How to Solve a Sudoku Puzzle: Step-by-Step Guide with Example

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"Sudoku is a fun and challenging puzzle game that tests your logical thinking. In this article, I'll explain the basic rules and demonstrate how to solve a Sudoku puzzle step by step."

In this game, you can see that the Sudoku grid consists of 9 rows and 9 columns. The grid is divided into nine 3×3 boxes (or blocks), each containing nine cells. In the example above, the boxes are coloured differently to make them easier to identify.

The main rule of Sudoku is that the numbers in each row, column, and 3×3 box must be unique. The objective of the game is to fill all the empty cells with numbers from 1 to 9.

To solve a Sudoku puzzle correctly, every row, every column, and every 3×3 box must contain each number from 1 to 9 exactly once, with no repeated numbers. Using logical thinking and careful observation, players can determine which numbers belong in the empty cells and gradually complete the puzzle.

Now, let us identify which number appears most frequently in the puzzle. Since there are nine 3×3 boxes (matrices) in a Sudoku grid, each number from 1 to 9 must appear exactly once in every box. Therefore, each number should appear in all nine boxes when the puzzle is completed.

In this example, the number 9 already appears in Boxes 1, 2, 5, 6, 7, 8, and 9.

This means that the number 9 is present in 7 out of the 9 boxes. Therefore, it still needs to be placed in the remaining 2 boxes:

9 − 7 = 2

By focusing on the boxes where the number 9 is missing and applying the Sudoku rules for rows and columns, we can determine the correct positions for these remaining 9s.




In the diagram below, we can see that in Matrix 4 (the lavender-coloured box), the number 9 must be placed in the cell next to 7 and not in the cell next to 5.

The reason is that a 9 already appears in Matrix 5 in the same row as 5. According to the rules of Sudoku, a number cannot be repeated in the same row or column.

Therefore, placing a 9 beside 5 in Matrix 4 would result in two 9s appearing in the same row, which is not allowed.

Since every number must be unique within a row, 9 cannot be placed in Cell 8 of Matrix 4. As a result, the only possible position for 9 in Matrix 4 is the cell beside 7.

By using this process of elimination and applying the Sudoku rules, we can determine the correct position of the number 9.

Similarly, in Matrix 3 (the lime-coloured box), we need to determine the correct position for the number 9.

We can see that a 9 already appears in the first row of Matrix 1 and in the second row of Matrix 2. Therefore, within Matrix 3, the number 9 cannot be placed in any cell that belongs to those rows.

After applying these row restrictions, the only valid position for 9 in Matrix 3 is the eighth cell, beside the number 6. This cell is located in a row and column that do not already contain a 9.

Therefore, we place 9 in the eighth cell of Matrix 3.

With this placement, the number 9 now appears exactly once in each of the nine 3×3 boxes (matrices). Thus, all the positions for the number 9 have been successfully completed, and we can move on to solving the remaining numbers in the Sudoku puzzle.




Next, let us analyse the number 8.

We can see that 8 already appears in Matrices 1, 2, 3, 4, 7, and 9. This means that the number 8 is present in 6 of the 9 matrices.

According to the rules of Sudoku, each number from 1 to 9 must appear exactly once in every 3×3 matrix. Therefore, the number 8 still needs to be placed in the remaining matrices.

Since 8 has already appeared in 6 matrices, the number of matrices where it is still missing is:

9 − 6 = 3

Therefore, the number 8 must be placed in Matrices 5, 6, and 8.

We can now examine these three matrices and use the row and column restrictions to determine the exact position of each remaining 8.

In Matrix 5 (the light-blue-coloured box), we need to find the correct position for the number 8.

The number 8 cannot be placed in the eighth or ninth cell of Matrix 5, beside the number 9, because there is already an 8 in Matrix 4 in the same row. According to the rules of Sudoku, a number cannot be repeated within the same row.

After eliminating these cells, we find that the first cell of Matrix 5 is the only valid position for the number 8. This cell does not conflict with any existing 8 in its row or column.

Therefore, 8 is placed in the first cell of Matrix 5. By applying the Sudoku rule that each number must be unique in every row, column, and 3×3 box, we can confidently determine the correct position of 8.

In Matrix 6 (the lavender-coloured box), we need to determine the correct position for the number 8.

The number 8 cannot be placed beside 9 in the first row of Matrix 6 because an 8 already exists in the same row in Matrix 5. According to Sudoku rules, a number cannot be repeated within a row.

Similarly, 8 cannot be placed beside 1 in Matrix 6 because there is already an 8 in Matrix 4 in the same row. Placing another 8 there would violate the rule that each row must contain unique numbers.

After eliminating these invalid positions, the only possible location for 8 is the cell beside 5 in Matrix 6.

Therefore, 8 is placed beside 5 in Matrix 6. This placement satisfies the Sudoku requirement that numbers must be unique in every row, column, and 3×3 box.

Similarly, in Matrix 8 (the peach-coloured box), the number 8 is placed in the cell beside 5 and not in the cell beside 9.

This is because placing 8 beside 9 would result in a duplicate 8 in the same row or column. According to the rules of Sudoku, the numbers within each row and column must be unique and must not be repeated.

Therefore, after applying the row and column restrictions, the only valid position for 8 in Matrix 8 is the cell beside 5.

Hence, 8 is entered beside 5 in Matrix 8, ensuring that the Sudoku rules are satisfied.

Now, as shown in the figure above, Matrices 2, 4, and 8 each have only one vacant cell remaining.

Since every 3×3 matrix in Sudoku must contain the numbers 1 to 9 exactly once, we can determine the missing number by identifying which number is absent from each matrix.

In Matrix 2, the only missing number is 3, so we fill the vacant cell with 3.

Similarly, in Matrix 4, the only missing number is 3, so we place 3 in the empty cell.

In Matrix 8, the only missing number is 6, so we fill the vacant cell with 6.

By completing these cells, all the numbers in Matrices 2, 4, and 8 become unique, satisfying the Sudoku rule that each 3×3 box must contain the numbers 1 to 9 exactly once.

Next, let us examine the number 7.

We can see that 7 is already present in Matrices 1, 2, 4, 8, and 9. Since each number must appear exactly once in every 3×3 matrix, the number 7 is still missing from Matrices 3, 5, 6, and 7.

Therefore, we need to place 7 in these four matrices by applying the same method used previously for the numbers 9 and 8. By checking the rows and columns that already contain a 7, we can eliminate invalid positions and identify the correct cell for 7 in each matrix.

Using this process of elimination, we can gradually determine the positions of all the remaining 7s while ensuring that every row, column, and 3×3 box contains unique number.

After applying the Sudoku rules and eliminating the invalid positions, we can determine the correct locations for the number 7 in the remaining matrices.

The number 7 is placed in the ninth cell of Matrix 3, the eighth cell of Matrix 5, the second cell of Matrix 6, and the fourth cell of Matrix 7.

These positions satisfy all the Sudoku requirements, as no row, column, or 3×3 matrix contains a duplicate 7. By carefully checking the existing numbers in the corresponding rows and columns, we can confirm that these are the only valid locations for the number 7.

With these placements, the number 7 now appears exactly once in each of the nine 3×3 matrices, completing all the entries for 7 in the puzzle.

Next, let us examine the number 6.

We can see that 6 is already present in Matrices 2, 3, 4, 6, and 8. Therefore, it still needs to be placed in Matrices 1, 5, 7, and 9.

Before determining the positions of 6 in these matrices, we notice that Matrix 5 has only one empty cell remaining. Since every 3×3 matrix must contain the numbers 1 to 9 exactly once, the missing number in Matrix 5 must be 6.

Therefore, we place 6 in the vacant cell of Matrix 5. With this placement, Matrix 5 is completed, and all its numbers are unique, satisfying the Sudoku rules.

We can now proceed to determine the positions of the remaining 6s in Matrices 1, 7, and 9 by applying the row and column restrictions.

After applying the Sudoku rules and eliminating the invalid positions, we can determine the remaining locations for the number 6.

In Matrix 1, the number 6 is placed in the first cell. In Matrix 7, the number 6 is placed in the fifth cell, and in Matrix 9, the number 6 is placed in the second cell.

With these placements, the number 6 now appears in all the required matrices and satisfies the Sudoku rule that numbers must be unique within each row, column, and 3×3 box.

Next, let us examine the number 5.

We can see that 5 is already present in Matrices 2, 3, 4, 5, 6, 7, 8, and 9. Therefore, the only matrix in which 5 is missing is Matrix 1.

By checking the available cells and applying the row and column restrictions, we find that the only valid position for 5 in Matrix 1 is the fourth cell.

Therefore, 5 is placed in the fourth cell of Matrix 1, completing the placement of the number 5 in all nine matrices.

Next, let us examine the number 4.

We can see that 4 is already present in Matrices 2, 3, 4, 5, 8, and 9. Therefore, it still needs to be placed in Matrices 1, 6, and 7.

In Matrix 6, only one cell is vacant. Since every 3×3 matrix must contain the numbers 1 to 9 exactly once, the missing number must be 4. Therefore, we place 4 in the empty cell of Matrix 6.

We are now left with Matrices 1 and 7. To determine the correct positions for 4 in these matrices, we apply the Sudoku rules that require all numbers in a row and column to be unique.

By checking the rows and columns that already contain a 4, we can eliminate invalid positions and identify the correct cells for the remaining 4s. This ensures that no row, column, or 3×3 matrix contains duplicate numbers.

At this stage, Matrix 1 still has two possible cells where the number 4 could be placed. Since both positions appear valid, we cannot determine the correct location immediately.

Therefore, we move on to another matrix and look for a more definite placement.

In Matrix 6, the Sudoku rules allow the number 4 to be placed only in the eighth cell. All other possible positions are eliminated because they would create a duplicate number in the corresponding row or column.

Hence, we place 4 in the eighth cell of Matrix 6.

By filling in numbers that have only one possible position, we gradually reduce the number of possibilities in the remaining matrices, making it easier to solve the puzzle step by step.

Next, we observe that Matrix 7 has only one empty cell remaining. Since every 3×3 matrix must contain the numbers 1 to 9 exactly once, the missing number must be 4. Therefore, we place 4 in the vacant cell of Matrix 7.

After placing 4 in Matrix 7, we can use this information to determine the position of 4 in Matrix 1.

Since 4 is now located in the third cell of Matrix 7, no other cell in the same column can contain a 4. Therefore, the ninth cell of Matrix 1 cannot be 4 because it lies in the same column.

As a result, the only remaining valid position for 4 in Matrix 1 is the eighth cell.

Therefore, 4 is placed in the eighth cell of Matrix 1, completing the placement of the number 4 in the puzzle.

Initially, there were two possible positions for the number 4 in Matrix 1. However, only one of these positions can be correct.

After placing 4 in the third cell of Matrix 7, we apply the Sudoku rule that a number cannot be repeated in the same column. Since the third cell of Matrix 7 and the ninth cell of Matrix 1 lie in the same column, the ninth cell of Matrix 1 cannot contain 4.

This eliminates one of the two possibilities in Matrix 1.

Therefore, the only remaining valid position for 4 is the eighth cell of Matrix 1. Hence, 4 is inserted in the eighth cell of Matrix 1.

By using the column restriction rule, we successfully determine the correct position of 4 and continue solving the puzzle.

Next, let us examine the number 3.

We can see that 3 is already present in Matrices 2, 4, 5, 6, 7, 8, and 9. Therefore, it still needs to be placed in Matrices 1 and 3.

In Matrix 1, the number 3 cannot be placed in the seventh or ninth cell because doing so would violate the Sudoku rule that numbers must be unique within each row and column.

By applying the row and column restrictions, we eliminate these invalid positions and narrow down the possible locations for 3 in Matrix 1.

We can now continue analysing Matrices 1 and 3 to determine the exact positions of the remaining 3s.

Next, we determine the position of 3 in Matrix 3.

The number 3 cannot be placed in the fourth or fifth cell of Matrix 3 because there is already a 3 in the same row of Matrix 1. According to the Sudoku rules, a number cannot be repeated within the same row.

After eliminating these invalid positions, the first cell of Matrix 3 becomes the only possible location for the number 3.

Therefore, 3 is placed in the first cell of Matrix 3.

By using the row restriction rule, we successfully identify the correct position of 3 in Matrix 3 while ensuring that all numbers remain unique within their respective rows, columns, and 3×3 boxes.

This placement helps reduce the remaining possibilities in Matrix 3 and brings us one step closer to completing the Sudoku puzzle.

Next, let us examine the number 2.

We can see that the number 2 is missing from Matrices 3 and 9. Therefore, it must be placed in these two matrices.

In Matrix 3, there are two possible positions where the number 2 can be inserted, so we cannot determine its exact location immediately.

However, in Matrix 9, the situation is clearer. After applying the  column restrictions, we find that the number 2 can be placed only in the eighth cell of Matrix 9.

Therefore, 2 is inserted in the eighth cell of Matrix 9.

This placement helps reduce the remaining possibilities in Matrix 3 and brings us one step closer to completing the Sudoku puzzle.

After placing 2 in the eighth cell of Matrix 9, we can determine the position of 2 in Matrix 3.

In Matrix 3, there were two possible positions for the number 2: the fourth cell and the fifth cell. However, placing 2 in the fifth cell would create a duplicate 2 in the same row or column, which is not allowed in Sudoku.

Therefore, the number 2 can only be placed in the fourth cell of Matrix 3.

Next, we examine Matrix 8. We find that only one cell remains empty in this matrix. Since every 3×3 matrix must contain the numbers 1 to 9 exactly once, the missing number must be 8.

Therefore, 8 is inserted in the vacant cell of Matrix 8, completing the matrix and ensuring that all numbers within it are unique.

Finally, we examine the number 1.

At this stage of the puzzle, the number 1 is missing from Matrices 1, 3, and 9. By applying the Sudoku rules and checking the remaining empty cells, we can determine the correct positions for the missing 1s.

Therefore, 1 is inserted in the vacant cells of Matrices 1, 3, and 9, completing the placement of the number 1 in the puzzle.

With these final entries, all the rows, columns, and 3×3 matrices contain the numbers 1 to 9 exactly once, with no duplicates. The Sudoku puzzle is now completely solved.

This step demonstrates an important Sudoku strategy: as the puzzle nears completion, many rows, columns, and matrices have only one missing number, making it easier to identify the remaining values and finish the puzzle.

This is the final solution to the Sudoku puzzle. I hope you enjoyed following the step-by-step explanation and learned some useful Sudoku-solving techniques.

If you would like to practise more Sudoku puzzles, you can check out my Sudoku book on Amazon. This puzzle is just a small sample of the many puzzles and detailed solutions included in the book.

I would greatly appreciate your feedback after reading or solving the puzzles in the book. Your valuable comments and reviews will help me improve future editions and create even better Sudoku resources for puzzle lovers.

Thank you for your support, and happy Sudoku solving!

                                        

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