# How to solve sudoku: ways, methods and strategy

Sudoku is a mathematical puzzle, the birthplace of which is considered to be the land of the rising sun - Japan. Time for an incredibly exciting and developing mystery flies unnoticed. The article will provide ways, methods and strategies for how to solve sudoku.

## The history of the name of the game

Oddly enough, Japan is not the birthplace of the game. In fact, the puzzle was invented by the famous mathematician Leonard Euler in the 18th century. Of the course in higher mathematics, many should remember the famous “Euler circles”. The scientist was fascinated by the combinatorial and logic of statements, he called his squares of various orders "Latin" and "Greek-Latin", as he used mainly letters to compose. But the real popularity of the puzzle acquired after regular publication in the Japanese magazine Nikoli, where it received the name Sudoku in 1986.

## What does a mystery 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 the specified number of squares filled. Sometimes beginners are interested in the question: "How many puzzle options 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, so it is necessary to calculate factorial 9. By simple calculations we get 9! = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 7 * 9 = 362 880 — variants of various combinations of strings. Next, you need to use the matrix permutation formula and count the number of possible positions of rows and columns. The calculation formula is quite complicated, it’s enough to point out that if you replace only in one triple column / row, 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 equal to 362,880 * 46,656 = 16,930,529,280 game options**-**decide not to decide.

However, according to Bertham Felgenhauer's calculations, the puzzle has many more solutions. Bertham formulas are very complex, but they give the total number of permutations in 6,670,903,752,021,072,936,960 - variants.

## Rules of the game

The rules of sudoku game vary depending on the type of puzzle. But for all the variations, the requirement of the classic sudoku is common: the numbers from 1 to 9 should not be repeated vertically and horizontally, and also in each marked area "three by three."

There are other types of games, such as sudoku, diagonal, vindoku, girandol, region, and Latin sudoku. In Latin, letters of the Latin alphabet are used instead of numbers. The variant even-odd should be solved as a sudoku usual, only taking into account the multi-colored areas. In the cells of the same color should be even numbers, and the second - odd. In the diagonal riddle to the classic rules "vertical, horizontal, three by three" two more diagonal fields are added, in which there should not be any repetitions either. A region variation is a type of color sudoku, in which there are no three-to-three divisions of the classic type of game. Instead, using color or oily boundaries, arbitrary areas of 9 cells are selected, in which numbers are to be placed.

## How to solve sudoku correctly?

The main rule of the puzzle says: there is only one correct version of the number for each cell of the field.If you select the wrong number at some stage, further solution will become impossible. The numbers on the vertical and horizontal will begin to repeat.

The simplest example of a statement is the situation with 8 known numbers horizontally, vertically, or in the "three by three" area. Ways to solve sudoku in this case are obvious - enter the missing digit 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 the "three by three" area remain unfilled. In this case, each cell has two possible fill 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 part of the vertical and horizontal. For example, in the “three by three” square there are not enough 2 and 3. One needs to select one cell and consider the vertical and horizontal intersection, which it is. Suppose there is already one 3 vertically, but in both sequences there is not enough 2. Then the choice is obvious.

Entry level puzzles are difficult, as a rule, they provide an opportunity to fill several cells with the only correct values right away.It is only necessary to carefully consider the playing field. But not always the choice of ways / methods how to solve sudoku is so simple.

## What does “predefined choice” mean in sudoku?

Sometimes the choice is not the only, but nevertheless predetermined. Let's call such a number a “unique candidate.” It is easy to find such an arrangement of numbers on the puzzle field, but it will require some experience in solving a puzzle. An example of how to correctly solve sudoku with a unique candidate is described in detail for the variation of the playing field in the image below.

In the highlighted red square, at first glance any number can stand, except 5. However, in fact, the unique candidate for a place is the number 4. It is necessary to consider all the verticals and horizons of the considered area “three by three”. So, in the verticals 2 and 3 there are fours, which means that 4 small fields can be in one of the three squares of the first column. The upper square is already occupied by the number 5, the number of locations of the symbol 4 is reduced. In the lower horizontal 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 considered example 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 for explaining the method of how to solve sudoku by searching for a unique candidate.

Although the description of the solution does not seem simple, its application in practice is not difficult. A unique candidate is always searched in a specific three-by-three area. In this regard, the player is interested only in three verticals and three horizontals of the playing field. All others are considered irrelevant and simply discarded. In the example, it is necessary to find the place of the unique candidate number 7 for the central region. The corner squares of the field under consideration 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 the 1st and 3rd cells of the middle row of the three-by-three area.

## How to solve difficult sudoku?

In each type of game 4 levels of difficulty are divided. They differ in the number of digits in the initial version of the field.The more of them, the easier it is to solve sudoku. As in other games, fans arrange competitions and whole sudoku championships.

The most difficult variants of the game involve a large number of options for filling each cell. Sometimes they can be the maximum possible number - 8 or 9. In such situations, it is recommended to write down all the options at the edges and corners of the cage with a pencil. Enumeration of all combinations, with a detailed study, can already help to eliminate overlapping numbers and reduce the number of variations for a single cell.

## Color strategies for solving the puzzle

A more difficult version of the game are sudoku puzzles with color. Difficult such puzzles are considered due to the introduction of additional conditions. In fact, color is not only an element of complication, but also a kind of hint, which should not be neglected in the decision. This also applies to the game of even-odd.

But color can also be used to solve ordinary sudoku, noting the more likely cases of substitution. In the above puzzle image, the number 4 can only be put in blue and orange cells, all the other options are obviously wrong.Selecting the specified areas will allow you to distract from the number 4 and switch to the search for other values, while forgetting about the cells will not work.

## Sudoku for kids

This may sound strange, but kids love to solve sudoku. The game is very well developed logic and imaginative thinking. Scientists have already proven that the game prevents the death of brain cells. People who solve the puzzle regularly have a higher IQ.

For very young children, who do not yet know the numbers, sudoku variants with symbols have been developed. The riddle is absolutely semantically independent. Parents should definitely teach kids to play sudoku if they want to develop the logic, concentration and thinking of children. The game is useful for maintaining mental abilities at any age. Researchers compare the action of the puzzle on the human brain with the effect of physical exercises for the development of muscles. Psychologists say that sudoku relieves depression and helps in the treatment of dementia.