The easiest way to solve Sudoku. Solving difficult sudoku
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.
Hello! In this article, we will analyze in detail the solution of complex Sudoku using a specific example. Before starting the analysis, we agree to call the small squares numbers, numbering them from left to right and from top to bottom. All the basic principles of solving Sudoku are described in this article.
As usual, we will first look at open singles. And there were only two such b5-5, e6-3. Next, we place possible candidates on all empty fields.
Candidates will be placed in small green print to distinguish them from the numbers already standing. We do this mechanically, simply sorting through all the empty cells and entering in them the numbers that can be in them.
The fruit of our labors can be seen in Figure 2. Let's turn our attention to the cell f2. She has two candidates 5 and 9. We will have to go with the guessing method, and in case of an error, return to this choice. Let's put number five. Let's remove the five from the candidates of row f, column 2 and square four.
We will constantly remove possible candidates after setting the number, and in this article we will no longer focus on that!
We look further at the fourth square, we have a tee - these are cells e1, d2, e3, which have candidates 2, 8 and 9. Let's remove them from the rest of the unfilled cells of the fourth square. Move on. In square six, the number five can only be on e8.
More at the moment there are no pairs, no tees, let alone fours. Therefore, let's go the other way. Let's go through all the verticals and horizontals in order to remove unnecessary candidates.
And so on the second vertical, the number 8 can only be on the cells -h2 and i2, let's remove the figure eight from the other unfilled cells of the seventh square. On the third file, the number eight can only be on e3. What we got is shown in Figure 3.
There is nothing more to grab on to. We got a pretty tough nut, but we'll crack it anyway! And so, consider again our pair e1 and d2, arrange it in this way d2-9, e1 -2. And in case of our mistake, we will return again to this pair.
Now we can safely write a deuce into the cell d9! And there are seven in the square, nine can only be on h1. After that, on the vertical 1, a five can only be on i1, which in turn gives the right to place a five on the h9 cell.
Figure 4 shows what we have done. Now consider the next pair, these are d3 and f1. They have candidates 7 and 6. Looking ahead, I will say that the arrangement variant d3-7, f1-6 is erroneous and we will not consider it in the article, so as not to waste time.
Figure 5 illustrates our work. What is left for us to do next? Of course, again go through the options for setting numbers! We put a triple in the cell g1. Save as always so you can come back. One is set on i3. now in the seventh square we get a pair of h2 and i2, with the numbers 2 and 8. This gives us the right to exclude these numbers from the candidates for the entire unfilled vertical.
Based on the last thesis, we arrange. a2 is a four, b2 is a three. And after that we can put down the entire first square. c1 - six, a1 - one, b3 - nine, c3 - two.
Figure 6 shows what happened. On i5 we have a hidden loner - the number three! And i2 can only have the number 2! Accordingly, on h2 - 8.
Now let's turn to the cells e4 and e7, this is a pair with candidates 4 and 9. Let's arrange them like this: e4 four, e7 nine. Now a six is placed on f6, and a nine is placed on f5! Further on c4 we get a hidden loner - the number nine! And we can immediately put four from 8, and then close the horizontal with: c6 eight.
Check if there are large squares on the field with one missing number. Check each large square and see if there is one missing just one digit. If there is such a square, it will be easy to fill it. Just determine which of the digits from one to nine is missing in it.
- For example, a square may contain numbers from one to three and from five to nine. In this case, there is no four there, which you want to insert into an empty cell.
Check for rows and columns that are missing just one digit. Go through all the rows and columns of the puzzle to find out if there are any cases where only one number is missing. If there is such a row or column, determine which number from the row from one to nine is missing, and enter it in an empty cell.
- If there are numbers from one to seven and a nine in the column of numbers, then it becomes clear that the eight is missing, which must be entered.
Carefully look at the rows or columns to fill in the large squares with the missing numbers. Look at the row of three large squares. Check it for two duplicate digits in different large squares. Swipe your finger over the rows that contain these numbers. This number must also be present in the third large square, but it cannot be located in the same two rows that you traced with your finger. It should be in the third row. Sometimes two of the three cells in this row of the square will already be filled with numbers and it will be easy for you to enter the number that you checked in its place.
- If there is an eight in two large squares of the row, it must be checked in the third square. Run your finger along the rows with two eights present, since in these rows the eight cannot stand in the third large square.
Additionally, view the puzzle field in the other direction. Once you understand the principle of looking at the rows or columns of a puzzle, add a look in the other direction to it. Use the above view principle with a little addition. Perhaps when you get to the third large square, in the row in question there will be only one finished number and two empty cells.
- In this case, it will be necessary to check the columns of numbers above and below the empty cells. See if one of the columns contains the same number that you are going to put. If you find this number, you cannot put it in the column where it already exists, so you need to enter it in another empty cell.
Work immediately with groups of numbers. In other words, if you notice a lot of the same numbers on the field, they can help you fill in the rest of the squares with the same numbers. For example, there may be many fives on the puzzle board. Use the above field scan technique to fill it with as many remaining fives as possible.
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 an alternative version of crossword puzzles - this 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 in all Sudoku cells 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.
- 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 one cannot be placed in the third in the top row - 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 exclusion, 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".
- 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 the following rule: write small in the corner of each cell those numbers that could be 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 many as possible - the fewer options remain, the easier it is to handle. 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, it may happen that a certain number in a given 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 identical numbers in any three cells of one 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 fits. 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:
- The sum of all the numbers in one area/column/row is forty-five.
- 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. Sudoku comes out for completely different platforms, so it doesn't matter what exactly is on your personal computer.
And more recently, mobile applications with the Sudoku game have appeared - both for Android and for iPhones, the puzzle is now available for download. And I must say that this application is very popular among cell phone owners.
- The minimum possible number of clues for a Sudoku puzzle is seventeen.
- There is an important recommendation on how to solve Sudoku: take your time. This game is considered relaxing.
- 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!
SUDOKU is a popular puzzle game, which is a number puzzle that can be overcome only by building logical conclusions. In the name Sudoku, translated from Japanese, “su” means “number”, and doku “doku” means “standing apart”. Therefore, "SUDOKU" roughly translates to "single digit".
The name "Sudoku" was given to this puzzle by the Japanese publisher Nicoli in 1984. Sudoku is an abbreviation for "Suuji wa dokushin ni kagiru", which means "there must be only one number" in Japanese. Publisher Nikoli not only came up with a sonorous name, but also for the first time introduced symmetry in tasks for their puzzles. The name of the puzzle was given by the leader of Nicoli - Kaji Maki. The whole world adopted this new Japanese name, but in Japan itself the puzzle is called "Nanpure". Nicoli has registered the word "Sudoku" as a trademark in its country.
Origins of SUDOKU
India is considered the birthplace of chess, England is considered the birthplace of football. The game of Sudoku (sudoku), which quickly spread throughout the world, has no homeland as such. The prototype of Sudoku can be considered the Magic Square puzzle, which appeared in China 2000 years ago.
The history of Sudoku as a game goes back to the famous Swiss mathematician, mechanic and physicist Leonhard Euler (1707 - 1783).
The papers in his archive, dated October 17, 1776, contain notes on how to form a magic square with a certain number of cells, especially 9, 16, 25 and 36. In another document entitled "Scientific study of new varieties of magic square" Euler placed in cells with Latin letters (Latin square), later he filled the cells with Greek letters and called the square Greco-Latin. Exploring various versions of the magic square, Euler drew attention to the problem of combining symbols in such a way that not one of them is repeated in any row and in any column.
In its modern form, Sudoku puzzles were first published in 1979 in Word Games magazine. The author of the puzzle was Harvard Garis of Indiana. Puzzle "Number Place" (translated into Russian - "the place of the number") - this can be considered one of the first releases of modern Sudoku. It added blocks of 3x3 cells, which was an important improvement, as it allowed to make the puzzle more interesting. He used the principle of Euler's Latin square, applied it to a 9x9 matrix and added additional restrictions, the numbers should not be repeated in internal 3x3 squares.
Thus, the idea of Sudoku did not come from Japan, as many people think, but the name of the game is really Japanese.
In Japan, this puzzle was published by Nicoly Inc., a major publisher of collections of various puzzles, in the Monthly Nicolist newspaper in April 1984 under the title "Number can only be used once". On November 12, 2004, The Times published the first Sudoku puzzle on its pages. This publication became a sensation, the puzzle quickly spread throughout Britain, Australia, New Zealand; gained popularity in the US.
Sudoku variants
So what is Sudoku? Currently, there are many upgrades for this popular type of puzzle, but the classic Sudoku is a 9x9 square, divided into sub-squares with sides of 3 cells each. Thus, the total playing field is 81 cells. In the appendix to my work, I will put different types of Sudoku and solutions (my parents helped me solve them).
Sudoku vary in level of difficulty depending on the size of the square:
- 1. For little lovers of puzzles, Sudoku is made with fields of 2x2, 6x6 cells.
- 2. For professionals, there are Sudoku 15x15 and 16x16 cells
Sudoku comes in different levels:
- easy
- average
- complicated
- very complicated
- super complex
Decision Rules
Sudoku puzzles have only one rule. It is necessary to fill in the free cells so that in each row, in each column and in each small 3X3 square, each number from 1 to 9 would occur only 1 time. Some cells in Sudoku are already filled with numbers, and it remains for you to fill in the rest. The more numbers are initially, the easier it is to solve the puzzle. By the way, a correctly composed Sudoku has only one solution.
Sudoku solution
Sudoku solving strategy includes three steps:
- learning the location of the numbers in the puzzle
- preliminary arrangement of numbers
- analysis
The best solution is to write the candidate numbers at the top left corner of the cell. After that, you can see exactly the numbers that should occupy this cell. Sudoku should be played slowly as it is a relaxing game. Some puzzles can be solved in minutes, but others can take hours or, in some cases, even days.
Mathematical basis. The number of possible combinations in 9x9 Sudoku is 6,670,903,752,021,072,936,960 according to Bertham Felgenhauer's calculations.