How to play Sudoku secrets. An example of solving problems is the most difficult sudoku

History of the game

The numerical structure was invented in Switzerland in the XVIII century, on its basis in the 20th century, a numerical crossword was developed. However, in the United States, where the game was directly invented, she did not get much spread, unlike Japan, where the puzzle not only gothes, but also gained great popularity. It was in Japan that she acquired the usual name "Sudoku", and then spread the world.

Rules of the game

Crossword has simple structure: The matrix of 9 squares called sectors is set. These squares are located three in a row and have a size of 3x3 cells. The matrix sudoku looks like a square consisting of 3 lines and 3 columns that divide it into 9 sectors containing 9 cells each. Part of the cells are filled with numbers - the more digits are known, the simpler puzzle.

Purpose of the game

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

Methods of solution

Consider a simple version of Sudoku. For example, only 1 empty cell remained in the sector or string, it is logical that it should be written in it that is not in the numerical row.

Next is to explore the rows and columns in which there are same numbers in 2 sectors. Since numbers should not be repeated, then it can be checked in which cells the same figure can be located in the 3 sector. Often there is only 1 cell, in which you just need to enter the figure.

Thus, part of the crossword field will be filled. Then you can proceed to the study of lines. Suppose, there are 3 free cells in the line, you understand what numbers should be inscribed there, but it is unknown where exactly. You need to try substitution. There are often options when in 2 other cells the number can not be located, because either it is in the corresponding column or in the sector.

Sophisticated Sudoku

In sophisticated sudoku, these methods work only half, occurs when it comes when it is absolutely impossible to determine which cell to enter the number. Then you need to make an assumption and check it out. If there are 2 cells in the row, column or sector, in which it is equally possible to enter the digit, then you need to enter it with a pencil and follow the filling logic on. If your assumption is incorrect, then at some point the crossword will show a mistake, and repeat figures. Then it becomes obvious that the figure should be in the second cell, you need to go back and correct the error. It is better to use a color pencil in such a case to find it easier to find the moment from which the crossword is renovated.

Little secret

It is easier and faster to solve Sudoku if initially outlaw a pencil, which numbers can be in each cell. Then it does not have to check all sectors every time, and in the process of filling immediately, those cells in which only 1 version of the permissible number remained.

Sudoku is not only fascinating gamewhich allows you to pass time, it is a puzzle that develops logical thinking, the ability to hold a large amount of information and care to details.

This Sudoku solution strategy is called "counting" because you must "calculate" all possible values \u200b\u200bfor the cell in the hope of finding the only possible number for it.

Thus, on the above example, it is calculated for the P4K4 cell (highlighted by color and marked with a question mark).

So: 9 - there is in the column, 8 - there is in the block, 7 - there is in the column, 6 - there is in line, 5 - there is in the block, 4 - there is a line, 3 - there is in the column, 2 - there is in the block. The only possible option is the number 1.

2. Outlet 1.

For a specific number (in this case, 5) we find cells that this number contains.

Pay attention to block 3 (highlighted by color). We know that rows and columns containing the number 5 in other cells cannot contain it. This allows us to highlight the cells in block 3 that do not contain the number 5.

In one of the two remaining cells (P3K8) there is a number 1.

The only cell that may contain the number 5 - P3K7 - is highlighted by color and marked with a question mark.

In the previous embodiment, we "exchanged" numbers in three lines. Now we give a simpler example - with two lines.

3. Hacking 2.

"Reverse" version of the previous strategy. If a specific number is contained in the block, then the strings and columns in the intersection places with the block are the same number to be kept.

Pay attention to the P8K6 cell (highlighted by color and marked a question mark).

At first glance (using the counting strategy), numbers 6, 9, 5, 8 may be located in it.

But, if we pay attention to the entire column 6, we will come to the following conclusions: 1) in the P5K6 cell 6 can not be - there is in the corresponding row 5; 2) in the P2K6 and P3K6 cells of the number 6 can not be - available in block 2. So, the only possible option is the P8K6 cell.

4. Blockade 1.

Sometimes in the process of solving the situation occurs when a certain number in the block can be located only in one row (column) within this block. As a result, this number cannot be in other cells of this series (column) outside the block.

The location of the number 7 in the P6K7 cell in addition to other numbers in block 4 gives us two variants of the number 7 in said block - cells P4K3 and P5K3.

It is clear that in one of these cells there should be a number 7. As a consequence, the number 7 should not be in the remaining column cells 3.

5. Blockade 2.

Reverse version of the previous strategy. If a certain number in a row or column can only be located within one block, the same number cannot be located in other cells of the block under consideration.

Thus, the number 2 in column 7 can only be in the P7K7 and P9K7 cells. Both cells are located in block 9, it means that in other cells of this block number 2 should not be.

• Tutorial

1. Basics

Most of us, habrarites, know what Sudoku is. I will not tell about the rules, and immediately go to the methods.
To solve a puzzle, it does not matter difficult or simple, cells are originally searched for filling.

1.1 " Last Hero»

Consider the seventh square. There are only four free cells, which means that something can be quickly fill.
"8 " on the D3. Blocks filling H3. and J3; similar " 8 " on the G5closes G1.and G2.
With a clean conscience set " 8 " on the H1.

1.2 "Last Hero" in the string

After viewing the squares for obvious solutions, go to columns and rows.
Consider " 4 "On the field. It is clear that it will be somewhere in the string A. .
We have " 4 " on the G3.What is poked A3., there is " 4 " on the F7.cleaner A7.. And another one " 4 "In the second square, prohibits her repeat A4.and A6..
"The last hero" for ours " 4 " this is A2.

1.3 "No choice"

Sometimes there are several reasons for a specific location. " 4 " in J8.it will be an excellent example.
Bluethe arrows show that this is the last possible number in the square. Red and bluearrows give us the last number in the column 8 . Greenarrows give the last possible number in the string J..
As you can see, we have no choice but to put this " 4 "In place.

1.4 "And who, what is not me?"

Filling numbers is easier to carry out the methods described above. However, checking the number as the last possible value, also gives results. The method should be applied when it seems that all the numbers are, but something is missing.
"5 " in B1.puts on the fact that all numbers from " 1 " before " 9 ", Besides " 5 "There is a row, column and square (marked green).

On the jargon it " Naked loner"If you fill in the field with possible values \u200b\u200b(candidates), then in the cell such a number will be the only possible. Developing this technique, you can search" Hidden single"- Numbers, unique for a specific string, 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 unit: row, column, square.
It is clear that the correct puzzle solutions will only be in these cells and only with these values, while all other candidates from the general block can be removed.



In this example, several "naked steam".
Red in line BUT Selected cells A2.and A3., both are containing " 1 "And" 6 ". I don't know how they are located here, but I can calmly remove all the others" 1 "And" 6 "From string A. (marked yellow). Also A2.and A3. Belong a common square, so we remove " 1 "Is C1..

2.2 "Threesome"

"Naked Troika" - Complete option "naked couples".
Any group of three cells in one block containing generally Three candidates are "Naked Troika". When such a group was found, these three candidates can be removed from other cell cells.

Combinations of candidates for "Naked Troika" Might be such:

// Three numbers in three cells.
// Any combination.
// Any combination.


In this example, everything is quite 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 there may be. This allows us to remove them from other candidates of the block. This trick gives us a solution " 3 "For cell E7..

2.3 "Magnificent Four"

"" Naked "Four" very rare phenomenon, especially in full formand still gives results when detected. The logic of solutions is the same as "Naked Trok".

IN specified example in the first square cell A1., B1., B2.and C1. In general, contain [ 1,5,6,8 ], so these numbers will take only these cells and no others. We remove the reflecting yellow candidates.

3. "All the secret becomes apparent"

3.1 Hidden couples

Great way to reveal the field will search hidden. This method allows you to remove unnecessary candidates from the cell and give the development of more interesting strategies.

In this puzzle we see that 6 and 7 There are in the first and second squares. Besides 6 and 7 Eating in column 7 . Combining these conditions, we can argue that in cells A8.and A9.there will be only these values \u200b\u200band all other candidates we remove.


More interesting and complex example hidden. Blue highlighted pair [ 2,4 ] in D3.and E3.cleaner 3 , 5 , 6 , 7 From these cells. Red two highlighted hidden couplesconsisting of [ 3,7 ]. On one side, they are unique for two cells in 7 column, on the other hand - for a string E.. Selected yellow candidates are cleaned.

3.1 Hidden Troika

We can develop hidden couples before hidden Trokor even hidden fours. Hidden Troika Consists of three pairs of numbers located in one block. Such as, and. However, as in the case of "Naked Troops"In each of the three cells it does not have to be three numbers. Worked total Three numbers in three cells. For example , , . Hidden Troikawill be disguised by other candidates in cells, so you must first make sure that troikaapplicable to a specific block.


In that complex example There are two hidden Troika. The first, marked 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 may be 2, 5 or 6, so they will only be there. Consequently remove the extra candidates.

Second, in the column 9 . [4,7,8 ] Unique for cells B9., C9. and F9.. Using the same logic, we clean the candidates.

3.1 Hidden Fours

Excellent 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 yellow).

4. "Non-resident"

If any of the numbers appears twice or three times in one block (row, column, square), then we can remove this number from the conjugate block. There are four types of pairing:

1. A pair or triple in a square - if they are located in one line, you can remove all other same values \u200b\u200bfrom the corresponding string.
2. Couple or triple in a square - if they are located in one column, you can remove all other same values \u200b\u200bfrom the corresponding column.
3. A pair or triple in the string - if they are located in one square, you can remove all other same values \u200b\u200bfrom the corresponding square.
4. A pair or triple in the column - if they are located in one square, then you can remove all the same values \u200b\u200bfrom the corresponding square.

4.1 Fixing Couples, Troika

As an example, I will show this puzzle. In the third square " 3 "is only in B7.and B9.. Following the statement №1 we remove candidates from B1., B2., B3.. Similarly, " 2 "From the eighth square removes the possible value from G2..


Special puzzle. Very difficult in solving, but if you look at, you can see several pointing par. It is clear that it is not always necessary to find them all to move into the decision, but each such find makes us a task.

4.2 Reducing inconvenient

This strategy includes a neat analysis and comparison of rows and columns with square content (rules №3 , №4 ).
Consider a string BUT. "2 "Possible only in A4.and A5.. Following the rule №3 , remove " 2 " them B5., C4., C5..


We will continue to solve the puzzle. We have the only location " 4 "within one square in 8 column. According to the rule №4 , we remove the extra edging and, in addition, we get a decision " 2 "For C7.

In this world there is great amount Different to help you in the development of one of the most important organs - brain. Of course, the widespread Japanese puzzles sudoku are one of them. With their help, you can be pretty "pumping up ameal", because in addition to the need to calculate a huge number of options for the location of the numbers, you also need to be able to do it for a couple of dozen moves forward. In short, this is a real paradise, if you want to not give your neurons to "dry up". And today we will look at the main techniques that are used by Sudoku's experts. It will be useful both beginners and long-standing fans of these puzzles. After all, someone needs to make their first steps in the art of Sudoku, and someone increase the effectiveness of their solutions!

rules

If you are not familiar with, then first you should familiarize yourself with the rules. Believe me, they are very simple.

The playing field is a square that has 9 × 9 sizes. At the same time, it is divided into smaller squares with 3 × 3 sizes. That is, the entire field consists of 81 cells.

The condition of the problem is those numbers that are already placed in these cells.

Block (cell block) - Small square, line, or lines.

What needs to be done: put all the other numbers, observing several rules. First, in each of the small squares there should be no repetitions. Secondly, in all columns and rows also should not be repetitions. That is, each number must occur only once in each of these blocks. In order for everything to become even clearer, pay attention to the resolved Sudoku:

Basic solution solution

As a rule, if you decide simple sudoku, then all you need to do is to paint all possible options for each of the 81 cells and gradually cross out unsuitable options. It is very simple.

But if you go to the level above, to a more complex sudoku, then everything becomes more interesting. It will often be so that putting new numbers there is no possibility, and you will have to go through assumptions: "Let such a number standing here", after which you will need to consider this hypothesis and or come to solve the problem or the contradiction of your assumption.

But of course there is special techniquesthat will help do all this more efficiently.

Receivers

1. Naked couples / Troika / Four

If you have two cells in one block (square, string or column), in which you can only put 2 digits, it is obvious that these numbers can be removed from possible options for other cells of this block.

Moreover, such a trick can be easily done with top three, and fours:

2. Hidden couples

Highly useful reception, in some way, inverse naked pairs. If in some two cells of one square in " possible options"You have numbers that are no longer repeated anywhere (within this square), then all other numbers from these two cells can be removed.

In order to become more clear, pay attention to the examples (one simple and more complicated):

Fortunately, it works for both triples, and for the fours, but it is worth mentioning a very important and very cool chip. It is not necessary that in the three / four cells there are the same 3 digits of the form (a; b; c) (a; b; c) (a; b; c). You will be enough such an option: (a; b) (B; c) (a; c).

3. Unnamed Rule

If you have a steam or triple in one column / string, which are located in one square, you can safely remove these numbers from other cells of this square.

4. Indicating couples

If in one line / column in "possible versions" there are two identical numbers, then such numbers can be removed from the corresponding column / string.

At times it happens very useful, especially if you find several such pairs:

Of course, these numbers should be absent in other squares of the square, but according to the nameless rule, it is not required.

Love Sudoku and other riddles, games, puzzles and tests aimed at development various aspects thinking? Get all interactive materials on the site to develop more efficiently.

Conclusion

We reviewed the main techniques that are used when solving Sudoku. I note that it is only the beginning and in the following articles we will look at more complex and more interesting chips, thanks to which the solution to such tasks will become even more interesting and easier.

As a workout, the editors of 4Brain invites you to familiarize yourself with the file in which Sudoku is contained. various levels difficulties. Do not take time on training, because if you give this lesson enough time, then at the end of this course of articles, believe me, you will become a real ASA in solving Japanese puzzles.

If you have any questions on these techniques or on Sudoku, which we apply to the article, you can safely ask them in the comments!

For those who like to solve the riddles of the Cupudoka independently and slowly, the formula that allows you to quickly calculate the answers, it may seem recognition of weakness or soulmatic

But for those whom to solve Sudoku is worth too much effort, it can be literally an ideal solution.

Two researchers have developed a mathematical algorithm that allows you to solve Sudoku very quickly, without assumptions and busting with the return.

Researchers of integrated networks Zoltov Torozhukai and Maria Erxi Ravaz from the University of Notre Dama were also able to explain why some riddles see more complicated than others. The only drawback is that in order to understand what they offer, you need a degree of doctor mathematics.

Can you solve this puzzle? It is created by the mathematician Arto Inkala, and, according to, this is the most difficult sudoku in the world. Photos from Nature.com

Torozhkay and Erxi Ravaz began to analyze Sudoku as part of their study of the theory of optimization and computational complexity. They say that most sudoku lovers use the "rough strength" approach to solving these tasks, based on the technique of assumptions. Thus, sudoku lovers are armed with a pencil and try everything possible combinations Numbers until the correct answer is found. This method will inevitably lead to success, but it is time consuming and takes a lot of time.

Instead, the baroji and Erxi Ravaz offered a universal analog algorithm, which is absolutely determined (does not use the assumption or bust) and always finds the correct solution of the problem, and quite quickly.

Researchers used the "deterministic analog solver" to fill this sudoku. Photos from Nature.com

The researchers also found that the time that is required to solve the puzzle using their analog algorithm is correlated with the degree of complexity of the task that is evaluated by a person. It inspired them to develop a ranking scale for the difficulty of riddles or problems.

They created a scale from 1 to 4, where 1 is "easy", 2 - "average degree of complexity", 3 - "difficult", 4 - "very difficult." To solve a puzzle with rating 2, it takes an average of 10 times longer than for a rating task 1. According to this system, the most complex mystery From the well-known still has a rating of 3.6; more complex tasks Sudoku is still unknown.

The theory begins with probability cartography for each individual square. Photos from Nature.com

"I was not interested in Sudoku, until we started working on more common class The feasibility of Boolean problems, says Torozhkai. - Since Sudoku is part of this class, the Latin square of the 9th order turned out to be a good test field for us, so I met with them. Me and many researchers who study such problems captures the question of how far we are people who are able to go to Sudoku's decision, determined, without a search, which is a choice of at random, and, if a guess is not correct, you need to go back to step or a few steps ago And start first. Our analog model solving is determined: there is no random choice or return in the dynamics. "

Chaos Theory: The degree of complexity of the riddles is shown here as a chaotic speaker. Photos from Nature.com

Torozhkay and Erxi Ravaz believe that their analog algorithm is potentially suitable for applying large number A variety of tasks and problems in industry, computer science and computing biology.

The experience of the research also made a hanging like a big fan of Sudoku.

"My wife and I have several sudoku applications on our iPhone, and we must have played thousands of times, competing in less time at every level," he says. - She often intuitively sees a combination of patterns that I do not notice. I have to withdraw them. For me it becomes impossible to solve many puzzles that our scale categorizes as difficult or very difficult, without changing the probabilities of the pencil. "

The peculiar and Erxi Ravaz methodology was first published in the Nature Physics magazine, and then in the magazine Nature Scientific Reports.