table for Art. 1: non-pair 13; CR2-21; nonpara 35. Lok. table for St.2: unpaired 19,89,48,14. Lock. table for St.3: unpaired 39,79,37. In Art. 6 we find unpair 23 (red), it forms a chain of pairs with a green pair; in this live station. we find pair 78, it reveals pair 58. Deadlock. By a volitional decision we open the chain of unpairs starting from 13(1,3), including the pairs: 28,78,23,34. We find 3-27. Dot. 1.11. Combined use of two techniques. C&S techniques can be used in conjunction with the “logical approach” technique; we will show this using the example of solving Sudoku in which the “logical approach” technique and the C&S technique are used together. Fig. 11101. Ref. comp. - 28 black numbers. We easily find: 1-1 - 8-5. Page 2. NC - 23569, cell (2,2) is marked with the numbers 259, if it were also marked with a six, then the matter would be in the bag. but such a six virtually exists in Q4, which is beaten by two sixes from Q5. and Kv6. Thus we find CR3(2,2)-6. We find a pair of 35 in Q4. and Page 5; 2-7; 8-8; pair 47. To find non-pairs, we analyze the lock. table: Page 4: NC - 789 - non-para 78; Page 2: NC - 2569 - unpaired 56.29; Page 5: NC - 679 - non-para 67; Square 5: NC - 369 - non-para 59; Quarter 7: nc - 3479 - unpaired 37.39; Dead end; We open the pair 47 by a strong-willed decision; we find 4-9,4-10,8-11 and a pair of 56; we find pairs 67 and 25; pair 69, which reveals non-pair 59 and a chain of pairs 35. Pair 67 reveals non-pair 78. Next we find 9-12; 9-13; 2-14; 2-15 reveals a pair of 25; we find 4-16 - 8-19; 6-20 reveals a pair of 67; 9-21; 7-22; 7-23 reveals unpaired 37, 39; 7-24; 3-25; 5-26 reveals pairs 56, 69 and unpaired 29; we find 5-27; 3-28 - 2-34. Dot. 1.12.Half-pairs* 1.12.1.If, when using MK or SiSa techniques, we fail to find that single cell for a certain CR in a given structure, and all we have achieved are two cells in which the desired CR will presumably be located (for example, 2 Fig, 1.12.1), then we enter the small required number 2 in one corner of these cells - this will be a half-pair. 1.12.2. A straight semi-pair, during analysis can sometimes be perceived as a CR (in the longitudinal direction). 1.12.3. With further search, we can determine that another number (for example 5) claims to be the same two cells in this structure - this will already be pair 25, we write it in a normal font. 1.12.4. If for one of the cells of the half-pair we found another CR, then in the second cell we update its own digit as a CR. 1.12.5.Example. Fig.1.12.1. Ref. comp. - 25 numbers in black. We begin the search for CR using the MK technique. We find semi-pairs 1 in Q.6 and Q.8. half-pair 2 - in Q4, half-pair 4 - in Q2 and Q4, half-pair from Q4 we use the “logical approach” and find CR4-1; Here half pair 4 from Q4 is represented for Q7 as CR4 (as mentioned above). half pair 6 - in Q2 and use it to find CR6-2; half-pair 8 - in square 1; half pair 9 - in Q4 and use it to find CR9-3. 1.12.6. If there are two identical half-pairs (in different structures), and one of them (straight line) is perpendicular to the other, and hits one of the cells of the other, then we install a CR in the unbeaten cell of the other half-pair. 1.12.7. If two identical straight half-pairs (not shown in Fig.) are located in the same way in two different squares relative to rows or columns and parallel to each other (suppose: Square 1. - half-pair 5 in cells (1,1) and ( 1,3), and in Q. 3. - semi-pair 5 in cells (7.1) and (7.3), these semi-pairs are located in the same way relative to the rows), then the desired CR, unambiguous with the semi-pairs in the second square, will appear in the row (or column ) not used (..om) in half-pairs. In our example, CR5 is in Kv.2. will be located on Page 2. The above is also true for the case when there is a half-pair in one square and a pair in the other. See picture: Pair 56 in Quarter 7 and half pair 5 in Quarter 8 (in Line 8 and Line 9), and the result of CR5-1 in Quarter 9 in Line 7. Considering the above, to successfully promote the solution to initial stage It is necessary to mark ABSOLUTELY ALL half-pairs! 1.12.8.Interesting examples related to semi-pairs. In Figure 1.10.2. small square 5 is completely empty, it contains only two half-pairs: 8 and 9 (red). In small squares 2,6 and 8, among other things, there are semi-pairs 1. In small square 4 there is pair 15. The interaction of this pair and the above semi-pairs gives CR1 in small square 5, which in turn also gives CR8 in the same square!
In Figure 1.10.3. in small square 8 there are CRs: 2,3,6,7,8. There are also four semi-pairs: 1,4,5 and 9. When CR 4 appears in square 5, it generates CR4 in square 8, which in turn begets CR9, which in turn generates CR5, which in turn generates CR1 (on not shown in the figure).
1.13. Solving Sudoku with a small initial number of digits. Nontriads. The minimum initial number of digits in a sudoku is 17. Such sudokus often require the volitional disclosure of a pair (or pairs). When solving them, it is convenient to use nontriads. A nontriad is a cell in any structure in which the three missing digits of the NC are located. Three non-triads in the same structure containing the same NCs form a triad. 1.14.Quadro. Quad - when four cells of any one structure contain four identical CNs. We cross out similar numbers in other cells of this structure. 1.15.Using the above techniques, you will be able to solve Sudoku different levels difficulties. You can start the solution by using any of the above techniques. I recommend starting from the very beginning simple method Small Squares MK (1.1), noting ALL half-pairs (1.12) that you discover. It is possible that these semi-pairs will eventually turn into pairs (1.5). It is possible that identical half-pairs interacting with each other will determine the CR. Having exhausted the possibilities of one technique, move on to using others, having exhausted them, return to the previous ones, etc. If you cannot make progress in solving Sudoku, try opening the pair (1.9) or using the tabular solution algorithm described below, find several CRs and continue solving using the above techniques. 2. TABLE ALGORITHM FOR SOLVING SUDOKU. This and subsequent chapters may not be read during initial reading. A simple algorithm for solving Sudoku is proposed; it consists of seven points. Here is the algorithm: 2.P1.Draw a Sudoku table in such a way that nine numbers can be entered into each small cell. If you draw on paper in a square, then each Sudoku cell can be made 9 cells in size (3x3) 2.P2.In each empty cell of each small square we enter all the missing numbers of this square. 2.P3.For each cell with missing numbers, look through its row and column and cross out the missing numbers that are identical to the result numbers found in the row or column outside the small square to which the cell belongs. 2.P4.Look through all the cells with missing numbers. If there is only one number left in any cell, then this is the RESULT NUMBER (DR). We circle it. Having circled all the CRs, we move on to step 5. If the next execution of step 4 does not produce results, then go to step 6. 2.P5.We look through the remaining cells of the small square and cross out the missing numbers in them that are identical to the newly obtained result number. Then we do the same with the missing numbers in the row and column to which the cell belongs. Let's move on to step 4. If the Sudoku level is easy, then the further solution is to alternately perform steps 4 and 5. 2.P6.If the next execution of step 4 does not produce results, then we look through all the rows, columns and small squares for the following situation: If in any row, column or small square one or more missing digits appear only once together with other digits appearing repeatedly, then she or they are RESULT NUMBERS (RD). For example, if a row, column or small square looks like: 1,279,5,79,4,69,3,8,79 Then Numbers 2 and 6 are CR because they are present in the row, column or small square in a single copy, circle them circle, and the numbers standing nearby cross it out. In our example, these are the numbers 7 and 9 near the two and the number 9 near the six. The row, column or small square will look like: 1,2,5,79,4,6,3,8,79. Let's move on to step 5. If the next execution of step 6 does not produce results, then go to step 7. 2.P7.a) We look for a small square, row, or column in which two cells (and only two cells) contain the same pair of missing digits, as in this line (pair-69): 8,5,69,4 ,69,7,16,1236,239. and we cross out the numbers that make up this pair (6 and 9), located in other cells - this way we can get the CR, in our case - 1 (after crossing out the six in the cell where the numbers were - 16). The line will look like: 8,5,69,4,69,7,1,123,23. After completing step 5, our line will look like this: 8,5,69,4,69,7,1,23,23. If there is no such pair, then you need to look for them (they can exist in an implicit form, as in this line): 9,45,457,2347,1,6,237,8,57 here pair 23 exists in an implicit form. Let’s “clear” it, the line will take the form: 9,45,457,23,1,6,23,8,57 Having carried out such a “cleaning” operation on all rows, columns and small squares, we will simplify the table and, perhaps, (see P. 6) we will receive a new CR. If not, then you will have to make a choice in some cell from two result values, for example, in the column: 1,6,5,8,29,29,4,3,7. Two cells each have two missing numbers: 2 and 9. You need to decide and choose one of them (circle it) - turn it into a CR, and cross out the second in one cell and do the opposite in the other. Even better, if there is a chain of pairs, then for greater effect it is advisable to use it. A chain of pairs is two or three pairs of identical numbers arranged in such a way that the cells of one pair belong to two pairs at the same time. An example of a chain of pairs formed by pair 12: Line 1: 3,5,12,489,489,48,12,7,6. Column 3: 12,7,8,35,6,35,12,4,9. Small square 7: 8,3,12,5,12,4,6,7,9. In this chain, the upper cell of the column pair also belongs to the first row pair, and the bottom cell of the column pair is part of the seventh small square pair. Let's move on to step 5. Our choice (p7) will be either correct and then we will solve the Sudoku to the end, or incorrect and then we will soon discover this (two identical digits of the result will appear in one row, column or small square), we will have to go back, make a choice opposite to the one previously made and continue the solution until victory. Before choosing, you must make a copy of the current state. The choice should be made last after b) and c). Sometimes the choice in one pair is not enough (after identifying several TAs, progress stops); in this case, it is necessary to reveal another pair. This happens in complex Sudoku. 2.P7.b) If the search for pairs is unsuccessful, we try to find a small square, a row or column in which three cells (and only three cells) contain the same triad of missing numbers, as in this small square (triad - 189): 139,2,189,7,189,189,13569,1569,4. and cross out the numbers that make up the triad (189), located in other cells - this way we can get the CR. In our case, this is 3 - after crossing out the missing numbers 1 and 9 in the cell where the numbers 139 were. The small square will look like: 3,2,189,7,189,189,356,56,4. After completing step 5, our small square will take the form: 3,2,189,7,189,189,56,56,4. 2.P7.c) If you are unlucky with triads, then you need to carry out an analysis based on the fact that each row or column belongs to three small squares, consisting, as it were, of three parts and if in some square some figure belongs to one row (or column) only in this square, then this figure cannot belong to the other two rows (columns) in the same small square. Example. Consider the small squares 1,2,3 formed by lines 1,2,3. Page 1: 12479.8.123479;1679.5.679;36.239.12369. Page 2: 1259.1235.6;189.4.89;358.23589.7. Page 3: 1579.15.179;3.179.2;568.4.1689. Quarter 3: 36.239.12369;358.23589.7;568.4.1689. It can be seen that the missing numbers 6 in Line 3 are found only in Quarter 3, and in Line 1 - in Quarter 2 and Quarter 3. Based on the above, we cross out the numbers 6 in the cells of Page 1. in Q3., we get: Line 1: 12479.8.123479;1679.5.679;3.239.1239. We got Tsr 3(7.1) in Q3. After completing P.5, the line will look like: Page 1: 12479,8,12479;1679,5,679;3,29,129. A Kv3. will look like: Q3: 3.29.129;58.2589.7;568.4.1689. We carry out this analysis for all numbers from 1 to 9 in rows sequentially for triplets of squares: 1,2,3; 4,5,6; 7,8,9. Then - in columns for triplets of squares: 1,4,7; 2,5,8; 3,6,9. If this analysis does not give a result, then we go to a) and make a choice in pairs. Working with a table requires great care and attention. Therefore, having identified several CRs (5 - 15), you need to try to move forward using simpler techniques outlined in I. 3. PRACTICAL INSTRUCTIONS. In practice, step 3 (crossing out) is performed not for each cell separately, but for the whole row or for the whole column at once. This speeds up the process. It is easier to control crossing out if crossing out is done in two colors. Crossing out rows is one color, and crossing out columns is another color. This will allow you to control the deletion not only for under-deletions, but also for its excess. Next we perform step 4. We look at all cells with missing result numbers only the first time step 4 is performed after step 3 is completed. During subsequent executions of step 4 (after performing step 5), we look through one small square, one row and one column for each newly obtained result digit (RD). Before performing step 7, in the event of a volitional disclosure of the pair, you need to make a copy of the current state of the table in order to reduce the amount of work if you have to return to the choice point. 4.EXAMPLE OF SOLVING SUDOKU BY TABLE METHOD. To consolidate the above, let's solve a Sudoku of medium difficulty (Fig. 4.3). The result of the solution is shown in Fig.4.4. START P.1.Draw a large table. P.2.In each empty cell of each small square we enter all the missing numbers of the result of this square (Fig. 1). For small square N1 it is 134789; for a small square N2 it is 1245; for small square N3 it is 1256789, etc. P.3. We carry out in accordance with the practical instructions for this paragraph (See). P.4. We look through ALL cells with missing result numbers. If there is only one number left in some cell, then this is the CR, circle it. In our case, these are CR5(6,1)-1 and CR6(5,7)-2. We transfer these numbers to the Sudoku playing field. The table after completing step 1, step 2, step 3 and step 4 is shown in Fig. 1. Two CRs discovered during step 4 are circled; these are 5(6,1) and 6(5,7). Those who want to get a complete understanding of the solution process should draw themselves a table with the original numbers, independently perform step 1, step 2, step 3, step 4 and compare your table with Fig. 1, if the pictures are the same, then you can move on. This is the first checkpoint. Let's continue with the solution. Those who wish to participate can mark its stages in their drawing. P.5. Cross out the number 5 in the cells of small square N2, row N1 and column N6, these are “fives” in cells with coordinates: (9,1), (4,2), (6,5) and (6,6 ); cross out the number 6 in the cells of small square N8, row N7 and column N5, these are “sixes” in cells with coordinates: (6,8), (2,7), (3,7), (5,4) and (5 ,5)(5,6). In Fig. 1 they are crossed out, but in Fig. 2 they are no longer there at all. In Fig. 2, all previously crossed out numbers have been removed, this is done to simplify the drawing. According to the algorithm, we return to A.4. P.4. CR9(5,5)-3 was discovered, circle it and move it. Step 5. Cross out the “nines” in the cells with coordinates: (5,6) and (9,5), go to step 4. P.4 No result. Let's move on to step 6. P.6. In the small square N8 we have: 78, 6, 9, 3, 5, 47, 47, 2, 1. The number 8 (4,7) appears only once - this is CR8-4, we circle it, and the next number is 7 cross it out. Let's move on to step 5. P.5. Cross out the number 8 in the cells of row N7 and column N4. Let's move on to point 4. P.4. No result. P.6. In the small square N9 we have: 257, 25, 4, 2789, 289, 1, 79, 6, 379. The number 3(9,9) appears once - this is CR3(9,9)-5, circle it, move it (see Fig. 4.4), and cross out the adjacent numbers 7 and 9. P.5. Cross out the number 3 in the cells of row N9 and column N9. P.4. No result. P.6. In the small square N2 we have: 6, 7, 5, 24, 8, 3, 9, 14, 24. Number 1 (5,3) - CR1-6, circle it. P.5. We cross it out. P.4 No result. P.6. In the small square N1 we have: 18, 2, 19, 6, 1479, 179, 5, 347, 37. Number 8 (1,1) - CR8-7, circle it. P.5. We cross it out. P.4. Numbers 9 (9,1) - TsR9-8, circle it. P.5. We cross it out. P.4. Number 1(3,1) - CR1-9. P.5. We cross it out. P.4. No result. P.6. Line N5, we have: 12, 8, 4, 256, 9, 26, 3, 7, 56. Number 1 (1.5) - CR1-10, circle. P..5. We cross it out. P.4. No result P.6. Column N2 we have: 2, 479, 347, 367, 8, 367, 137, 4679, 5. Number 1 (2,7) - CR1-11. This is the second checkpoint. If your drawing uv. reader, this place completely coincides with Fig. 2, then you are on the right track! Continue to fill it out further yourself. P.5. We cross it out. P.4. No result P.6. Column N9 We have: 9, 57, 678, 56, 56, 2, 4, 1, 3. The number 8 (9,3) is CR8-12. P.5. Cross out, P.4. Number 2(8,3) - CR2-13. P.5. We cross it out. P.4 CR5(8.7)-14, CR4(6.3)-15. P.5. We cross it out. P.4. TsR2(4,2)-16, TsR7(6,8)-17, TsR1(8,2)-18. P.5. We cross it out. P,4. TsR4(8,4)-19, TsR4(4,9)-20, TsR6(6,6)-21. P.5. We cross it out. P.4. TsR3(5.4)-22, TsR7(1.9)-23, TsR2(6.5)-24. P.5. We cross it out. P.4 TsR3(1,6)-25, TsR9(7,9)-26, TsR4(5,6)-27. P.5. We cross it out. P.4. CR: 2(1.7)-28, 8(8.8)-29, 5(4.5)-30, 7(2.6)-31. P.5. We cross it out. P.4. CR: 3(3.7)-32, 7(7.7)-33, 4(1.8)-34, 9(8.6)-35, 2(7.8)-36, 6(9 ,5)-37, 7(4,4)-38, 3(2,3)-39, 6(2,4)-40, 5(3,6)-41. P.5. We cross it out. P.4. CR: 7(3.3)-42, 6(7.3)-43, 5(7.2)-44, 5(9.4)-45, 2(3.4)-46, 8(7 ,6)-47, 9(2,8)-48. P.5 Cross out. P.4. CR: 9(3.2)-49, 7(9.2)-50, 1(7.4)-51, 4(2.2)-52, 6(3.8)-53. END! Solving Sudoku using the tabular method is a troublesome task and there is no need in practice to bring it to the very end, just as there is no need to solve Sudoku using this method from the very beginning. 5..shtml

Feb 27, 2015 —

Sudoku is a number puzzle. Today it is so popular that most people are very familiar with it or have simply seen it in printed publications. 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” first published Sudoku on its pages. 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 square field with a side of 9 cells, divided in turn into squares measuring 3 by 3. Thus, the large square is divided into 9 small ones, the total number of cells of which is 81. 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.

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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 of an exciting game, to develop and train logical thinking, memory, and 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.)

Don't let the mention of an electronic game confuse you in the future. You can find the game on the pages of newspapers or magazines, the basic principle remains the same.

Electronic version of the game provides great opportunities, by choosing 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 numbers, as a rule, in the electronic version of the game 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, it is possible to use a hint. 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. There is no doubt about the correctness of this solution option and the fact that all the numbers we considered and excluded were originally given in the task. 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 right decision 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 it will be useful for you, and will allow you to join a large number 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") It has 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.

Twin Sudoku ("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.

Good day to you, dear fans of logic games. In this article I want to outline the basic methods, methods and principles of solving Sudoku. There are many types of this puzzle presented on our website, and even more will undoubtedly be presented in the future! But here we will consider only the classic version of Sudoku, as the main one for all others. And all the techniques outlined in this article will also apply to all other types of Sudoku.

Loner or the last hero.

So, where do you start solving Sudoku? It doesn't matter whether the difficulty level is easy or not. But always at the beginning there is a search for obvious cells to fill.

The figure shows an example of a single figure - this is the number 4, which can be safely placed on cell 2 8. Since the sixth and eighth horizontal lines, as well as the first and third verticals, are already occupied by a four. They are shown by green arrows. And in the lower left small square we have only one unoccupied position left. In the picture the number is marked in green. The rest of the singles are arranged in the same way, but without arrows. They are painted blue. There can be quite a lot of such singletons, especially if there are a lot of numbers in the initial condition.

There are three ways to search for singles:

• Single player in a 3 by 3 square.
• Horizontally
• Vertically

Of course, you can randomly browse and identify singles. But it is better to stick to a specific system. The most obvious thing to do is start with number 1.

• 1.1 Check the squares where there is no unit, check the horizontal and vertical lines that intersect the given square. And if they already contain ones, then we eliminate the line completely. Thus, we are looking for the only possible place.
• 1.2 Next, we check the horizontal lines. In which there is a unit, and in which there is not. We check in small squares that include this horizontal line. And if they contain a 1, then we exclude the empty cells of this square from possible candidates for the desired number. We will also check all verticals and exclude those that also contain a single. If the only possible empty space remains, then put the required number. If there are two or more empty candidates left, then we leave this horizontal line and move on to the next one.
• 1.3 Similar to the previous point, we check all horizontal lines.

"Hidden Units"

Another similar technique is called “who, if not me?!” Look at Figure 2. Let's work with the upper left small square. First, let's go through the first algorithm. After which we managed to find out that in cell 3 1 there is a single figure - the number six. We put it, and in all the other empty cells we put in small print all the possible options in relation to the small square.

After which we discover the following: in cell 2 3 there can only be one number 5. Of course, at the moment the 5 can also appear on other cells - nothing contradicts this. These are three cells 2 1, 1 2, 2 2. But in cell 2 3 the numbers 2,4,7, 8, 9 cannot appear, since they are present in the third row or in the second column. Based on this, we rightfully put the number five on this cell.

Naked couple

Under this concept I combined several types of Sudoku solutions: naked pair, three and four. This was done due to their similarity and the only difference is in the number of numbers and cells involved.

So, let's figure it out. Look at Figure 3. Here we put all the possible options in fine print in the usual way. And let's take a closer look at the upper middle small square. Here in cells 4 1, 5 1, 6 1 we have a series of identical numbers - 1, 5, 7. This is a naked three in its true form! What does this give us? And the fact is that only in these cells will these three numbers 1, 5, 7 be located. Thus, we can exclude these numbers in the middle upper square on the second and third horizontal lines. Also in cell 1 1 we will exclude the seven and immediately put four. Since there are no other candidates. And in cell 8 1 we will exclude one; we should think further about four and six. But that's a different story.

It should be said that only a special case of a bare triple was considered above. In fact, there can be many combinations of numbers

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

hidden couple

This method of solving Sudoku will reduce the number of candidates and give life to other strategies. Look at Figure 4. The top middle square is filled with candidates as usual. The numbers are written in small print. Two cells are highlighted in green - 4 1 and 7 1. Why are they remarkable to us? Only these two cells contain candidates 4 and 9. This is our hidden pair. By and large, it is the same couple as in point three. Only in cells there are other candidates. These others can be safely crossed out from these cells.