Principles for solving complex Sudoku. Example of problem solving - the most difficult Sudoku
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, 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 these are 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!
Sudoku is a mathematical puzzle whose homeland is considered to be the land of the rising sun - Japan. Time flies with this incredibly exciting and educational mystery. The article will provide ways, methods and strategies on how to solve Sudoku.
History of the game name
Oddly enough, Japan is not the birthplace of the game. In fact, the puzzle was invented by the famous mathematician Leonhard Euler in the 18th century. From the course of higher mathematics, many should remember the famous “Euler circles”. The scientist was fascinated by the fields of combinatorics and propositional logic; he called his squares of various orders “Latin” and “Greco-Latin”, since he mainly used letters to compose them. But the puzzle gained real popularity after regular publication in the Japanese magazine Nikoli, where it received the name Sudoku in 1986.
What does a riddle look like?
The puzzle is a square field with dimensions of 9 by 9 cells. Depending on the complexity and type of the puzzle, the computer leaves a given number of square cells filled. Sometimes beginners are interested in the question: “How many variations of a puzzle can you make?”
According to the rules of combinatorics, the number of permutations can be found by calculating the factorial of the number of elements. So, Sudoku uses numbers from 1 to 9, which means it is necessary to calculate the factorial of 9. With some simple calculations we get 9! = 1*2*3*4*5*6*7*7*9 = 362,880 - options for various string combinations. Next, you need to use the matrix permutation formula and calculate the number of possible positions of rows and columns. The calculation formula is quite complex; you just need to point out that by replacing only one column/row triple, you can increase the total number of options by 6 times. Multiplying the values we get 46,656 - ways of permutations in the riddle matrix for only 1 combination. It is not difficult to guess that the final number will be 362,880 * 46,656 = 16,930,529,280 game options - decide not to overdecide.
However, according to Bertham Felgenhauer's calculations, the puzzle has many more solutions. Bertham's formulas are very complex, but they give a total number of permutations of 6,670,903,752,021,072,936,960 options.
Rules of the game
The rules of Sudoku vary depending on the type of puzzle. But all options have in common the requirement of classical Sudoku: numbers from 1 to 9 should not be repeated vertically and horizontally of the field, as well as in each selected three-by-three section.
There are other types of games, such as odd-even, diagonal, windoku, girandole, area and Latin sudoku. In Latin, letters of the Latin alphabet are used instead of numbers. The even-odd variant should be solved like a regular Sudoku, only taking into account the multi-colored areas. Cells of one color should contain even numbers, and cells of the second color should contain odd numbers. In the diagonal puzzle, to the classic rules “vertical, horizontal, three by three,” two more diagonals of the field are added, which also should not have repetitions. A variation of the area is a type of colored Sudoku that lacks the three-by-three divisions of the classic type of game. Instead, using color or bold borders, arbitrary areas of 9 cells are selected in which numbers must be placed.
How to solve Sudoku correctly?
The main rule of the riddle is: there is only one correct number for each cell of the field. If you choose the wrong number at some stage, further decision will become impossible. The numbers will begin to repeat vertically and horizontally.
The simplest example of a statement is a situation with 8 known numbers horizontally, vertically or in a three-by-three area. The ways to solve Sudoku in this case are obvious - enter the missing number of the sequence from 1 to 9 into the required square. In the example in the image above, this will be the number 4.
Sometimes two cells of a three-by-three area remain unfilled. In this case, each cell has two possible filling options, but only one is correct. You can make the right choice by considering empty areas not only as part of the area, but also as part of the vertical and horizontal. For example, in a three-by-three square, 2 and 3 are missing. You need to select one cell and consider the vertical and horizontal intersection of which it is. Let's say there is already one 3 vertically, but both sequences are missing 2. Then the choice is obvious.
Entry-level riddles are difficult, as a rule, they provide the opportunity to fill several cells with the only correct values at once. You just need to carefully examine the playing field. But the choice of methods/methods for solving Sudoku is not always so simple.
What does "predetermined choice" mean in Sudoku?
Sometimes the choice is not the only one, but nevertheless predetermined. Let's call this number “unique candidate”. Finding such an arrangement of numbers on the puzzle field is not difficult, but it will require some experience in solving the puzzle. An example of how to correctly solve Sudoku with a unique candidate is described in detail for the playing field option in the image below.
At first glance, the highlighted red square could contain any number except 5. However, in fact, the unique candidate for the location is the number 4. It is necessary to consider all the verticals and horizontals of the three-by-three area in question. So, in verticals 2 and 3 there are fours, which means 4 of the small field can be in one of the three squares of the first column. The top square is already occupied by the number 5, the number of locations for the symbol 4 is reduced. In the lower horizontal line of the area it is also not difficult to find a four, therefore, out of 3 options for the location of the number, only one remains.
Search for a unique candidate on the playing field
The example considered was obvious, since there were simply no other numbers on the field. Finding a unique candidate in a particular puzzle is not easy. The playing field in the image below will serve as a clear example to explain the method of solving Sudoku by searching for a unique candidate.
Although the description of the solution option does not seem simple, its application in practice does not cause difficulties. A unique candidate is always sought in a specific three-by-three area. In this regard, the player is only interested in three verticals and three horizontals of the playing field. All others are considered unimportant and are simply discarded. In the example, you need to find the location of the unique candidate number 7 for the central region. The corner squares of the field in question are occupied by numbers, and the number 7 is already present in the central vertical. This means that the only possible squares for placing the unique candidate 7 are cells 1 and 3 of the middle row of the three-by-three area.
How to solve difficult Sudoku?
Each type of game has 4 difficulty levels. They differ in the number of digits in the initial version of the field. The more there are, the easier it is to solve Sudoku. As in other games, fans organize competitions and entire Sudoku championships.
The most complex versions of the game involve a large number of options for filling each cell. Sometimes there may be the maximum possible number - 8 or 9. In such situations, it is recommended to write down all the options in pencil along the edges and corners of the cell. Listing all combinations, with a detailed study, can already help eliminate overlapping numbers and reduce the number of variations for a single cell.
Color Puzzle Solving Strategies
A more complex version of the game is color Sudoku riddles. Such puzzles are considered difficult due to the introduction of additional conditions. In fact, color is not only an element of complication, but also a kind of hint that should not be neglected when deciding. This also applies to the odd-even game.
But color can also be used when solving ordinary Sudoku, marking more likely cases of substitution. In the above picture of the puzzle, the number 4 can only be placed in the blue and orange squares, all other options are obviously wrong. Highlighting these areas will allow you to distract yourself from the number 4 and switch to searching for other values, but you won’t be able to completely forget about the cells.
Sudoku for kids
It may sound strange, but children love solving Sudoku. The game develops logic and imaginative thinking very well. Scientists have already proven that playing prevents the death of brain cells. People who regularly solve puzzles have higher IQ levels.
For very young children who do not yet know numbers, variants of Sudoku with symbols have been developed. The riddle is absolutely semantically independent. Parents should definitely teach their kids to play Sudoku if they want to develop their children’s logic, concentration and thinking. The game is useful for maintaining mental abilities at any age. Researchers compare the effect of puzzles on the human brain to the effect of physical exercise on muscle development. Psychologists say that Sudoku relieves depression and helps treat dementia.
How to play Sudoku?
Sudoku is a very popular number puzzle. Once you understand how to play Sudoku, you won’t be able to put it down!
The essence of the game:
The cells of the playing field must be filled with numbers from 1 to 9. There should not be repeated numbers in each vertical and horizontal line. Also, they cannot be repeated in small squares (3x3 cells). At the very beginning of the game there are already numbers (depending on the difficulty of the level, the number of initially given numbers may differ).
Rules for playing Sudoku:
- Select a row, column or square with the maximum number of given numbers. Fill in what is missing (it is better to use a pencil). In almost all cases, there is a place where only 1 number fits.
- Next, look through each column in turn, compare which numbers can fit into each cell. You can write down options on a separate piece of paper.
- When also looking at lines and squares, eliminate numbers that are repeated.
- As you fill the puzzle with numbers, it will become easier to solve.
Start playing Sudoku with easy tasks, because the ability to solve the puzzle comes with experience. Or play Sudoku online - incorrect numbers will be highlighted in a different color. This will help you get used to the game. During this lesson, logic develops, so you can gradually complicate the level. Also watch the video attached to the article.
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 all the 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 cases, a 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, and therefore you can download similar puzzles from various sites, solve them online, or buy paper versions at newsstands. 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.
Sudoku is a very interesting puzzle. It is necessary to arrange the numbers from 1 to 9 in the field so that each row, column and block of 3 x 3 cells contains all the numbers, and at the same time they should not be repeated. Let's look at step-by-step instructions on how to play Sudoku, basic methods and solution strategy.
Solution algorithm: from simple to complex
The algorithm for solving the Sudoku mind game is quite simple: you need to repeat the following steps until the problem is completely solved. Gradually move from the simplest steps to more complex ones, when the first ones no longer allow you to open a cell or exclude a candidate.
Single candidates
First of all, for a more clear explanation of how to play Sudoku, we will introduce a system for numbering blocks and cells of the field. Both cells and blocks are numbered from top to bottom and left to right.
Let's start looking at our field. First, you need to find single candidates for a place in the cell. They can be hidden or obvious. Let's consider the possible candidates for the sixth block: we see that only one of the five free cells contains a unique number, therefore, the four can be safely entered into the fourth cell. Considering this block further, we can conclude: the second cell must contain the number 8, since after eliminating the four, the eight does not appear anywhere else in the block. With the same justification we put the number 5.
Review all possible options carefully. Looking at the central cell of the fifth block, we find that besides the number 9 there cannot be any more options - this is a clear single candidate for this cell. Nine can be crossed out from the remaining cells of this block, after which the remaining numbers can be easily entered. Using the same method, we go through the cells of other blocks.
How to detect hidden and obvious “naked pairs”
Having entered the necessary numbers in the fourth block, we return to the unfilled cells of the sixth block: it is obvious that the number 6 should be in the third cell, and 9 in the ninth.
The concept of "naked couple" is present only in the game Sudoku. The rules for their detection are as follows: if two cells of the same block, row or column contain an identical pair of candidates (and only this pair!), then the remaining cells of the group cannot have them. Let's explain this using the eighth block as an example. Having placed possible candidates in each cell, we find a clear “naked pair”. The numbers 1 and 3 are present in the second and fifth cells of this block, and there are only 2 candidates in both, therefore, they can be safely excluded from the remaining cells.
Completing the puzzle
If you have learned the lesson on how to play Sudoku and followed the instructions above step by step, then you should end up with a picture something like this:
Here you can find single candidates: a one in the seventh cell of the ninth block and a two in the fourth cell of the third block. Try to solve the puzzle to the end. Now compare the result with the correct solution.
Happened? Congratulations, because this means that you have successfully learned the lessons of how to play Sudoku and learned how to solve simple puzzles. There are many varieties of this game: Sudoku of different sizes, Sudoku with additional areas and additional conditions. The playing field can vary from 4 x 4 to 25 x 25 cells. You may come across a puzzle in which the numbers cannot be repeated in an additional area, for example, diagonally.
Start with simple options and gradually move on to more complex ones, because with training comes experience.
