How to solve complex sudoku ways methods. How to Solve Sudoku: Ways, Methods and Strategy

SUDOKU SOLVING ALGORITHM (SUDOKU) columns.* 1.5.Local tables. Couples. Triads..* 1.6. Logical approach.* 1.7. Reliance on unopened pairs.* 1.8. An example of solving a complex Sudoku 1.9. Volitional opening of pairs and Sudoku with ambiguous solutions 1.10. Non-pairs 1.11. Joint use of two techniques 1.12. Half-pairs.* 1.13. Sudoku solution with a small initial number of digits. Non-triads. 1.14.Quadro 1.15.Recommendations 2.Tabular algorithm for solving Sudoku 3.Practical instructions 4.An example of solving Sudoku in a tabular way 5.Test your skills Note: items not marked with an asterisk (*) can be omitted during the first reading. Introduction Sudoku is a digital puzzle game. The playing field is a large square consisting of nine rows (9 cells in a row, the cells in a row are counted from left to right) and nine columns (9 cells in a column, the cells in a column are counted from top to bottom) in total: (9x9 = 81 cells), broken into 9 small squares (each square consists of 3x3 = 9 cells, the count of squares is from left to right, top to bottom, the count of cells in a small square is from 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 digits: 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 row - (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 - 8(3,2), etc., and the cell in the lower right corner of the playing field has coordinates (9,9). Solve Sudoku - fill in all the empty cells of the playing field with numbers from 1 to 9 in such a way that the numbers are not repeated in any row, column, or small square. The numbers in the filled cells are the result numbers (CR). The numbers that we need to find are the missing numbers - TsN. If three numbers are written in some small square, for example, 158 is CR (commas are omitted, we read: one, two, three), then - NC in this square is - 234679. In other words - solve Sudoku - find and correctly place all the missing numbers, each CN, the place of which is uniquely determined, becomes the CR. In the figures, the CRs are drawn with indices, index 1 determines the CR found first, 2 - the second, and so on. 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 Sudoku solving process. In "diagonal" Sudoku, one more condition is imposed, namely: in both diagonals of the large square, the numbers must also not be repeated. Sudoku usually has one solution, but there are exceptions - 2, 3 or more solutions. Solving Sudoku requires attention and good lighting. Use ballpoint pens. 1. SUDOKU SOLVING TECHNIQUES* 1.1.Small squares method - MK.* This is the simplest Sudoku solving method, it is based on the fact that in each small square, each of the nine possible digits can appear only once. You can start solving the puzzle with it. You can start searching for the CR with any number, usually we start with one (if they are present in the task). We find a small square in which this figure is absent. The search for a cell in which the number we have chosen in this square should be located is as follows. We look through all the rows and columns passing through our small square for the presence of the number we have chosen in them. 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", or are 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 triples (broken cells are indicated by red crosses), and in this "unbeaten" cell we will enter the result number - ЦР3 (5,5). 1.1.2. An example with an empty square. Analysis: Fig.11A. Square 4 is empty, but all its cells, except for one, are "bits" with numbers 7 (broken cells are marked with red crosses). In this one "unbeaten" cell with coordinates (3.5) we will enter the result number - ЦР7 (3.5). 1.1.3. We analyze the following small squares in the same way. Having worked with one digit (successfully or unsuccessfully) all the squares that do not contain it, we move on to another digit. If some figure is found in all small squares, we make a note about it. Having finished working with the nine, we go back to the one and work through all the numbers again. If the next pass does not give results, then proceed to other methods described below. The MK method is the simplest, with its help you can solve only the simplest Sudokus in their entirety Fig.11B. Black color - ref. comp., green color- first circle, red color - second, third circle - empty cells for Tsr2. For a better insight into the essence of the matter, I recommend drawing the initial state (black numbers) and going through the entire solution path. 1.1.4. To solve complex Sudokus, 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, whether they are straight, diagonal, or angular. 1.2. Method of rows and columns - C&S. * St - column; Str - string. When we see that in a particular column, small square or row there is only one empty cage, then easily fill it. If things don’t come to this, and the only thing we managed to achieve is two free cells, then in each of them we enter the two missing numbers - this will be a “pair”. If three empty cells are in the same row or column, then in each of them we enter the three missing numbers. If all three empty cells were in one small square, then it is considered that they are now filled and do not participate in the further search in this small square. If there are more empty cells in any row or column, then we use the following methods. 1.2.1.SiCa. For each missing digit, we check all free cells. If there is only ONE "unbroken" cell for this missing digit, then we set this digit in it, this will be the digit of the result. Fig.12a: An example of solving a simple Sudoku using the CCa method.
The red color shows the TAs found as a result of column analysis, and the green color - as a result of row analysis. Solution. Art.5 there are three empty cells in it, two of them are bits of two, and one is not a bit, we write 2-1 into it. Next we find 6-2 and 8-3. Page 3 there are five empty cells in it, four cells are beaten by fives, and one is not, and we write 5-4 into it. St.1 there are two empty cells in it, one bit is a unit, and the other is not, we write 1-5 into it, and 3-6 into the other. This sudoku can be solved to the end using only one CC move. 1.2.2.SiSb. If, however, the use of the CuCa criterion does not allow finding more than a single digit of the result (all rows and columns are checked, and everywhere for each missing digit there are several “unbroken” cells), then you can search among these “unbroken” cells for one that is “beaten” by all the others missing digits, except for one, and put this missing digit in it. We do it in the following way. We write down the missing digits of any line and check all columns crossing this line by empty cells for compliance with criterion 1.2.2. Example. Fig.12. Line 1: 056497000 (zeroes indicate empty cells). The missing digits of line 1: 1238. In line 1, 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 "beats" only two missing digits of the line: 28. Column 7 - "beats" three digits: 128, this is what we need, the missing number 3 remained unbeaten, and we will write it in the seventh empty cell of line 1, this is will be the digit of the result of CR3 (7,1). Now NTs Str.1 -128. St.1 "beats" the two missing digits (as mentioned earlier) -28, the number 1 remains unbeaten, and we write it in the first poached cell of Page 1, we get CR1 (1,1) (it is not shown in Fig. 12) . With some skill, checks of SiSa and SiSb are performed 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 digits of the columns). 1.2.3.Fig. 12B: An example of solving a more difficult Sudoku using MK - green, SiCa - red and SiSb - blue. Consider the application of the CSB technique. Search 1-8: Page 7, there are three empty cells in it, cell (8,7) is a two and a nine, and a unit is not, a unit 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 one, two and nine, and seven is not, it will be the CR in this cell: 7-11. With 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 digits 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 digit 7 in this cell is "unbite" and it will be the result in this cell is CR7(1,2). 1.3.1. Empty cells are also capable of "beating". If only one small line (three digits) or one small column is empty in a small square, then it is easy to calculate the numbers that are implicitly present in this small line or small column and use their "beat" property for your own purposes. 1.4. Joint analysis of a square, a row and a column. * Example. Fig.14. Square 1: 004109060. Missing digits in 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 in 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 some kind of structure - a row, column or small square, and in applying the techniques described in the above chapter . 1.5.1.Local table for a column. Couples. We will show 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 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 make a table for column 5, for which we write all its missing numbers in all empty cells of the column, table 1 will take the form: We 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 the numbers in other cells identical to the numbers of the pair (23), we get table 3: In its fourth line is the figure of the result CR9 (5,4). With this in mind, column 5 will now look like: Column 5: 070900005 Row 4: 710090468 Further solution of 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 digits. Using the MK technique, we find the CR 2-1 - 7-14. Local table for Quarter 5. NC - 1345789; Fill in the table, cross out ( in green) and we get a triad (triad - when there are three identical CIs in three cells of any one structure) 139 in cells (4.5), (6.5) and in cell (6.6) after cleansing from the five (cleansing , 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; we cross out the five in the cell (4.6) - we get CR7 (4.6) -16; we cross out the sevens - we get a pair of 48. We continue the solution. Small example for cleansing. Let's assume lok. tab. for Quarter 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 CR7 and continue working. If, as a result of our choice, we come to a contradiction, then we will return to the choice point, take another cell for purification and continue the solution. In practice, if the number of missing digits in a small square is small, then we do not draw a table, we perform the necessary actions in the mind, or we simply write out the NC in a line to facilitate work. When performing this technique, you can enter up to three numbers in one Sudoku cell. Although I have no more than two numbers in my drawings, I did this for better legibility of the drawing! 1.6. Logical approach * 1.6.1. A simple example. There was a situation in the decision. Fig. 161, without the red six.
Analysis Q6: CR6 must be either in the upper right cell or in the lower right cell. Square 4: there are three empty cells in it, the lower right of them is a bit with a six, and in some of the upper six there may be. This six will beat the top cells in Q6. This means that the six will be in the lower right cell Q6 .: CR6 (9,6). 1.6.2. A beautiful example. Situation.
In Q2, CR1 will be in cells (4.2) or (5.2). In Kv7 CR1 will be in one of the cells: (1.7); (1.8); (1.9). As a result, all cells in Kv1 will be beaten except for the cell (3,3), in which there will be CR1(3,3). Then we continue the solution to the end using the techniques described in 1.1 and 1.2. Track. CR: CR9(3.5); CR4(3.2); CR4(1.5); Cr4(2,8), etc. 1.7. Reliance on unopened pairs.* An unopened pair (or simply - a pair) is two cells in a row, column or small square, in which there are two identical missing digits, unique for each of the structures described above. A pair can appear naturally (there are two empty cells left in the structure), or as a result of a purposeful search for it (this can happen even in an empty structure). After opening, the pair contains one digit of the result in each cell. An undisclosed pair can: 1.7.1. Already by its mere presence, occupying two cells simplifies the situation by reducing the number of missing digits in the structure by two. When analyzing rows and columns, unexpanded pairs are perceived as expanded if they are entirely in the body of the analyzed Page. (St.) (in Fig.1.7.1 - pairs E and D, which are entirely in the body of the analyzed Page 4), or are entirely in one of the small squares through which the anal passes. Page (St.) not being a part of it (him) (in the figure - pairs B, C). Either the couple is partially or completely outside of such squares, but is located perpendicular to the anal. Page (St.) (in Fig. - pair A) and can even cross it (it), again without being part of it (it) (in Fig. - pairs G, F). IF ONE cell of an undisclosed couple belongs to anal, Pg. (St.), then in the analysis it is considered that in this cell there can only be numbers of this pair, and for the rest NC. Page (St.) this cell is occupied (in the Fig. - pairs K, M). A diagonal unopened pair is perceived as open if it is entirely in one of the squares through which the anal passes. (Art.) (in Fig. - pair B). If such a pair is 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 the analysis of small squares. 1.7.2. Participate in the generation of a new pair. 1.7.3. Open another pair if the pairs are perpendicular to each other, or the pair being opened is diagonal (the cells of the pair are not on the same horizontal or vertical line). The technique is good for use in empty squares, and when solving minimal sudoku. Example, fig.A1.
The original figures are black, without indices. Kv.5 - empty. We find the first CRs with indices 1-6. Analyzing Q. 8 and P. 9, we see that in the upper two cells there will be a pair of 79, and in the bottom line of the square - the numbers 158. The lower right cell of the bit is numbered 15 from Art. 6 and CR8 (6,9 )-7, and in two adjacent cells - a pair of 15. In Page 9, the numbers 234 remain undefined. Looking at Art. Now empty Apt.5. The sevens beat the two left columns and the middle row in it, the sixes do the same. The result is a pair of 76. Eights beat the top and bottom rows and the right column - a pair of 48. We find CR3 (5,6), index 9 and CR1 (4,6), index 10. This unit reveals a pair of 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 beaten 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 a pair of 79 - CR7(5,7) and CR9(5,8) indices 35 and 36. Next, we find the CR with indices 37 - 52. Four with index 52 and eight with index 53 reveal a pair of 48 - CR4 (4.5) ind.54 and CR8 (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. For a better perception of the text and benefit from reading it, 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; couples: 57, 15, 47; 7(3.5)-12; 2-13; 3-14; 4-15; 4-16 reveals the pair 47; pair 36(Square 4); 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 Q.6 the five will appear after the opening of pair 15 in one of the two cells of the pair, based on the above, the five in Q.9 will be in the middle cell of the top row: 5(8,7)- 17 (green). Couple 19 (Art. 8); Page 9 two empty cells of its Q8 bits are three and six, we get a chain of pairs 36 We build a local table for st. The result is a chain of pairs 19. 7(5,9)-18 reveals the pair 57; 4-19; 3-20; pair 26; 6-21 reveals the string of pairs 36 and pair 26; pair 12(Page 2); 3-22; 4-23; 5-24; 6-25; 6-26; pair 79 (Art. 2) and pair 79 (Q. 7; pair 12 (Art. 1) and pair 12 (Art. 5); 5-27; 9-28 reveals pair 79 (Q. 1), a chain of pairs 19, a chain par 12; 9-29 reveal pair 79(Q7); 7-30; 1-31 reveal pair 15. End 1.9. Volitional opening of pairs and sudoku with ambiguous solution. 1.9.1. This paragraph and paragraph 1.9.2 These points can be used to solve Sudokus that are not quite correct, which is now rare when you notice that in any structure you have two same digits, or you are trying to do so. In this case, you need to change your choice when opening the pair to the opposite one and continue the solution from the point of opening the pair.
Example Fig.190. Solution. Ref. comp. 28 black numbers, we use techniques - MK, SiSa and once - SiSb - 5-7; after 1-22 - para37; after 1-24 - pair 89; 3-25; 6-26; couple 17; two pairs of 27 - red and green. dead end. We reveal the voluntarists 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, during the solution all the pairs were opened correctly, otherwise, we would have to go back, alternatively open the pairs. To simplify the process, the volitional disclosure of pairs and the further decision must be done with a pencil, so that in case of failure, write new numbers in ink. 1.9.2 Sudoku with an ambiguous solution has not one, but several correct solutions.
Example. Fig.191. Solution. Ref. comp. 33 black digits. We find green CRs up to 7 (9.5) -21; four green pairs - 37,48,45,25. Dead end. Randomly opened a chain of pairs 45; find new red pairs59,24; open a pair of 25; new pair 28. We 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 one Square 2, where there are fives; pairs are revealed next24,28; 9-3; 5-4; 8-5. In fig.192 I will give 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 for this structure. if there are two cells with a given combination of numbers in the structure, then this is a pair. Non-pairs appear as a result of using local tables or as a result of their targeted search. Revealed as a result of the prevailing conditions, or a strong-willed decision. Example. Fig.1.101. Solution. Ref. comp. - 26 black digits. We find CR (green): 4-1 - 2-7; couples 58,23,89,17; 6-8; 2-9; Square 3 bits in pairs 58 and 89 - we find 8-10; 5-11 - 7-15; pair 17 is revealed; pair 46 opens with a six from Art.1; 6-16; 8-17; pair 34; 5-18 - 4-20; Lok. tab. for St.1: non-pair 13; CR2-21; unpara 35. Loc. tab. for Art.2: non-pairs 19,89,48,14. Lok. tab. for Art.3: non-pairs 39,79,37. In Art.6 we find non-pair 23 (red), it forms a chain of pairs with a green pair; in this wv St. we find a pair of 78, it reveals a pair of 58. Dead end. We open the chain of non-pairs starting from 13(1,3), including pairs: 28,78,23,34 by a strong-willed decision. We find 3-27. Dot. 1.11. Joint use of two techniques. SiS techniques can be used in conjunction with the "logical approach" technique; we will show this on the example of a Sudoku solution in which the "logical approach" technique and the C&S technique are used together. Fig.11101. Ref. comp. - 28 black digits. Easy to find: 1-1 - 8-5. Page 2. NTs - 23569, cell (2,2) is bitten with numbers 259, if it was also bitten with a six, then it would be in the bag. but such a six virtually exists in Quarter 4, which is beaten by two sixes from Quarter 5. and Q6. 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 lok. table: Page 4: NTs - 789 - non-pair 78; Page 2: NTs - 2569 - non-pairs 56.29; Page 5: NC - 679 - non-pair 67; Quarter 5: NTs - 369 - non-para 59; Quarter 7: nc - 3479 - non-pairs 37.39; Dead end; Opening a strong-willed decision couple 47; we find 4-9,4-10,8-11 and a pair of 56; 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; find 4-16 - 8-19; 6-20 reveals the pair 67; 9-21; 7-22; 7-23 reveals the non-pair 37, 39; 7-24; 3-25; 5-26 reveals pairs 56, 69 and non-pair 29; find 5-27; 3-28 - 2-34. Dot. 1.12. Half-pairs * 1.12.1. If, using the methods of MK or SiSa, we cannot find that single cell for a certain CR in this structure, and all we have achieved is two cells in which the desired CR will presumably be located (for example, 2 Fig. 1.12.1), then we enter in one corner of these cells the small required number 2 - this will be a half-pair. 1.12.2. A straight half-pair, in the analysis can sometimes be perceived as a CR (in the direction along). 1.12.3. With further search, we can determine that another number (for example, 5) claims the same two cells in this structure - this will already be a pair of 25, we write it in a normal font. 1.12.4. If for one of the cells of the half-pair we have found another CR, then in the second cell we update its own digit as the CR. 1.12.5 Example. Fig.1.12.1. Ref. comp. - 25 black digits. We begin the search for the CR using the MK technique. We find half-pairs 1 in Q.6 and Q.8. half-pair 2 - in Q.4, half-pair 4 - in Q.2 and Q.4, half-pair from Q.4 we use the "logical approach" in the technique and find TsR4-1; Here semi-pair 4 from Q4 is represented for Q7 as CR4 (which was mentioned above). half-pair 6 - in Quarter 2 and use it to find CR6-2; half-pair 8 - in square 1; half-pair 9 - in Quarter 4 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 beats one of the cells of the other, then we set the CR in the unbeaten cell of the other half-pair. 1.12.7. If two identical straight half-pairs (not shown in the 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 required one-to-one with the semi-pairs CR in the second square will be in the row (or column ) not used (..om) in semi-pairs. In our example, TA5 is in Quarter 2. will be in 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 Q7 and semi-pair 5 in Q8 (in Page 8 and Page 9), and result CR5-1 in Q9 in Page 7. Considering the above, in order to successfully promote the solution on initial stage it is necessary to mark ABSOLUTELY ALL semi-pairs! 1.12.8. Interesting examples related to semi-pairs. Figure 1.10.2. small square 5 is absolutely empty, it contains only two half-pairs: 8 and 9 (red color). In the small squares 2,6 and 8, among other things, there are half-pairs 1. In the small square 4 there is a pair 15. The interaction of this pair and the above half-pairs gives CR1 in the small square 5, which in turn also gives CR8 in the same square!
Figure 1.10.3. in the small square 8 are CR: 2,3,6,7,8. There are also four half-pairs: 1,4,5 and 9. When CR 4 appears in square 5, it generates CR4 in square 8, which in turn generates CR9, which in turn generates CR5, which in turn generates CR1 (on not shown).
1.13. Sudoku solution with a small initial number of digits. Non-triads. The minimum initial number of digits in a Sudoku is 17. Such Sudokus often require the willful opening of a pair (or pairs). When solving them, it is convenient to use nontriads. A non-triad is a cell in some structure in which there are three missing numbers of NC. Three non-triads in one structure containing the same NC form a triad. 1.14.Quad. Quadro - when four identical CNs are located in four cells of any one structure. 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 methods. I recommend starting from the very beginning simple method Small Squares MK (1.1), marking ALL half-pairs (1.12) that you find. It is possible that these semi-pairs will turn over time 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, proceed to the use of others, having exhausted them, return to the previous ones, etc. If you can't get ahead in sudoku solving, try opening a pair (1.9) or using the table solution algorithm described below, find several DOs and continue the solution using the above techniques. 2. TABLE ALGORITHM FOR SOLVING SUDOKU. This and subsequent chapters can not be read at the initial acquaintance. A simple algorithm for solving Sudoku is proposed, it consists of seven points. Here is the algorithm: 2.P1. We draw a Sudoku table in such a way that nine numbers can be entered in each small cell. If you draw on paper in a cell, then each Sudoku cell can be made 9 cells (3x3) in size. 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 digits, we look through its row and column and cross out the missing digits that are identical to the result digits found in the row or column outside the small square to which the cell belongs. 2.P4. We look through all the cells with the missing numbers. If there is only one digit left in a cell, then this is the RESULT NUMBER (CR), We circle it. Having circled all the CRs, we proceed to step 5. If the next execution of step 4 does not give a result, then go to step 6. 2.P5. We look through the remaining cells of the small square and cross out the missing digits in them that are identical to the newly obtained digit of the result .. Then we do the same with the missing digits in the row and column to which the cell belongs. We pass to item 4. If the Sudoku level is easy, then the further solution is the alternate execution of paragraphs 4 and 5. 2.P6.If the next execution of step 4 does not give a result, then we look through all the rows, columns and small squares for the presence of the following situation: If in any row, column or small square one or more missing digits appear only once together with other numbers appearing repeatedly, then she or they are RESULT NUMBERS (TR). 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 a row, column or small square in a single copy, circle them circle, and the numbers standing side by side strike out. In our example, these are the numbers 7 and 9 near the two and the number 9 near the six. A row, column or small square will look like: 1,2,5,79,4,6,3,8,79. We pass to item 5. If the next execution of item 6 does not give a result, then go to item 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 the numbers that make up this pair (6 and 9), located in other cells, are crossed out - 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 string will take the form: 8,5,69,4,69,7,1,123,23. After 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 implicitly, as in this line): 9,45,457,2347,1,6,237,8,57 here the pair 23 exists implicitly. 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, possibly, (see P. 6) get a new CR. If not, then you will have to make a choice in some cell from two result values, for example, in a column: 1,6,5,8,29,29,4,3,7. Two cells have two missing numbers each: 2 and 9. you must 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 another. 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 pair of the first row, and the lower cell of the column pair is part of the pair of the seventh small square. We pass to item 5. Our choice (n7) will either be correct and then we will solve the Sudoku to the end, or wrong and then we will soon find it out (two identical digits of the result will appear in one row, column or small square), we will have to return, make the choice opposite to the one made earlier and continue the solution until victory. Before choosing, you must make a copy of the current state. Making a choice is the last thing after b) and c). Sometimes choosing in one pair is not enough (after determining several TAs, progress stops), in this case it is necessary to open one more pair. This happens in difficult sudoku. 2.P7.b) If the search for pairs was 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 digits, as in this small square (triad - 189): 139.2.189.7.189.189.13569.1569.4. and the numbers that make up the triad (189) located in other cells are crossed out - 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 not lucky with triads, then you need to carry out an analysis based on the fact that each row or column belongs to three small squares, consists of three parts, and if in some square some number 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 small squares 1,2,3 formed by rows 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. Q3: 36.239.12369;358.23589.7;568.4.1689. It can be seen that the missing numbers 6 in Page 3 are only in Quarter 3, and in Str. 1 - in Quarter 2 and Quarter 3. Based on the foregoing, cross out the numbers 6 in the cells of Page. 1. in Q3., we get: P.1: 12479.8.123479;1679.5.679;3.239.1239. We got CR 3(7,1) in Q3. After the execution of P.5, the line will take the form: Page 1: 12479.8.12479;1679.5.679;3.29.129. A Kv3. will look like: Square 3: 3.29.129; 58.2589.7; 568.4.1689. We carry out such an analysis for all numbers from 1 to 9 in rows sequentially for triples of squares: 1,2,3; 4,5,6; 7,8,9. Then - in columns for triples of squares: 1,4,7; 2.5.8; 3,6,9. If this analysis did not give a result, then we go to a) and make a choice in pairs. Working with the table requires great care and attention. Therefore, having identified several TAs (5 - 15), you need to try to move further more simple tricks set out in I. 3. PRACTICAL INSTRUCTIONS. In practice, item 3 (deletion) is performed not for each cell separately, but immediately for the whole row, or for the whole column. This speeds up the process. It is easier to control the strikeout if the strikeout is done in two colors. Strike out by rows in one color, and strike out by columns in another. This will allow you to control the strikeout not only for undershooting, but also for its excess. Next, we perform step 4. All cells with missing digits of the result are viewed only at the first execution of step 4 after the execution of step 3. On subsequent executions of paragraph 4 (after the execution of paragraph 5), we look at one small square, one row and one column for each newly obtained digit of the result (CR). Before performing step 7, in case of a volitional disclosure of a pair, it is necessary 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 selection point. 4. EXAMPLE OF SOLUTION OF SUDOKU IN A TABLE METHOD. To consolidate the above, we will solve a Sudoku of medium complexity (Fig. 4.3). The solution result is shown in Fig.4.4. START P.1. We draw a large table. A.2. In each empty cell of each small square we enter all the missing numbers of the result of this square (Fig. 1). For the small square N1, this is 134789; for the small square N2, this is 1245; for the small square N3 it is 1256789, and so on. P.3. We carry out in accordance with the practical instructions for this item (See). P.4. We look through ALL cells with the missing numbers of the result. If in some cell there is one digit left, then this is - CR we 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 performing p.1, p.2, p.3 and p.4 is shown in Fig.1. Two CRs found during step 4 are circled, these are 5(6.1) and 6(5.7). Those who want to get a complete picture of the solution process should draw themselves a table with the initial numbers, independently complete step 1, step 2, step 3, step 4 and compare their 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. A.5. We cross out the number 5 in the cells of the small square N2, row N1 and column N6, these are the "fives" in the cells with coordinates: (9.1), (4.2), (6.5) and (6.6) ); cross out the number 6 in the cells of the small square N8, row N7 and column N5, these are the "sixes" in the cells with coordinates: (6.8), (2.7), (3.7), (5.4) and (5 .5)(5.6). In Fig. 1 they are crossed out, and in Fig. 2 they are no longer there at all. In Fig. 2, all previously crossed-out figures are removed, this is done to simplify the figure. According to the algorithm, we return to P.4. P.4. CR9(5,5)-3 was found, circle it, transfer it. A.5. Cross out the "nines" in the cells with coordinates: (5.6) and (9.5), go to step 4. P.4 No result. We pass to item 6. P.6. In the small square N8 we have: 78, 6, 9, 3, 5, 47, 47, 2, 1. The number 8 (4,7) occurs only once - this is TsR8-4, circle it, and next to it is the number 7 strike out. We pass to item 5. P.5. We cross out the number 8 in the cells of row N7 and column N4. Let's move on to item 4. Item 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) occurs once - this is CR3 (9.9) -5, circle it, transfer (see Fig.4.4), and cross out the adjacent numbers 7 and 9. P.5. We 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. The number 1 (5,3) - TsR1-6, circle it. P.5. We strike out. P.4 No result. P.6. In the small square N1 we have: 18, 2, 19, 6, 1479, 179, 5, 347, 37. The number 8 (1,1) is TsR8-7, circle it. P.5. We strike out. P.4. Numbers 9 (9,1) - TsR9-8, circle it. P.5. We strike out. P.4. Digit 1 (3,1) - TsR1-9. P.5. We strike out. P.4. No result. P.6. Line N5, we have: 12, 8, 4, 256, 9, 26, 3, 7, 56. Number 1 (1.5) - TsR1-10, circled. P..5. We strike 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, in this place it completely coincides with Fig. 2, then you are on the right track! Continue to fill it further on your own. P.5. We strike out. P.4. No result P.6. Column N9 We have: 9, 57, 678, 56, 56, 2, 4, 1, 3. Digit 8 (9.3) - ЦР8-12. P.5. We strike out, P.4. Number 2 (8.3) - TsR2-13. P.5. We strike out. Clause 4 CR5(8.7)-14, CR4(6.3)-15. P.5. We strike out. P.4. CR2(4.2)-16, CR7(6.8)-17, CR1(8.2)-18. P.5. We strike out. P,4. CR4(8.4)-19, CR4(4.9)-20, CR6(6.6)-21. P.5. We strike out. P.4. CR3(5.4)-22, CR7(1.9)-23, CR2(6.5)-24. P.5. We strike out. Clause 4 CR3(1.6)-25, CR9(7.9)-26, CR4(5.6)-27. P.5. We strike out. P.4. CR: 2(1.7)-28, 8(8.8)-29, 5(4.5)-30, 7(2.6)-31. P.5. We strike 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 strike 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 We 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. THE END! Solving Sudoku in a tabular way is troublesome and there is no need in practice to bring it to the very end, as well as solving Sudoku in this way from the very beginning. 5.shtml

I will not talk about the rules, but immediately move on to the methods.
To solve a puzzle, no matter how complex or simple, cells that are obvious to fill are initially searched for.

1.1 "The Last Hero"

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

1.2 "Last Hero" in a row

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, eat " 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 A2

1.3 "No Choice"

Sometimes there are several reasons for specific location. "4 " in J8 would be a great example.
Blue the arrows indicate that this is the last possible number squared. Red And blue the arrows give us the last number in the column 8 . Greens the arrows give the last possible number in the line J.
As you can see, we have no choice but to put this " 4 "in place.

1.4 "And who, if not me?"

Filling in numbers is easier to do using the methods described above. However, checking the number as the last possible value also yields results. The method should be used when it seems that all the numbers are there, but something is missing.
"5 " in B1 is set based on the fact that all numbers from " 1 " before " 9 ", except " 5 " is in the row, column and square (marked in green).

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.

2. "Naked Mile"

2.1 Naked couples

""Naked" couple" - a set of two candidates located in two cells belonging to one common block: row, column, square.
It is clear that the correct solutions of the puzzle will be only in these cells and only with these values, while all other candidates from the general block can be removed.


In this example, there are several "naked pairs".
red in line BUT cells are highlighted A2 And A3, both containing " 1 " And " 6 ". I don't know exactly how they are located here yet, but I can safely remove all the others " 1 " And " 6 " from string A(marked in yellow). Also A2 And A3 belong to a common square, so we remove " 1 " from C1.

2.2 "Threesome"

"Naked Threes"- a complicated version of "naked couples".
Any group of three cells in one block containing all in all three candidates is "naked trio". When such a group is found, these three candidates can be removed from other cells of the block.

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.

2.3 "Fab Four"

"Naked Four" very a rare thing, especially in full form, and still produces results when found. The solution logic is the same as "naked triplets".

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.

3. "Everything hidden becomes clear"

3.1 Hidden pairs

A great way to open the field is to search hidden pairs. This method allows you to remove unnecessary candidates from the cell and give rise to more interesting strategies.

In this puzzle we see that 6 And 7 is in the first and second squares. Besides 6 And 7 is in the column 7 . Combining these conditions, we can assert that in the cells A8 And A9 there will be only these values ​​and we remove all other candidates.

More interesting and complex example hidden pairs. The pair [ 2,4 ] in D3 And E3, cleaning 3 , 5 , 6 , 7 from these cells. Highlighted in red are two hidden pairs consisting of [ 3,7 ]. On the one hand, they are unique for two cells in 7 column, on the other hand - for a row E. Candidates highlighted in yellow are removed.

3.1 Hidden triplets

We can develop hidden couples before hidden triplets or even hidden fours. The Hidden Three consists of three pairs of numbers located in one block. Such as, and. However, as in the case with "naked triplets", each of the three cells does not have to contain three numbers. will work Total three numbers in three cells. For example , , . Hidden triplets will be masked by other candidates in the cells, so first you need to make sure that troika applicable to a specific block.

In that complex example there are two hidden triplets. The first, marked in red, in the column BUT. Cell A4 contains [ 2,5,6 ], A7 - [2,6 ] and cell A9 -[2,5 ]. These three cells are the only ones where there can be 2 , 5 or 6, so they will be the only ones there. Therefore, we remove unnecessary candidates.

Second, in a column 9 . [4,7,8 ] are unique to cells B9, C9 And F9. Using the same logic, we remove candidates.

3.1 Hidden fours

Perfect example hidden fours. [1,4,6,9 ] in the fifth square can only be in four cells D4, D6, F4, F6. Following our logic, we remove all other candidates (marked in yellow).

4. "Non-rubber"

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:

1. Pair or Three in a square - if they are located in one line, then you can remove all other similar values ​​​​from the corresponding line.
2. Pair or Three in a square - if they are located in one column, then you can remove all other similar values ​​​​from the corresponding column.
3. Pair or Three in a row - if they are located in the same square, then you can remove all other similar values ​​​​from the corresponding square.
4. Pair or Three in a column - if they are located in the same square, then you can remove all other similar values ​​\u200b\u200bfrom the corresponding square.

4.1 Pointing pairs, triplets

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 " removes from the eighth square possible meaning 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.

4.2 Reducing the irreducible

This strategy involves carefully parsing and comparing rows and columns with the contents of the squares (rules №3 , №4 ).
Consider the line BUT. "2 "are possible only in A4 And A5. following the rule №3 , remove " 2 " them B5, C4, C5.

Let's continue to solve the puzzle. We have a single location 4 "within one square in 8 column. According to the rule №4 , we remove unnecessary candidates and, in addition, we obtain the solution " 2 " for C7.

Game history

The numerical structure was invented in Switzerland in the 18th century; on its basis, a numerical crossword puzzle was developed in the 20th century. However, in the United States, where the game was directly invented, it did not become widespread, unlike Japan, where the puzzle not only took root, but also gained great popularity. It was in Japan that it acquired the familiar name "Sudoku", and then spread throughout the world.

Rules of the game

The crossword puzzle has a simple structure: a matrix of 9 squares, called sectors, is given. These squares are arranged three in a row and have a size of 3x3 cells. The Sudoku matrix looks like a square, consisting of 3 rows and 3 columns, which divide it into 9 sectors containing 9 cells each. Some of the cells are filled with numbers - the more numbers you know, the easier the puzzle.

Purpose of the game

You need to fill in all the empty cells, while there is only 1 rule: the numbers should not be repeated. Each sector, row and column must contain numbers from 1 to 9 without repetition. It is better to fill in empty cells with a pencil: it will be easier to make changes in case of a mistake or start over.

Solution Methods

Consider a simple version of Sudoku. For example, in a sector or line there is only 1 empty cell left - it is logical that you need to enter in it the number that is not in the number series.

Next, it is worth examining the rows and columns that have the same numbers in 2 sectors. Since the numbers should not be repeated, it is possible to check in which cells the same number can be located in the 3rd sector. Often there is only 1 cell in which you just need to enter the number.

Thus, part of the crossword field will be filled. Then you can start learning strings. Let's say there are 3 free cells in a line, you understand what numbers should be entered there, but you don't know where exactly. You need to try the substitution. Often there are options when a number cannot be located in 2 other cells, because either it is in the corresponding column or in the sector.

Difficult Sudoku

In complex sudoku, these methods only work halfway, there comes a point when it is completely impossible to determine in which cell to enter the number. Then you need to make an assumption and check it. If there are 2 cells in a row, column or sector in which it is equally possible to enter a number, then you need to enter it with a pencil and follow the filling logic further. If your assumption is wrong, then at some point the crossword puzzle will show an error, and there will be a repetition of numbers. Then it becomes obvious that the number should be in the second cell, you need to go back and correct the mistake. In this case, it is better to use a colored pencil to make it easier to find the moment from which you need to solve the crossword puzzle again.

Little secret

It’s easier and faster to solve Sudoku if you first outline with a pencil what numbers can be in each cell. Then you do not have to check all the sectors every time, and in the process of filling, those cells in which only 1 variant of the valid number remains will be immediately obvious.

Sudoku is not only an exciting game that allows you to pass the time, it is a puzzle that develops logical thinking, the ability to retain a large amount of information and attention to detail.

VKontakte Facebook Odnoklassniki

For those who like to solve Sudoku puzzles on their own and slowly, a formula that allows you to quickly calculate answers may seem like an admission of weakness or cheating.

But for those who find Sudoku too hard to solve, this can be literally the perfect solution.

Two researchers have developed a mathematical algorithm that allows you to solve Sudoku very quickly, without guesswork or backtracking.

Complex network researchers Zoltan Torozhkai and Maria Erksi-Ravaz of the University of Notre Dame were also able to explain why some Sudoku puzzles are more difficult than others. The only downside is that you need a PhD in Mathematics to understand what they offer.

Can you solve this puzzle? Created by mathematician Arto Incala, it is claimed to be the hardest Sudoku in the world. Photo from nature.com

Torozhkay and Erksi-Rawaz began to analyze Sudoku as part of their research into optimization theory and computational complexity. They say that most sudoku enthusiasts use a brute-force approach based on the guessing technique to solve these problems. Thus, Sudoku lovers arm themselves with a pencil and try all possible combinations of numbers until the correct answer is found. This method will inevitably lead to success, but it is laborious and time consuming.

Instead, Torozhkay and Erksi-Ravaz proposed a universal analog algorithm that is absolutely deterministic (does not use guessing or enumeration) and always finds correct solution tasks, and quite quickly.

The researchers used a "deterministic analog solver" to complete this sudoku. Photo from nature.com

The researchers also found that the time it takes to solve a puzzle using their analog algorithm correlates with the degree of difficulty of the task, as judged by the person. This inspired them to develop a ranking scale for the difficulty of a puzzle or problem.

They created a scale from 1 to 4, where 1 is "easy", 2 is "average", 3 is "difficult", 4 is "very difficult". A puzzle rated 2 takes on average 10 times longer to solve than a puzzle rated 1. According to this system, the most difficult riddle of those known so far has a rating of 3.6; more challenging tasks Sudoku is not yet known.

The theory starts with a probability mapping for each individual square. Photo from nature.com

"I wasn't interested in Sudoku until we started working on more common class satisfiability of Boolean problems, says Torozhkay. - Since sudoku is part of this class, the Latin square of the 9th order turned out to be a good field for us to test, so I got to know them. I and many researchers who study such problems are fascinated by the question of how far we humans can go in solving Sudoku, deterministically, without busting, which is a choice at random, and if the guess is not correct, you need to go back a step or several steps. and start over. Our analog decision model is deterministic: there is no random choice or recurrence in dynamics.”

Chaos Theory: The degree of complexity of puzzles is shown here as chaotic dynamics. Photo from nature.com

Torozhkay and Erksi-Ravaz believe that their analog algorithm is potentially suitable for application to the solution a large number a variety of tasks and problems in industry, computer science and computational biology.

The research experience also made Torozhkay a big fan of Sudoku.

“My wife and I have several Sudoku apps on our iPhones and we must have played thousands of times already, competing in less time on each level,” he says. - She often intuitively sees combinations of patterns that I do not notice. I have to take them out. It becomes impossible for me to solve many of the puzzles that our scale categorizes as difficult or very difficult without writing the probabilities in pencil.”

The Torozhkay and Erksi-Ravaz methodology was first published in Nature Physics and later in Nature Scientific Reports.

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

What is Sudoku?

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

The history of the puzzle

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

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

Terminology

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

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

How to solve Sudoku: rules

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

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

Varieties of Sudoku

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

Basic solution algorithms: rules and secrets

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

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

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

How to solve Sudoku: secrets and methods

It is recommended to adopt next rule: write down small in the corner of each cell those numbers that could stand there. As new information is obtained, the extra numbers must be crossed out, and then in the end the correct solution will be seen. In addition, first of all, you need to pay attention to those columns, rows or areas where there are already numbers, and as much as possible in more- how fewer options remains, the easier it is to deal with. This method will help you quickly solve Sudoku. As experts recommend, before entering the answer into the cell, you need to double-check it again so as not to make a mistake, because because of one incorrectly entered number, the whole puzzle can “fly”, it will no longer be possible to solve it.

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

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

Locked Candidate Method

When you work with any one particular block, a situation may arise that certain number in this area can only be in one row or in one column. This means that in other rows/columns of this block there will be absolutely no such number. The method is called “locked candidate” because the number is, as it were, “locked” within one row or one column, and later, with the advent of new information, it becomes clear exactly in which cell of this row or this column this number is located.

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

Method "open pairs"

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

Hidden Pair Method

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

x-wing method

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

Difficult Sudoku: Solving Methods

How to solve difficult sudoku? The secrets, in general, are the same, that is, all the methods described above work in these cases. The only thing is that in complex sudoku situations are not uncommon when you have to leave logic and act by the “poke method”. This method even has its own name - "Ariadne's Thread". We take some number and substitute it in the right cell, and then, like Ariadne, we unravel the ball of threads, checking whether the puzzle will fit. There are two options here - either it worked or it didn't. If not, then you need to “wind up the ball”, return to the original one, take another number and try all over again. In order to avoid unnecessary scribbling, it is recommended to do all this on a draft.

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

Math Rules

Mathematicians do not remain aloof from this problem. Mathematical Methods how to solve sudoku are as follows:

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

How to increase guessing speed?

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

Game Versions

More recently, Sudoku remained only a printed game, published in magazines, newspapers and individual books. Recently, however, all sorts of versions of this game have appeared, such as board sudoku. In Russia, they are produced by the well-known company Astrel.

There are also computer variations of Sudoku - and you can either download this game to your computer or solve the puzzle online. Come out sudoku for perfect different platforms, so it doesn't matter what exactly is on your personal computer.

And more recently, there have been mobile applications with the Sudoku game - for both Android and iPhones, the puzzle is now available for download. And it must be said that this application is very popular among cell phone owners.

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

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