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## Introduction to Sudoku

This fun, easy-to-understand puzzle game has captured the fascination of humans for nearly 3,000 years.

The object of the game is to fill in the Sudoku grid with a series of one-digit numbers, each of which appears only once in a given row, column, or 3x3 block. At the start, certain numbers are provided in the Sudoku grid, which serve as clues to help the player gradually solve the entire puzzle.

Anyone can play Sudoku. There are no calculations involved; it is entirely a game of logic, so you don't have to be a mathematician to solve Sudoku grids. This explains why Sudoku has become a true global phenomenon. Millions of people play Sudoku every day!

By definition, a valid Sudoku grid must have one (and only one) solution. We guarantee that all of our Sudoku grids have a unique solution. Although it is easier to design grids with multiple solutions (or no solution at all), these cannot be considered true Sudoku puzzles. As in the case in many logic games, there can only be one answer. Designing a grid therefore requires careful attention, because even one misplaced number would make the puzzle impossible to solve.

There is also an unwritten rule that the beauty of a Sudoku grid lies in the symmetrical distribution, on either side of the grid's two diagonals, of the numbers provided at the start. This visual harmony is highly sought after among the most avid Sudoku players. Even though it is infinitely more complicated to create symmetrical grids, especially those guaranteed to have a unique solution, we only design Sudoku grids that feature this type of symmetry. Some of our grids took weeks of computing to develop and we are therefore proud to be able to offer you these symmetrical, single-solution puzzles. After all, Sudoku isn't just a game; it's a philosophy and a lifestyle where beauty and harmony are the top priority!

The numerals in Sudoku puzzles are used solely for convenience; arithmetic relationships between them are irrelevant. Any set of distinct symbols will do; letters, shapes or colors may be used without altering the rules of the game.

The attraction of the game is that the rules are simple, yet the line of reasoning to solve the puzzle is complex. The grids we publish are ranked in terms of their difficulty from 1 (easiest) to 5 (most difficult). In general, the more numbers that are provided at the outset, the easier the puzzle will be to solve, and vice versa, though there are some exceptions.

In recent years, Sudoku's incredible rise in popularity and rapid introduction in international newspapers have made it the favorite puzzle game of the 21st century. Moreover, many governments encourage people to play Sudoku because the game is considered to have a significant role in preventing age-related diseases (especially Alzheimer's).

## Basic Method for Solving a Sudoku Puzzle

#### Principle

Start by scanning the Sudoku grid for each number from 1 to 9. In each block:

• Check whether the number appears;
• If the number does appear, determine which other squares in the same row or column cannot accept that number;
• If the number does not appear, determine which other squares cannot accept that number, given the position of other appearances of that same number in other blocks in the same row and column.

When there is only one possible value for a row, column, or block, this is where the number must appear. With a bit of experience, you will be able to visualize the squares where the number could appear as though they were "lit up" on the Sudoku grid. This will allow you to detect more advanced configurations.

If a Sudoku can be solved using only basic strategies, experienced players may not find it necessary to write down candidate numbers in the squares.

#### Singleton

A "singleton" is a trivial case where there is only one empty cell in a "region" (row, column, or block). In this case, the number value of that cell has to be the number that is missing in the region: it is both the only place where the missing number can go (hidden singleton) and the only value that the empty cell can accept (naked singleton).

This configuration occurs most often once a puzzle is close to being solved, when nearly all the Sudoku squares are filled in.

More generally, the term "singleton" refers to a situation where there is only one solution to a specific square, whether this is because it can only accept a single value (naked singleton) or because a value can only be in a single square (hidden singleton), as any other choice would lead to an immediate mismatch. Singletons differ from "pairs," "triplets," and "quads," where there could be several potential values in play simultaneously.

#### Direct Elimination: Hidden Singleton

When searching for a "hidden singleton", the question to ask is: "In this region (row, column, or block), which squares could potentially accept a 1 (2, 3 ... 9)?" If a number candidate appears only once in the region in question, then this must be the value for the cell.

The more frequently a value appears in the Sudoku grid, the easier it is to search for the hidden singleton; as position constraints increase, the number of possible positions decreases.

Marking potential values in the cells is of limited assistance when looking for hidden singletons; you will still need to scan the entire "region" to check that the value being sought appears as a candidate value only once. This is why these singletons are called "hidden."

Conversely, the "hidden singleton" is often easy to find by systematically scanning the numbers and blocks, as the position is dependent solely on the position of the number in question in the neighboring blocks and on whether the squares of the block in question are available or filled in.

#### Indirect Elimination

Indirect elimination is an extension of direct elimination.

While scanning the Sudoku grid to locate the potential squares for a particular candidate, you may find that all the available squares in a block are in the same row (or column). In such a case, regardless of the ultimate position of the candidate value in the block, the value cannot appear in any other available squares in the same row (or column) in the other blocks. In other words, if the candidates within a block are all in the same row, that value can be excluded from the other available squares throughout the row.

Similarly, when candidates are limited to two rows (or columns) in two contiguous blocks, the candidate values of the third block can appear only in the third row (or column).

This restriction can lead to the identification of a hidden singleton. In a more subtle way, it can also lead to the conclusion that, in another block along the same row (or column), the candidate values can be located only within a single row or column. This will produce a chain reaction of indirect eliminations. Therefore, this initial process of indirect elimination can be performed without marking the squares; however, it requires more logical thinking.