It often happens that you need to occupy yourself with something, entertain yourself - while waiting, or on a trip, or simply when there is nothing to do. In such cases, various crossword puzzles and scanword puzzles can come to the rescue, but their disadvantage is that the questions there are often repeated and remembering the correct answers and then entering them “automatically” is not difficult for a person with a good memory. Therefore, there is an alternative version of crossword puzzles - 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 - it is a number 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.
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, as a result of his research, developed the game “Latin Square”. It was on its basis that in the seventies of the last century in the USA they came up with number square puzzles. From America they came to Japan, where they received, firstly, their name, and secondly, unexpected wild popularity. This happened in the mid-eighties of the last century.
Already from Japan, the numerical problem went to travel around the world and also reached 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 correctly solve Sudoku, you should devote some time to studying the terminology of this game in order to be confident in the future that you correctly understand what is happening. So, the main element of the puzzle is the cell (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 (a square of 9 cells). A row can also be called a row, a column can be called a column, and an area can be called 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 are competing to get into that cell). There can be several candidates in a cell - from one to five. If there are two of them, they are called a pair, if there are three, they are called a trio, if there are four, they are called 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. So that's all there is to it large field for sudoku there are nine small blocks. The player’s task is to enter numbers from one to nine into all Sudoku cells so that they are not repeated horizontally, vertically, or in a small area. Initially, some numbers are already in place. These are hints given to make solving Sudoku easier. According to experts, a correctly composed puzzle can only be solved in one correct way.
Depending on how many numbers are already in Sudoku, the degrees of difficulty of this game vary. In the simplest ones, accessible even to a child, there are a lot of numbers, in the most complex ones there are practically none, but that makes it all the more interesting to solve.
Varieties of Sudoku
The classic type of puzzle is a large nine by nine square. However, lately, different versions of the game have become increasingly common:
Basic solution algorithms: rules and secrets
How to solve Sudoku? There are two basic principles that can help solve almost any puzzle.
- We remember that each cell contains a number from one to nine, and these numbers should not be repeated vertically, horizontally or in one small square. Let's try to use the method of elimination to find a cell only in which it is possible to find a number. Let's look at an example - in the figure above, take the ninth block (lower right). Let's try to find a place in it for one. There are four free cells in the block, but you cannot place a unit in the third one 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 number, in the area next door. Thus, for a given block it is permissible for a unit to be in only one cell - the first one in the last row. Thus, using the method of elimination, cutting off unnecessary 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 singles”.
- 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 into the central cell. This method is called “singles”.
Very often, the two methods described above are enough to quickly solve Sudoku.
How to solve Sudoku: secrets and methods
It is recommended to adopt the following rule: write down in fine detail in the corner of each cell the numbers that could appear there. As new information is obtained, extra numbers need to be crossed out, and then in the end the correct solution will be visible. In addition, first of all you need to pay attention to those columns, rows or areas where there are already numbers, and as much as possible more- how fewer options remains, the easier it is to cope. This method will help you quickly solve Sudoku. As experts recommend, before entering the answer into a cell, you need to double-check it again so as not to make a mistake, because because of one incorrectly entered number, the entire puzzle can “fly” and 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 the third cell will definitely contain the number six. After all, if there were a four in it, then there could only be five in the first two cells, but this is impossible.
Below are other rules and secrets on how to solve Sudoku.
Locked Candidate Method
When you are working with one specific block, a situation may arise 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 absolutely not be such a 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 appearance of new information, it becomes clear exactly in which cell of a given row or column this number is located.
In the figure above, consider block number six - central right. The number nine in it can only be in the column in the middle (in cells five or eight). This means that in other cells of this area there will definitely not be a nine.
Open Pairs Method
The next secret of how to solve Sudoku is: if in one column/one row/one area two cells can contain only any two identical numbers (for example, two and three), then they can be found in no other cells of this block/row/column will not. This often makes the task much easier. The same rule applies in a situation with three identical numbers in any three cells of the same row/block/column, and with four - respectively, in four.
Hidden pairs method
It differs from the above in the following way: if in two cells of the same row/area/column, among all possible candidates, there are two identical numbers that do not appear in other cells, then they will be located in these places. However, other numbers can be excluded from these cells. For example, if there are five free cells in one block, but only two of them contain the numbers one and two, then that is where they are located. This method works for three and four numbers/cells.
x-wing method
If a specific number (for example, five) can only be located in two cells of a certain row/column/area, then that is where it is located. Moreover, if in an adjacent row/column/area the placement of a five is allowed in the same cells, then this number is not found in any other cell of the row/column/area.
Difficult Sudoku: solution methods
How to solve difficult Sudoku? The secrets, in general, are still the same, that is, all the methods described above work in these cases. The only thing is that in complex Sudoku there are often situations when you have to abandon logic and act at random. This method even has its own name - “Ariadne’s Thread”. We take a number and insert it into the correct cell, and then, like Ariadne, we unravel a ball of thread, checking whether the puzzle fits together. 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 scribbles, it is recommended to do this all on a draft.
Another way to solve complex Sudoku is to analyze three blocks horizontally or vertically. You need to choose a number and see if you can substitute it in all three areas at once. In addition, in cases of solving complex Sudoku, it is not only recommended, but absolutely necessary, to recheck all the cells, return to what you missed before - after all, new information appears that needs to be applied to the playing field.
Mathematical rules
Mathematicians do not remain aloof from this problem. Mathematical methods for solving Sudoku are as follows:
- The sum of all numbers in one area/column/row is forty-five.
- If in some area/column/row three cells are not filled, and it is known that two of them must contain certain numbers (for example, three and six), then the desired third number is found using the example 45 - (3+6+ S), where S is the sum of all filled cells in this area/column/row.
How to increase your guessing speed?
The following rule will help you solve Sudoku faster. You need to take a number that is already in its place in most blocks/rows/columns, and by eliminating 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 in separate books. However, recently all kinds of versions of this game have appeared, for example 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 is being released for completely different platforms, so it doesn’t matter what exactly is installed on your personal computer.
And just recently they appeared mobile applications with the game Sudoku - both for Android and iPhone, 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.
- Eat important recommendation, how to solve Sudoku: take your time. This game is considered relaxing.
- It is recommended to solve the puzzle with a pencil, not a pen, so that you can erase the wrong number.
This puzzle is truly an addicting 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!
Feb 27, 2015 —
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 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.
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Sumdoku
Sumdoku is also known as killer sudoku or killer sudoku. In this type of puzzle, numbers are arranged in the same way as in classic Sudoku. But the field additionally contains colored blocks, for each of which the sum of numbers is indicated. Please note that sometimes numbers may be repeated in these blocks!
How to solve sumdoku?
Consider sumdoku (in the picture on the right). To solve it, remember that the sum of the numbers in any row, any column and any small rectangle is the same. For our case, this is 1+2+3+…+9+10 = 55. For sumdoku 9x9 it would be 45.
Let's pay attention to the highlighted gray blocks. They almost completely (except for one number) cover the two lower rectangles. Let's calculate the sum of the numbers in all marked blocks: 13 + 8 + 13 + 15 + 13 + 7 + 14 + 12 + 5 = (13+13+14) + (13+7) + (12+8) + (15+5 ) = 40 + 20 + 20 + 20 = 100. So, the sum of the numbers in the marked blocks is 100. But if we take the two lower rectangles completely, then the sum of the numbers in them should be 55 + 55 = 110. This means that in the only unmarked cell the number is 10.
As you can see, by constantly solving sumdoku, you will become a master of arithmetic. You can, of course, use a calculator, but this dark and slippery path is not for real samurai
Let us now consider the blocks highlighted in the figure on the right. They cover one penultimate horizontal line of the Sudoku and two “extra” cells. Let's calculate the sum of numbers in blocks: 13 + 8 + 15 + 13 + 10 + 14 = (13+13+14) + (10+15) + 8 = 40 + 25 + 8 = 73. But we know that the sum of numbers in horizontal line is 55, which means you can find out the sum of the numbers in two “extra” cells: 73 - 55 = 18.
Let's write down all possible combinations of numbers in these “extra” cells: 10+8, 9+9, 8+10.
History of Sudoku
9+9 - eliminated, since the cells are located on the same horizontal line, leaving 10+8 and 8+10. But if you put 8 in the first “extra” cell, then in the penultimate horizontal line you will get two fives, and the numbers in the horizontal lines should not be repeated. Thus, we find that the first “extra” cell can only contain 10. We immediately arrange the remaining obvious numbers.
06/15/2013 How to solve Sudoku, rules with example.
I would like to say that Sudoku is a really interesting and exciting task, a riddle, a puzzle, a puzzle, a digital crossword, you can call it whatever you like. The solution of which will not only bring real pleasure to thinking people, but will also allow in the process exciting game develop and train logical thinking, memory, perseverance.
For those who are already familiar with the game in any of its manifestations, the rules are known and understandable. And for those who are just thinking about starting, our information may be useful.
The rules for playing Sudoku are not complicated; they are found on the pages of newspapers or can be found quite easily on the Internet.
The main points are laid out in two lines: the main task of the player is to fill all the cells with numbers from 1 to 9. This must be done in such a way that in a row, column and mini-square 3x3, none of the numbers are repeated twice.
Today we offer you several versions of the Sudoku-4tune electronic game, including more than a million built-in puzzle options in each game player.
For clarity and a better understanding of the process of solving the riddle, consider one of simple options, first difficulty level Sudoku-4tune, 6** series.
And so, a playing field is given, consisting of 81 cells, which in turn make up: 9 rows, 9 columns and 9 mini-squares measuring 3x3 cells. (Fig.1.)
Do not be confused by the further mention of an electronic game. You can find the game on the pages of newspapers or magazines, the basic principle remains the same.
The electronic version of the game provides great opportunities to choose the difficulty level of the puzzle, options for the puzzle itself and their number, at the request of the player, depending on his preparation.
When you turn on the electronic toy, key numbers will be given in the cells of the playing field. Which cannot be transferred or changed. You can choose the option that is more suitable for the solution, in your opinion. Reasoning logically, starting from the given numbers, it is necessary to gradually fill the entire playing field with numbers from 1 to 9.
An example of the initial arrangement of numbers is shown in Fig. 2. Key numbers, as a rule, in the electronic version of the game are marked with an underscore or a dot in the cell. In order not to confuse them in the future with the numbers that will be set by you.
Looking at the playing field. It is necessary to decide where to start the solution. Typically, you need to determine the row, column, or mini square that has the minimum number of empty cells. In the version we have presented, we can immediately select two lines, top and bottom. These lines are missing just one digit. Thus, a simple decision is made, having determined the missing numbers -7 for the first line and 4 for the last, we enter them into the free cells of Fig. 3.
The resulting result: two completed lines with numbers from 1 to 9 without repetitions.
Next move. Column number 5 (from left to right) has only two free cells. After some thought, we determine the missing numbers - 5 and 8.
To achieve a successful result in the game, you need to understand that you need to navigate in three main directions: column, row and mini-square.
In this example, it is difficult to navigate only by rows or columns, but if you pay attention to the mini-squares, it becomes clear. It is impossible to enter the number 8 in the second (from the top) cell of the column in question, otherwise there will be two eights in the second mine-square. Likewise with the number 5 for the second cell (bottom) and the second lower mini-square in Fig. 4 (wrong location).
Although the solution seems correct for a column, nine digits, in a column, without repetition, it contradicts the basic rules. In mini-squares, numbers should also not be repeated.
Accordingly, for the correct solution, you need to enter 5 in the second (top) cell, and 8 in the second (bottom) cell. This decision fully complies with the rules.
For the correct option, see Figure 5. 
Further solution to a seemingly simple task requires careful consideration of the playing field and the use of logical thinking.
How to solve Sudoku - ways, methods and strategy
You can again use the principle of the minimum number of free cells and pay attention to the third and seventh columns (from left to right). There were three cells left unfilled. Having counted the missing numbers, we determine their values - these are 2,3 and 9 for the third column and 1,3 and 6 for the seventh. Let's leave filling out the third column for now, since there is no certain clarity with it, unlike the seventh. In the seventh column you can immediately determine the location of the number 6 - this is the second free cell from the bottom. What is this conclusion based on?
When examining the mini-square, which includes the second cell, it becomes clear that it already contains the numbers 1 and 3. Of the digital combinations 1,3 and 6 we need, there is no other alternative. Filling the remaining two free cells of the seventh column is also not difficult. Since the third row already contains a filled 1, 3 is entered into the third cell from the top of the seventh column, and 1 is entered into the only remaining free second cell. For an example, see Figure 6.
Let's leave the third column for now for a clearer understanding of the moment. Although, if you wish, you can make a note for yourself and enter the expected version of the numbers required for installation in these cells, which can be corrected if the situation becomes clearer. Electronic games Sudoku-4tune, 6** series allow you to enter more than one number in the cells for a reminder.
Having analyzed the situation, we turn to the ninth (lower right) mini-square, in which, after our decision, there were three free cells left.
Having analyzed the situation, you can notice (an example of filling a mini-square) that the following numbers 2.5 and 8 are missing to completely fill it. Having examined the middle, free cell, you can see that of the necessary numbers only 5 fits here. Since 2 is present in the top cell column, and 8 in a row, which, in addition to the mini-square, includes this cell. Accordingly, in the middle cell of the last mini-square we enter the number 2 (it is not included in either the row or the column), and in the top cell of this square we enter 8. Thus, we have the lower right (9th) mini-square completely filled. a square with numbers from 1 to 9, while the numbers are not repeated in columns or rows, Fig. 7.
As free cells are filled, their number decreases, and we are gradually getting closer to solving our puzzle. But at the same time, solving a problem can be both simplified and complicated. And the first method of filling the minimum number of cells in rows, columns or mini-squares ceases to be effective. Because the number of explicitly defined digits in a particular row, column, or mini-square decreases. (Example: the third column we left). In this case, you need to use the method of searching for individual cells, setting numbers that do not raise any doubts.
In electronic games Sudoku-4tune, 6** series, it is possible to use a hint. Four times per game you can use this function and the computer itself will set the correct number in the cell you have chosen. In the 8** series models there is no such function, and the use of the second method becomes the most relevant.
Let's look at the second method in the example we're using.
For clarity, let's take the fourth column. The empty number of cells in it is quite large, six. Having calculated the missing numbers, we determine them - these are 1,4,6,7,8 and 9. You can reduce the number of options by taking as a basis the average mini-square, which has a fairly large number of specific numbers and only two free cells in a given column. Comparing them with the numbers we need, we can see that 1,6, and 4 can be excluded. They should not be in this mini-square to avoid repetition. That leaves 7,8 and 9. Please note that in the row (fourth from the top), which includes the cell we need, there are already numbers 7 and 8 from the three remaining ones that we need. Thus, the only option left for this cell is number 9, Fig. 8 Doubts about the correctness this option The fact that all the figures we considered and excluded were originally given in the assignment does not cause a decision. That is, they are not subject to any change or transfer, confirming the uniqueness of the number we have chosen for installation in this particular cell.
Using two methods simultaneously depending on the situation, analyzing and thinking logically, you will fill in all the empty cells and come to the right decision any Sudoku puzzle, and this riddle in particular. Try to complete the solution to our example in Fig. 9 yourself and compare it with the final answer shown in Fig. 10.
Perhaps you will determine for yourself any additional key points in solving puzzles, and develop your own system. Or take our advice, and it will be useful for you, and will allow you to join a large number lovers and fans of this game. Good luck.
Sudoku ("Sudoku") is a number puzzle. Translated from Japanese, “su” means “digit”, and “doku” means “standing alone”. In the traditional Sudoku puzzle, the grid is a square of size 9 x 9, divided into smaller squares with a side of 3 cells ("regions"). Thus, the entire field has 81 cells. Some of them already contain numbers (from 1 to 9). Depending on how many cells have already been filled, the puzzle can be classified as easy or difficult.
The Sudoku puzzle has only one rule. It is necessary to fill in the empty cells so that in each row, in each column and in each small square 3 x 3 each digit from 1 to 9 would appear only once.
Program Cross+A knows how to solve a large number of varieties of Sudoku.
The task can be complicated: the main diagonals of the square must also contain numbers from 1 to 9. This puzzle is called sudoku diagonals ("Sudoku X"). To solve these tasks you need to check the box Diagonals.
Sudoku-argyle (Argyle Sudoku) contains a pattern of lines arranged diagonally.
Sudoku rules
Argyle pattern consisting of multi-colored diamonds same size, was present on the kilts of one of the Scottish clans. Each of the marked diagonals must contain non-repeating numbers.
The puzzle may contain free-form regions; these are called sudoku geometric or curly ("Jigsaw Sudoku", "Geometry Sudoku", "Irregular Sudoku", "Kikagaku Nanpure").
Letters can be used instead of numbers in Sudoku; these types of puzzles are called Godoku ("Wordoku", "Alphabet Sudoku"). After the solution, you can read the keyword in any row or column.
Sudoku-asterisk ("Asterisk") is a variation of Sudoku that contains an additional area of 9 squares. These cells must also contain numbers from 1 to 9.
Sudoku girandole ("Girandola") also contains an additional area of 9 cells, with numbers from 1 to 9 (a girandole is a fountain of several jets in the form of fireworks, a “fire wheel”).
Sudoku with center points ("Center Dot") is a variant of Sudoku, where the central cells of each region 3 x 3 form an additional area.
The cells in this additional area must contain numbers from 1 to 9.
Sudoku can contain four additional regions 3 x 3. This type of puzzle is called sudoku window ("Windoku", "Four-Box Sudoku", "Hyper Sudoku").
Sudoku puzzle ("Offset Sudoku", "Sudoku-DG") contains additional 9 groups of 9 cells. Cells within a group do not touch each other and are highlighted in the same color. In each group, each number from 1 to 9 should appear only once.
Not a horse's step ("Anti-Knight Sudoku") has an additional condition: identical numbers cannot “beat” each other with a knight’s move.
IN sudoku hermits ("Anti-King Sudoku", "Touchless Sudoku", "Sudoku without touching") identical numbers cannot be in adjacent cells (both diagonally, horizontally and vertically).
IN sudoku-antidiagonal ("Anti Diagonal Sudoku") each diagonal of the square contains no more than three different digits.
Killer Sudoku ("Killer Sudoku", "Sums Sudoku", "Sums Number Place", "Samunamupure", "Kikagaku Nampure"; another name - Sum-do-ku) is a variation of regular Sudoku. The only difference: additional numbers are specified - the sums of values in groups of cells. Numbers contained in a group cannot be repeated.
Sudoku more less ("Greater Than Sudoku") contains comparison signs (">" and "<«), которые показывают, как соотносятся между собой числа в соседних ячейках. Еще одно название — Compdoku.
Sudoku even-odd ("Even-Odd Sudoku") contains information about whether the numbers in the cells are even or odd. Cells containing even numbers are marked in gray, cells containing odd numbers are marked in white.
Sudoku neighbors ("Consecutive Sudoku", "Sudoku with partitions") is a variation of regular Sudoku. It marks the boundaries between adjacent cells that contain consecutive numbers (that is, numbers that differ from each other by one).
IN Non-Consecutive Sudoku numbers in adjacent cells (horizontally and vertically) must differ by more than one. For example, if a cell contains the number 3, adjacent cells should not contain the numbers 2 or 4.
Sudoku points ("Kropki Sudoku", Dots Sudoku, "Sudoku with dots") contains white and black dots at the boundaries between cells. If the numbers in neighboring cells differ by one, then there is a white dot between them. If in neighboring cells one number is twice as large as the other, then the cells are separated by a black dot. Between 1 and 2 there can be a dot of any of these colors.
Sukaku ("Sukaku", "Suuji Kakure", "Pencilmark Sudoku") is a square of size 9 x 9, containing 81 groups of numbers. It is necessary to leave only one number in each cell so that in each row, in each column and in each small square 3 x 3 each number from 1 to 9 would appear only once.
Sudoku chains ("Chain Sudoku", "Strimko", "Sudoku-convolutions") is a square consisting of circles.
It is necessary to arrange the numbers in the circles so that in each horizontal and each vertical all the numbers are different. In the links of one chain, all numbers must also be different.
The program can solve and create puzzles ranging in size from 4 x 4 before 9 x 9.
Sudoku-rama ("Frame Sudoku", "Outside Sum Sudoku", "Sudoku - sums on the side", "Sudoku with sums") is an empty square of size. The numbers outside the playing field indicate the sum of the nearest three digits in a row or column.
Skyscraper Sudoku ("Skyscraper Sudoku") contains key numbers along the sides of the grid. It is necessary to arrange the numbers in a grid; each number indicates the number of floors in the skyscraper. Key numbers outside the grid indicate exactly how many houses are visible in the corresponding row or column when viewed from that number.
Sudoku tripod (Tripod Sudoku) is a type of Sudoku in which the boundaries between regions are not indicated; instead, points are specified at the intersections of the lines. The dots indicate where regional boundaries intersect. Only three lines can extend from each point. It is necessary to restore the boundaries of the regions and fill the grid with numbers so that they are not repeated in each row, each column and each region.
Sudoku mines ("Sudoku Mine") combines the features of Sudoku and “minesweeper” puzzles.
The task is a square in size, divided into smaller squares with a side of 3 cells. You need to place the mines in the grid so that there are three mines in each row, each column and each small square. The numbers show how many mines are in neighboring cells.
Sudoku-half ("Sujiken") was invented by the American George Heineman. The puzzle is a triangular grid containing 45 cells. Some cells contain numbers. It is necessary to fill in all the cells of the grid with numbers from 1 to 9 so that the numbers are not repeated in each row, in each column and on each diagonal. Also, the same number cannot appear twice in each of the regions separated by thick lines.
Sudoku XV ("Sudoku XV") is a variation of regular Sudoku. If the border between adjacent cells is marked with a Roman numeral "X", the sum of the values in these two cells is 10, if the Roman numeral "V" is the sum is 5. If the border between two cells is not marked, the sum of the values in these cells cannot be equal to 5 or 10.
Sudoku Edge ("Outside Sudoku") is a variation of the regular Sudoku puzzle. Outside the grid are numbers that must be present in the first three cells of the corresponding row or column.);
- 16 x 16(size of regions 4 x 4).
Cross+A can solve and create variations of Sudoku consisting of several squares 9 x 9.
Such puzzles are called "Gattai"(translated from Japanese: "connected", "connected"). Depending on the number of squares, the puzzles are designated "Gattai-3", "Gattai-4", "Gattai-5" and so on.
Samurai Sudoku ("Samurai Sudoku", "Gattai-5") is a type of Sudoku puzzle. The playing field consists of five squares of size 9 x 9. The numbers 1 to 9 must be placed correctly in all five squares.
Sudoku flower ("Flower Sudoku", Musketry Sudoku) is similar to Samurai Sudoku. The playing field consists of five squares of size 9 x 9; the central square is entirely covered by four others. The numbers 1 to 9 must be placed correctly in all five squares.
Sudoku-sohei ("Sohei Sudoku") named after warrior monks in medieval Japan. The playing field contains four squares of size 9 x 9
Sudoku mill ("Kazaguruma", "Windmill Sudoku") consists of five squares of size 9 x 9: one in the center, the other four squares almost completely cover the central square. The numbers 1 to 9 must be placed correctly in all five squares.
Butterfly Sudoku ("Butterfly Sudoku") contains four intersecting squares of size 9 x 9, which form a single square of size 12 x 12. The numbers 1 to 9 must be placed correctly in all four squares.
Sudoku cross ("Cross Sudoku") consists of five squares. The numbers 1 to 9 must be placed correctly in all five squares.
Sudoku three ("Gattai-3") consists of three squares of size 9 x 9.
Double Sudoku ("Twodoku", "Sensei Sudoku", "DoubleDoku") consist of two squares of size 9 x 9. The numbers 1 to 9 must be placed correctly in both squares.
The program can solve double sudokus in which the regions have arbitrary shapes:
Triple Sudoku ("Triple Doku") are a puzzle of three squares of size 9 x 9. The numbers 1 to 9 must be placed correctly in all squares.
Twin Sudoku ("Twin Corresponding Sudoku") is a pair of regular Sudoku puzzles, each of which contains several starting numbers. Both puzzles must be solved; in this case, each type of numbers in the first grid corresponds to the same type of numbers in the second grid. For example, if the number 9 is in the upper left corner of the first Sudoku puzzle, and the number 4 is in the upper left corner of the second puzzle, then in all cells where there is a 9 in the first grid, there is a 4 in the second grid.
Hoshi ("Hoshi") consists of six large triangles; The numbers 1 to 9 must be placed in the triangular cells of each large triangle. Each line (of any length, even dashed) contains non-repeating numbers.
Unlike Hoshi, in sudoku star ("Star Sudoku") a row on the outer edge of the grid includes a cell located at the nearest sharp end of the figure.
Tridoku ("Tridoku") was invented by Japheth Light from the USA. The puzzle consists of nine large triangles; each one contains nine small triangles. The numbers from 1 to 9 must be placed in the cells of each large triangle. The field contains additional lines, the cells of which must also contain non-repeating numbers. Two touching triangular cells must not contain the same numbers (even if the cells touch each other by only one point).
Online assistant for solving Sudoku.
If you can't solve a difficult Sudoku, try this with a helper. It will highlight possible options for you.
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!
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. He spent three months creating this complex puzzle.
According to him, the crossword puzzle he created cannot be solved using logic alone. Arto Incala claims that even the most experienced players will spend at least several days on the solution. 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 suppose that if nine had ever really been the right choice, then Incala would hardly have left such an obvious option in sight, although the mechanism of his program could allow 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 number 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 the admissibility of 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 capabilities of an ordinary person - this is how it is intended - but it still allows us to notice some promising options for searching through possible substitutions of numbers and significantly reducing 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 allow one 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.
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 shape of this particular geometric figure, each side of which consists of 9 cells. The large square is filled with small square blocks, sub-squares, with a side of three squares. At the beginning of the game, some of them already contain “hint” numbers.
It is necessary to fill all remaining empty cells with 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 a large number of 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.
