Sudoku

Sudoku is a logic-based puzzle game. The puzzle is about digit placements on a 9x9 grid. The goal is to fill the entire 9x9 grid with digits from 1 to 9 according to the Sudoku rules. As the rules of Sudoku are simple, it has become one of the most popular puzzle games in the world. People love to solve Sudoku puzzles to exercise their brains or just play some casual Sudoku puzzle games for fun. Let's learn more about Sudoku and see if you will love this game.

Classic Sudoku Grid

A Sudoku puzzle is typically played on a 9x9 grid. In a classic Sudoku grid (see Figure 1 below), there are nine horizontal rows R1 to R9, nine vertical columns C1 to C9, and nine 3x3 sub-grids (called "blocks") B1 to B9. Each row, each column, and each 3x3 block have nine squares (called "cells"), respectively. In total, there are 81 cells in the entire grid.

C1
C2
C3
C4
C5
C6
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C9
R1
R2
R3
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Figure 1: Classic 9x9 Sudoku Grid

Sudoku Rules

In order to play Sudoku, we need to know the rules of the game. In fact, the Sudoku rules are very simple:

On a 9x9 Sudoku grid with nine rows, nine columns, and nine 3x3 blocks, all 81 cells are filled with digits from 1 to 9, so that

  1. Each row must contain each of the nine digits from 1 to 9 once, and only once.
  2. Each column must contain each of the nine digits from 1 to 9 once, and only once.
  3. Each 3x3 block must contain each of the nine digits from 1 to 9 once, and only once.

Based on these rules, we notice that each row, each column, or each 3x3 block has exactly 9 cells, and it must contain each of the nine digits from 1 to 9. As a result, a 9-cell unit (a row, a column, or a 3x3 block) cannot have the same digit more than once. That means there are no repeated digits in a row, in a column, or in a 3x3 block. This is a very important fact about Sudoku. Let's always keep this in mind when playing Sudoku.

More Details About Sudoku Rules

Sudoku Puzzle

What is a Sudoku puzzle? A Sudoku puzzle is a partially completed Sudoku grid, and the objective is to fill the entire grid with digits from 1 to 9 so that each of these 9 digits appears once in each row, each column, and each 3x3 block. Below is an example of a Sudoku puzzle.

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Figure 2: A Sudoku Puzzle

In Figure 2, the Sudoku puzzle has 24 pre-filled digits in the grid. To solve this puzzle, you need to fill up all the empty cells in the entire grid with the right digits according to the Sudoku rules. If all your entries in the grid match the unique solution for the puzzle as shown in Figure 3 below, then you have solved the puzzle.

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Figure 3: The Unique Solution to the Sudoku Puzzle in Figure 2

Notice that the solution must obey the Sudoku rules. Each row, each column, and each 3x3 block has each of the 9 digits from 1 to 9, and there are no repeated digits in any rows, in any columns, or in any 3x3 blocks. You may check that the solution shown in Figure 3 satisfies all these requirements.

Now, there is a question: Is any partially filled Sudoku grid a valid Sudoku puzzle? The answer is simply no. A valid Sudoku puzzle has only one unique solution. That means, given a valid Sudoku puzzle, there will be only one combination of digit placements that can satisfy the Sudoku rules with the pre-filled digits. So, the following cases are not valid Sudoku puzzles:

  1. The pre-filled digits violate the Sudoku rules. For example, the same digit appears more than once in a single row, in a single column, or in a single 3x3 block.
  2. The pre-filled digits do not violate the Sudoku rules, but there is no solution for the pre-filled grid. That means, based on the pre-filled digit positions, there is no combination of digit placements that can satisfy the Sudoku rules.
  3. The pre-filled digits do not violate the Sudoku rules, but there is more than one solution for the pre-filled grid. That means, based on the pre-filled digit positions, there are at least two combinations of digit placements that can satisfy the Sudoku rules.

The puzzles in the first two cases have no solutions. Therefore, it is meaningless to play with them. No one wants to play a puzzle without solutions.

Although the puzzles in the third case have solutions, they cannot be solved by logic alone, and therefore it is not really a logic-based game any more. So, those puzzles are also considered invalid. Figure 4 below is an example of an invalid puzzle that has multiple solutions.

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Figure 4: A Invalid Sudoku Puzzle With Multiple Solutions

At first glance, all the pre-filled digits for the puzzle in Figure 4 do not violate any Sudoku rules. So, it is impossible to tell that it is an invalid puzzle. But when you try to solve the puzzle, you cannot do it with logic alone. In fact, the puzzle has more than one solution, and Figure 5 below demonstrates two of them. So, there is no way you can solve it with logic alone.

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Figure 5: Two Distinct Solutions to the Puzzle in Figure 4

Since Sudoku is a logic-based game and those invalid puzzles cannot be solved with logic alone, we do not include any invalid puzzles in our database and only provide valid Sudoku puzzles. When you play a Sudoku puzzle on our website, you will never get an invalid puzzle. All the puzzles here are guaranteed to have only one solution and can be solved with logic.

More Details About Valid Puzzles and Invalid Puzzles

Tips to Solve a Sudoku Puzzle for Beginners

A Sudoku puzzle is a partially completed Sudoku grid with pre-filled digits in some cells. The objective is to fill the entire grid with digits from 1 to 9 so that each row, each column, and each 3x3 block have all 9 digits from 1 to 9 once and only once. For those who are new to this puzzle game, you may have no clue how to determine the correct digit for each empty cell of the grid. Here are some tips for beginners.

When playing Sudoku, always keep these two things in mind. First, each 9-cell unit (a row, a column, or a 3x3 block) must contain each of the nine digits from 1 to 9. Second, no digit can appear more than once in any 9-cell unit (a row, a column, or a 3x3 block). With these in mind, you may try to determine the correct digit for each empty cell.

To complete a Sudoku puzzle, each digit from 1 to 9 needs to be filled nine times for the entire Sudoku grid. Start with the digit that has been filled with the most times in the Sudoku grid, and then in a 3x3 block without this digit, look for the empty cells that are possible positions for the digit. If there is only one such empty cell, then it can be determined that the digit is the correct answer to that empty cell.

The reason to start with the most filled digit is that it will usually give you a better chance to find the right positions for the digit. In fact, you can go through this process with the other digits. But working with the digits that have appeared more times in the grid will have a higher success rate.

Also, instead of checking the 3x3 blocks, you can check the rows and columns to look for a unique possible position for the digit.

With this method, you can solve most of the very easy Sudoku puzzles, such as those in our "Kindergarten" level.

More Tips for Beginners

Difficulty Level of Sudoku Puzzles

A Sudoku puzzle is a partially completed Sudoku grid with pre-filled digits in some cells. Those pre-filled digits are called the clues for the Sudoku puzzle. Usually, the difficulty level of a Sudoku puzzle depends on the number of clues in the grid. The fewer clues in the Sudoku grid, the harder this Sudoku puzzle will be.

Is it true that if a Sudoku puzzle has many clues in the grid, then this puzzle must be easy? In most cases, it is. However, in some cases, even if many digits have been filled in the grids, it still needs some advanced solving techniques to solve the puzzles. Therefore, these puzzles cannot be classified as easy puzzles.

Sudoku Solving Techniques

To solve a Sudoku puzzle, always keep these two things in your head. First, each 9-cell unit (a row, a column, or a 3x3 block) must contain each of nine digits from 1 to 9. Second, no digits can be repeated in any 9-cell unit. It can help you make the correct decisions and avoid simple mistakes.

Two basic techniques can help you find the right digit for an empty cell:

  1. For each of the 9-cell units (rows, columns, and 3x3 blocks), look for how many possible positions (cells) there are for a specific digit. If there is only one, then that position is the correct place for the digit.
  2. For any empty cell, check how many digits can be candidates for that cell. If there is only one candidate, then that must be the right digit for that cell. How do you determine the candidates for a cell? If a digit appears in the same row, the same column, or the same 3x3 block of a cell, then it cannot be a candidate for that cell. So, only digits that do not appear in the same row, in the same column, or in the same 3x3 block can be candidates for that cell.

Just using these two techniques, you can solve many Sudoku puzzles at a low difficulty level. But in some cases, you cannot complete a Sudoku puzzle by just using these two techniques. You will need more advanced solving techniques to complete the Sudoku puzzle.

In general, advanced techniques can eliminate a few candidate digits from some cells, and then we may reach a situation where we can use those two basic techniques again to continue the game. We will not go into detail about those advanced techniques here. But we have prepared separate tutorials for you. After learning those techniques, you can challenge yourself with more difficult Sudoku puzzles.

Tutorials on Sudoku Solving Techniques

Now you know about Sudoku. Go play some Sudoku and enjoy the fun it brings!

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