Sudoku how to solve complex rules. How to solve Sudoku - ways, methods and strategy

Use numbers from 1 to 9

Sudoku is played on a playing field consisting of 9 by 9 cells, with a total of 81 cells. Inside the playing field there are 9 "squares" (consisting of 3 x 3 cells). Each horizontal row, vertical column and square (9 squares each) must be filled with numbers 1-9, without repeating any numbers in a row, column or square. Does this sound complicated? As you can see from the image below, each Sudoku game board has several cells that are already filled. The more cells are initially filled, the easier the game. The fewer cells are initially filled, the more difficult the game.

Don't repeat any numbers

As you can see, in the top left square (circled in blue) 7 of the 9 cells are already filled. Singular numbers The ones that are missing from that square are the numbers 5 and 6. By seeing which numbers are missing from each square, row, or column, we can use the process of elimination and deductive reasoning to decide which numbers should be in each square.

For example, in the top left square we know that to complete the square we need to add the numbers 5 and 6, but looking at the adjacent rows and squares we cannot yet clearly determine which number to add to which cell. This means that we must now skip the top left square for now and instead try to fill in the gaps in some other places on the playing field.

No need to guess

Sudoku is logic game, so no need to guess. If you don't know what number to put in a certain space, keep scanning other areas of the game board until you see the option to put the number you want. But don't try to "force" anything - Sudoku rewards patience, understanding and solving different combinations, not blind luck or guessing.

Use the elimination method

What do we do when we use the "method of elimination" in Sudoku? Here's an example. In this Sudoku grid (shown below), only a few numbers are missing from the left vertical column (outlined in blue): 1, 5, and 6.

One way to figure out which numbers can be inserted into each square is to use the "method of elimination", checking what other numbers are already in each square, since numbers 1-9 are not allowed to be duplicated in each square, row or column.

In this case, we can quickly notice that there is already a 1 in the top left and center left squares (the 1's are circled in red). This means that in the leftmost column there is only one place where the number 1 can be inserted (circled in green). This is how the elimination method works in Sudoku - you find out which cells are empty, which numbers are missing, and then eliminate the numbers that are already present in the square, columns and rows. Accordingly, fill in the empty cells with the missing numbers.

The rules of Sudoku are relatively simple - but the game is incredibly varied, with millions of possible number combinations and a wide range of difficulty levels. But it's all based on the simple principles of using numbers 1-9, filling in the blanks using deductive reasoning, and never repeating numbers in each square, row, or column.

Solving Sudoku is a creative process. The rules of the puzzle are very simple, although logical reasoning while searching for a solution can be of varying degrees of complexity. Experience comes only with time, and each player develops his own strategy. And so that you can better understand how to solve puzzles and get a taste for them, we present some recommendations.

Start your solution with one.

1. First, “look around” on the playing field, finding all the cells with the number “1”.

2. Check each of the 3x3 blocks sequentially to see if it already contains a unit. If it does, consider the following.

3. If there is no unit in the block yet, try to find all the cells within this block that could contain a unit. Remember the rule: each number can appear in each row, each column and each block only once. Eliminate from consideration all cells in the block in which the number “1” cannot be found because the column or row is already “occupied.” It is likely that there will be a block in which there will be only one cell left in which a unit can be located. Write it in.

4. If you are not sure of the uniqueness of the solution, it is better to leave this block and try with another one. You will definitely find a suitable block.

After you “go through” all the blocks with the number “1”, repeat the search with a different number. For example, with a deuce. Then with three and so on. Until you check all the numbers from 1 to 9. And you will see that you have already filled in a lot of cells. After which we advise you to repeat the entire “procedure” again from the very beginning - again from 1 to 9. The second time things will go easier, because many of the cells are already filled. And where you doubted, you can confidently enter a number.

Using the recommendations, solving a simple puzzle will not be difficult. From our experience, we know that people who can easily solve simple Sudoku puzzles may have difficulty with complex ones. Therefore, we will consider in detail the solution to one of the problems.

For convenience of explanation, we will use the numbering of rows, columns and 3x3 blocks from 1 to 9. Numbering order: left - right and top - down.

Designations:

1. The gray block, row or column is the “zone” that we analyze in search of a solution;

2. The highlighted “bold” number (blue) is the desired number found during the analysis process;

3. The lines show that in this direction the number from which this line begins cannot be placed.

We find the number "1" in the 2nd block. The lines coming from the units of the 5th and 8th blocks cross out the remaining empty cells.

We find the number "1" in the 4th block. For this project, we will determine where there can be ones in the 6th block by drawing lines from the units of the 5th and 9th blocks - two ones in the top row. Already from them we draw a line towards the 4th block and a line from the unit of the 5th block.

The search for possible twos was unsuccessful, but it is possible to find a three in the 9th block by drawing lines from the threes in the 3rd and 6th blocks. There were no options for the numbers “4”, “5”, “6”, “7”. But the number “8” was found in the 8th square: lines from the eights of the 2nd, 5th and 7th blocks. Nine was also not found.

Let's get started new search units. A unit was found in the first block: the lines from the units in the 2nd and 9th blocks determined the possible positions of the unit in the 3rd block, and from them the lines extended into the 1st block. The remaining lines are visible in the figure. The next unit was found in block 7.

The first two was found in block 4, after which the first five was also determined there. The numbers "3", "4", "6", "7" were not found.

The number “8” of block 1 is determined by the lines from the eights from blocks 4 and 7. Then we find the nine of the 9th row: since it cannot be in blocks 7 and 8 (see the lines from the corresponding nines), it is in the block 9.

The number “9” in the 1st line: it cannot be in block 2, which means it is in block 3. We enter “5” in the remaining cell of the line. Two numbers "9" were found in blocks 5 and 6. We start again with the number "1".

The first to be found was the quarter of the 6th block. Then the quadruple of the 5th column - it cannot be in the 4th and 7th row. Three cannot be in the 7th line, so it is in the 4th. Then the remaining cell contains a six.

In the next step, the queue is not necessary: ​​first we find the eight, and then the one in block 6, or vice versa.

We continue to place the eights: first we find “8” in block 9, and from it we draw a line, defining the eight in block 3.

The next numbers found were “1” and “6” in block 3, the order in which they were found is not important.

Then let's decide on the number "7" in the 9th column: it cannot be in block 6, then it is in the 2nd line. From the five in block 1 we draw a line - we find a place for the number “5” in the 3rd block. In the empty cell we enter the last number - “2”.

In the second row we find the number “2”, then “4” and finally “9”.

Then we find the number "4" in block 8. In the remaining cell - "7". We draw a line from it up to block 5 - a new seven. In the empty cell of the 9th line - "7".

Let's find sequentially the numbers "5", "2", "6" in block 5 and the numbers "7", "3" in the 6th row. Then we get "5" and "6" in the 6th block. The last digit is "6" in the 4th block.

The next "7" and "3" are in the 1st block; the numbers “7” and “2” in the 7th column and “5” in block 9. We analyze the 7th line, 2nd column and place first “9”, then “3” and “2”. The final touch is "4" and "6".

The solution is complete.

Very complex tasks There is one more trick. It is used when it is impossible to calculate a single move. There are at least two cells for one digit in a block (row/column). It is extremely difficult to sort through in your mind all the consequences of a position chosen at random. Then you should enter the number at random, but with a pencil. In this case, the only options can be entered directly with a ballpoint pen. If after a few moves an error is discovered, for example, it is impossible to enter any number into the block - there is no suitable place, then the entire pencil version is erased and the second option is written in the initial cells. You can also use the notation in cells of all possible numbers on this moment, this helps to quickly navigate the search for a solution. In any case, start with easy puzzles and good luck to you!

Many people like to force themselves to think: for some - to develop intelligence, for others - to keep their brains in good shape (yes, not only the body needs exercise), and the best simulator for the mind is various logic games and puzzles. One of the options for such educational entertainment can be called Sudoku. However, some have never even heard of such a game, let alone knowing the rules or other interesting points. Thanks to the article you will learn everything necessary information, for example, how to solve Sudoku, as well as their rules and types.

General

Sudoku is a puzzle. Sometimes complex, difficult to solve, but always interesting and addictive to anyone who decides to play this game. The name comes from Japanese: “su” means “digit”, and “doku” means “standing alone”.

Not everyone knows how to solve Sudoku. Complex puzzles, for example, can be solved either by smart, well-thought-out beginners, or by professionals who have been practicing the game for more than one day. It will not be possible for everyone to just take it and solve the problem in five minutes.

Rules

So, how to solve Sudoku. The rules are very simple and clear, easy to remember. However, do not think that simple rules promise a “painless” solution; you will have to think a lot, apply logical and strategic thinking, and strive to recreate the picture. You probably have to love numbers to solve Sudoku.

First, a 9 x 9 square is drawn. Then, with bolder lines, it is divided into so-called “regions” of three squares each. The result is 81 cells, which should eventually be completely filled with numbers. This is where the difficulty lies: the numbers from 1 to 9 placed along the entire perimeter should not be repeated either in “regions” (3 x 3 squares) or in lines vertically and/or horizontally. In any Sudoku, there are initially some filled cells. Without this, the game is simply impossible, because otherwise the result will not be solving, but inventing. The complexity of the puzzle depends on the number of numbers. Complex sudokus contain a few numbers, often arranged in such a way that you have to rack your brain quite a bit before solving them. In the lungs, about half of the numbers are already in place, making it much easier to figure out.

Fully disassembled example

It is difficult to understand how to solve Sudoku if there is no specific example showing step by step how, where and what to insert. The provided picture is considered simple, since many of the mini-squares are already filled in with the necessary numbers. By the way, it is on them that we will rely for the solution.

To begin with, you can look at the lines or squares, where there are especially many numbers. For example, the second column from the left fits perfectly; there are only two numbers missing. If you look at those that are already there, it becomes obvious that 5 and 9 are missing in the empty cells on the second and eighth lines. With the five, not everything is clear yet, it can be both here and there, but if you look at the nine, everything becomes clear. Since there is already a number 9 on the second line (in the seventh column), it means that in order to avoid repetitions, the nine must be placed down, on the 8th line. Using the elimination method, we add 5 to the 2nd row - and now we already have one filled column.

You can solve the entire Sudoku puzzle in a similar way, but in more complex versions, when one column, row or square is missing not just a couple of numbers, but much more, you will have to use a slightly different method. We will also analyze that now.

This time we will take as a basis the middle “region”, in which five numbers are missing: 3, 5, 6, 7, 8. We fill each cell not with large effective numbers, but with small, “draft” ones. We simply write in each square the numbers that are missing and that may be there due to their lack. In the top cell it is 5, 6, 7 (3 on this line is already in the “region” on the right, and 8 on the left); the cell on the left can contain 5, 6, 7; in the very middle - 5, 6, 7; right - 5, 7, 8; from below - 3, 5, 6.

So, now we look at which mini-digits contain different numbers from the others. 3: it is only in one place, it is not in the rest. This means that it can be corrected to be larger. 5, 6 and 7 are in at least two cells, which means we leave them alone. There is 8 in only one, which means that the remaining numbers disappear and you can leave the eight.

Alternating these two methods, we continue to solve Sudoku. In our example, we will use the first method, but it should be recalled that in complex variations the second is necessary. Without him it will be extremely difficult.

By the way, when a middle seven is found in the upper “region”, it can be removed from the mini-digits of the middle square. If you do this, you will notice that there is only one 7 left in that region, so you can only leave it.

That's all; finished result:

Kinds

There are different types of Sudoku puzzles. In some prerequisite is the absence of identical numbers not only in rows, columns and mini-squares, but also diagonally. Some contain other figures instead of the usual “regions,” which makes solving the problem much more difficult. One way or another, you know how to solve Sudoku, at least the basic rule that applies to any kind. This will always help you cope with a puzzle of any complexity, the main thing is to try your best to achieve your goal.

Conclusion

Now you know how to solve Sudoku, you can download similar puzzles from various sites, solve them online or buy them at newsstands paper options. In any case, now you will have something to do for long hours, or even days, because Sudoku is unrealistically drawn out, especially when you have to actually figure out the principle of their solution. Practice, practice and practice again - and then you will crack this puzzle like nuts.

In previous articles, we looked at different approaches to solving problems using Sudoku puzzles as examples. The time has come to try, in turn, to illustrate the capabilities of the considered approaches using a fairly complex example of problem solving. So, today we will start with the most “incredible” version of Sudoku. Please look at the terminology and preliminary information, otherwise it will be difficult for you to understand the content of this article.

Here is the information I found about this super complex option on the Internet:

University of Helsinki professor Arto Inkala claims (2011) that he created the world's most difficult Sudoku crossword puzzle. This the most difficult puzzle he created for three months.

According to him, the crossword puzzle he created cannot be solved using logic alone. Arto Incala argues that even the most experienced players It will take at least a few days to decide. The professor’s invention was called AI Escargot (AI – the initials of the scientist, Escargot – from the English “snail”).

To solve this difficult problem, according to Arto Incala, you need to keep eight sequences in your head at the same time, unlike ordinary puzzles, where you need to remember one or two sequences.

Well, “sequences of searches” – this still smacks of a machine version of problem solving, and those who solved Arto Incal’s problem with their own brains talk about it differently. Someone solved it for a couple of months, someone announced that it only took 15 minutes. Well, the world chess champion could probably cope with the task in such a time, and a psychic, if such a thing lives on our plane, perhaps even faster. And the problem could also be quickly solved by someone who accidentally picked up a few successful numbers to fill in the empty cells the first time. Let's say, one out of a thousand solvers of the problem might be similarly lucky.

So, about brute force: if you successfully choose two or three correct digits, then you may not need to brute force eight sequences (which means dozens of options). This was my thought when I decided to begin solving this problem. To begin with, I, having already been prepared within the framework of the methods of previous articles, decided to forget about what I knew so far. There is such a technique that the search for a solution should proceed freely, without schemes and ideas imposed on it. And the situation was new for me, so I needed to look at it in a new way. I have placed (in Excel) the original table (on the right) and the work table, the meaning of which I already had the opportunity to talk about in my first article about Sudoku:

Let me remind you that the worksheet contains pre-allowed combinations of numbers in initially empty cells.

After the usual almost routine processing of tables, the situation became a little simpler:

I began to study this situation. Well, since I’ve already forgotten how exactly I solved this problem a few days earlier, I’m starting to think about it anew. First of all, I paid attention to the two numbers 67 in the cells of the fourth block and combined them with the mechanism of rotation (movement) of cells, which I talked about in the previous article. After going through all the options for rotating the first three columns of the table, I came to the conclusion that numbers 6 and 7 cannot be in the same column and cannot rotate asynchronously; during the rotation process, they can only follow one another. Also, if you look closely, the seven and four seem to move synchronously along all three columns. Therefore, I make a plausible assumption that the number 7 should be placed in the lower left cell of block 4, and the number 6 in the upper right cell, respectively.

But for now I accept this result only as a possible guideline for testing other options. And I pay main attention to the number 59 in the cell of the 4th block. There can be either the number 5 or 9. Nine promises to destroy a lot of extra numbers, i.e. simplify the further course of solving the problem, and I start with this option. But quite quickly I reach a “dead end”, i.e. Then I have to make some choice again and who knows how long my choice will be checked. I guess if there really was a nine at one time the right choice, then Incala would hardly have left such an obvious option in plain sight, although the mechanism of his program could have allowed such a blunder. In general, one way or another, I decided to first thoroughly check the option with the number 5 in the cell with the number 59.

But later, when I solved the problem, I, so to speak, to clear my conscience, nevertheless returned to the option with the number 9 in order to determine how long it would take to check it. It didn't take very long to check. When I had the number 6 in the upper right cell of block 4, as expected according to the pre-selected reference point, then the number 19 appeared in the right middle cell (6 out of 169 was removed). I chose the number 9 in this cell for further testing and quickly came to a contradictory result, i.e. the choice of nine is incorrect. Then I choose number 1 and again check what comes out of it.

At some step I come to the situation:

where again I have to make a choice - the number 2 or 8 in the upper middle cell of block 4. I check both options (2 and 8) and in both cases I end up with a contradictory (not meeting the Sudoku condition) result. So I could check the option with the number 9 in the middle bottom cell of block 4 from the very beginning and it wouldn’t take much time. But I still, as I already said, settled on the number 5 in the mentioned cell. This led me to the following result:

The location of the numbers 4 and 7 in the first three columns (columns) indicates that they rotate synchronously, which is what was actually expected when choosing the number 7 for the lower left cell of the 4th block. In this case, a two or a nine, whether any of them is the required digit in the middle left cell of this block, must accordingly move asynchronously with the pair 4 and 7. In this case, I gave preference to the number 2, since it “promised” to eliminate many extra digits from the cell numbers and, accordingly, a quick check of validity this option. And nine quickly led to a dead end - it required the selection of new numbers. Thus, in the left middle cell of the block with the number 29, I put down, in my opinion, the more preferable number - 2. The result came out as follows:

Next, I had to once again make a semi-arbitrary choice: I chose two in the cell with the number 26 in the ninth block. To do this, it was enough to notice that 5 and 2 in the three lower lines rotate synchronously, since 5 did not rotate synchronously with either 1 or 6. True, 2 and 1 could also rotate synchronously, but for some reason - definitely not I remember - I chose 2 instead of the number 26, perhaps because this option, in my opinion, was quickly checked. However, there were already few options left, and it was possible to quickly check any of them. It was also possible, instead of the option with two, to assume that the numbers 7 and 8 rotate synchronously in the last three columns (columns), and from this it followed that in the upper left cell of the 9th block there could only be the number 8, which also leads to a quick solution to the problem .

It must be said that Arto Incal's problem does not allow for a purely logical solution within the framework of possibilities ordinary person– this is how it is intended, but still allows us to notice some promising options for searching through possible substitutions of numbers and significantly reduce this search. Try to start the search from positions other than those in this article, and you will see that almost all options very quickly lead to a dead end and you need to make more and more new assumptions regarding the further selection of suitable substitutions of numbers. About two months ago I already tried to solve this problem, without having the preparation that I described in previous articles. I checked ten options for her solution and abandoned further attempts. The last time, already being more prepared, I solved this problem for half a day or a little more, but at the same time thinking about the choice from my point of view of the most indicative options for readers and also with preliminary thinking about the text of the future article. And the final result of the solution was as follows:

Actually, this article has no independent significance; it is written only to illustrate how the acquired skills and theoretical considerations described in previous articles make it possible to solve quite complex problems. And the articles, let me remind you, were not about Sudoku, but about mechanisms for solving problems using Sudoku as an example. The subjects, as for me, are completely different. However, since Sudoku is of interest to many, I thus decided to draw attention to a more significant issue that concerns not Sudoku itself, but problem solving.

For the rest, I wish you success in solving all your problems.

Sudoku is an interesting puzzle for training logic, unlike scanword puzzles, which require erudition and memory. Sudoku has many countries of origin, one way or another, it was played in Ancient China, in Japan, North America... In order for you and me to learn the game, we have made a selection How to solve Sudoku from easy to difficult.

To begin with, let's tell you that Sudoku is a square measuring 9x9, which in turn consists of 9 squares measuring 3x3. Each square must be filled with numbers from one to nine so that each number is used only once along a vertical and horizontal line, and only in a 3x3 square.

When you fill in all the cells, you should have all the numbers from 1 to 9 in each of the 9 squares. So, along the horizontal line all the numbers are from 1 to 9. And along the vertical line the same thing, see the picture:

It would seem that, simple rules, but in order to answer the question of how to solve Sudoku, and even more so, if you want to know how to solve complex Sudoku (especially for those who are just starting their journey), you need to solve at least a couple of easy problems. Then it will be clear what we are talking about. Below are the games. Try printing them out and filling them out so that everything fits together:

How to solve difficult Sudoku

I hope you have read the text above and solved the task that you need in order to understand what will be discussed next. If yes, then let's continue.

This part of the article will answer the questions:

How to solve difficult Sudoku?

How to solve Sudoku: methods?

How to solve Sudoku: methods and methods of cells and fields?

So, you were given two games, by solving which you acquired skills and got a general idea. In order to save your time, I will tell you a couple of life hacks for quickly solving Sudoku.

1. Always start with number 1 and go first along the lines and then along the squares. This way you will definitely not get confused and will prevent yourself from making many mistakes.

2. Always check which number is missing where there are fewer empty cells left. This will save time. And be sure to pay attention to how many and what numbers are missing in the 3 by 3 square (both horizontal and vertical lines).

3. If there are a lot of empty cells in a square and you reach a dead end, try dividing the square along lines in your mind. Think about what numbers might be there, and from this you can understand what numbers will be on the same lines in other squares (and perhaps even understand what numbers will be in other squares on another line).

4. Don’t be afraid of anything, it’s better to make a mistake and understand why than to do nothing!

5. More practice and you will become a master.

And if people who solve Sudoku also have abstract intelligence, which gives powerful potential to its owner, then one can move far forward. Read more about such people.

Below you will find a selection of “How to solve difficult Sudoku”, after which you will be able to do a lot!