Tips for solving Sudoku. How to play Sudoku: step by step puzzle solution

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

Loner or the last hero.

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

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

There are three ways to search for singles:

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

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

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

"Hidden Units"

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

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

Naked couple

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

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

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

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

hidden couple

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

Sudoku is a number puzzle. Today it is so popular that most people are very familiar with it or have simply seen it in printed publications. In our article we will tell you where this game came from, as well as who invented Sudoku.

Despite the Japanese name, the history of Sudoku does not begin in Japan. The prototype of the puzzle is considered to be the Latin squares of Leonhard Euler, a famous mathematician who lived in the 18th century. However, in the form in which it is known today, it was invented by Howard Garnes. Being an architect by training, Garnes simultaneously invented puzzles for magazines and newspapers. In 1979, an American publication called “Dell Pencil Puzzles and Word Games” first published Sudoku on its pages. However, then the puzzle did not arouse interest among readers.

It was the Japanese who were the first to appreciate the rebus. In 1984, a Japanese publication published the puzzle for the first time. It immediately became widespread. It was then that the puzzle got its name - Sudoku. In Japanese, “su” means “number” and “doku” means “standing alone.” Some time later, this rebus appeared in many printed publications in Japan. In addition, separate collections of Sudoku were published. In 2004, the puzzle began to be published in UK newspapers, which marked the beginning of the game's spread outside Japan.

The puzzle is a square field with a side of 9 cells, divided in turn into squares measuring 3 by 3. Thus, the large square is divided into 9 small ones, the total number of cells of which is 81. Some cells initially contain clue numbers. The essence of the rebus is to fill empty cells with numbers so that they are not repeated in rows, columns, or squares. Sudoku only uses numbers from 1 to 9. The difficulty of the puzzle depends on the location of the clue numbers. The most difficult, of course, is the one that has only one solution.

The history of Sudoku continues in our time, and successfully. The game is becoming an increasingly common puzzle game, largely due to the fact that it can now be found not only on the pages of the newspaper, but also on your phone or computer. In addition, various variations of this rebus have appeared - letters are used instead of numbers, the number of cells and the shape change.

In this article we will look in detail at how to solve complex Sudoku using the example of diagonal Sudoku.

We get condition number 437, which is shown in Figure 1. And the first square immediately catches your eye, it is the most saturated with open numbers. The numbers 1, 3,4,9 are missing. But since the horizontal line a already contains three, the number three is placed on c1. We cannot accurately place the rest. So let’s look at what else we have. For example, the vertical is 4 and here the number four can only be on b4, due to the presence of a four in the fifth square and on the horizontal c. We will not put the remaining numbers for now.

All the techniques and methods that we will use further apply to solving both simple and complex Sudoku.

What do we have on horizontal b? There is not enough three here and it can only stand on b8. (In the second square it is already there and on vertical 9). And if we carefully examine the horizontal line b further, we will find that we have a hidden single - the number 9 on cell b9. Because the other candidates (these are 1 and 5) cannot stand on this square!

What can we do next? If we consider square five. Here the numbers 3 and 5 can be either on d5 or e6. This means that we do not consider these cells for the remaining numbers. Based on this, there is only one place left for the one - cell d6.

The result of our actions is shown in Figure 2. Thanks to our analysis, row b is filled in completely. One on b5, five on b6. What gives us the right to place 3 and 5 in the fifth square!

Let's continue the analysis of the fifth square. It lacks the number 7, it is not on the main diagonals, and what is most interesting is on the vertical 4. Thanks to this very vertical, we can say for sure that the number seven in the fifth square can be either on f4 or e4. Since the horizontal lines c and d already contain seven. And she cannot stand on e5 because of vertical 4. Next, let’s turn to the main horizontals. And then the sevens are immediately placed! On i9 and f4.

What we got can be seen in Figure 3. Next, we will continue the analysis of the main diagonals. If we look at the one coming from square a1, then it lacks a two, which is placed only on h8. This diagonal also lacks 1, 8 and 9. The 1 can only be placed on a1, put it quickly! But the eight cannot stand on d4, since it is already on the horizontal d. We arrange - d4 -9, e5 -8.

But now we can completely fill the fifth and first squares! What we got is shown in Figure 4.

Pay attention to vertical 3. Here you need to place 1, 6, 7. The unit is placed only on f3, and based on this the rest are placed - e3 -7, h3-6. Next in line we have vertical 9, as its placement is simply fabulous. d9-2, g9-6, h9-8.

What if we check for open singles?! For example, the number three is safely placed on cells d2 and h5. Although further analysis of singletons does not yield anything. Then let's turn to the remaining diagonal. She is missing 6, 2, 4. The number six can only be on c7. The rest is easy to fill out.

Why is vertical 4 not set to the end? Let's fix it. s4 -8.

The result of our research is shown in Figure 5. Now let's fill the horizontal line c. s8-1, s5-9, s6-2. And this is all based on the presence of these numbers in other verticals. Based on the horizontal c, it is easy to fill the horizontal d. d1-6, d7 -4. Then the third square is quite simply filled in. But the second square has not yet been filled, although there are also only two candidates - six and seven. But they do not occur along verticals five and six, and therefore we will put them aside for now.

Having analyzed all the verticals and horizontals, we come to the conclusion that it is impossible to put a single number unambiguously. Therefore, let's move on to considering squares. Let's turn to the sixth square. 5,6,8,9 are missing here. But we can definitely put numbers 6 and 8 on cells f7 and f8. Thanks to our analysis, the entire f horizontal line is marked! f1 -9, f2 -5. And what we see here is that the fourth square is completely filled! e1- 4, e2 -2.

What we got can be seen in Figure 6. Now let’s turn to square nine. Here we have one open single - number one on i7. Thanks to which we can put a one in the seventh square on g2. Eight on i2.

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

The puzzle is published in newspapers and magazines; there are computer versions of the game and mobile applications. The essence and rules in any of them are the same.

How to play

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

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

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

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

Thus, in each row and each column of the large square there will be numbers from one to ten, any small square will also contain these numbers without repetition.

Difficulty levels

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

Solution techniques

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

Elimination method

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

First of all, we take on the square most filled with numbers - the bottom left one. It is missing one, seven, eight and nine. To find out where to put the one, let's look at the columns and rows where this number is: it is in the second column, so our empty cell (the lowest one in the second column) cannot contain it. This leaves three possible options. But the bottom line and the second line from the very bottom also contain a 1 - therefore, by the method of elimination, we are left with the upper right empty cell in the subsquare in question.

Similarly, fill all empty cells.

Writing candidate numbers to a cell

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

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

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

But for those who find solving Sudoku too much effort, this could literally be the perfect solution.

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

Complex network researchers Zoltan Torozkay 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? It was created by mathematician Arto Incala and is claimed to be the hardest Sudoku in the world. Photo from nature.com

Torozkay and Erksi-Ravaz began analyzing 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 guessing techniques to solve these problems. Thus, Sudoku fans 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 labor-intensive and time-consuming.

Instead, Torozkay and Erksi-Ravaz proposed a universal analog algorithm that is completely deterministic (does not use guesswork or brute force) and always finds the correct solution to the problem, and quite quickly.

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

The researchers also found that the time it took to solve a puzzle using their analog algorithm correlated with the difficulty level of the task as judged by humans. 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 “moderately difficult,” 3 is “difficult,” and 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 complex riddle of the known ones still has a rating of 3.6; More complex Sudoku problems are not yet known.

The theory begins by mapping the probabilities for each individual square. Photo from nature.com

"I wasn't interested in Sudoku until we started working on more general class feasibility of Boolean problems, says Torozkay. - Since Sudoku is part of this class, the 9th order Latin square turned out to be for us good field for testing, that’s how I met 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 brute force, which is a choice at random, and if the guess is wrong, we need to go back a step or several steps back and start over. Our analogue decision model is deterministic: there is no random selection or return in the dynamics.”

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

Torozkay and Erksi-Ravaz believe that their analog algorithm has the potential to be applied to the solution large quantity various tasks and problems in industry, computer science and computational biology.

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

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

Torozkai and Erksi-Ravaz's methodology was first published in Nature Physics and later in Nature Scientific Reports.