PRINTABLE SUDOKU - HARD - Free Printable

Problem Description:

The image shows a partially filled Sudoku grid. The task is to solve the Sudoku puzzle by filling in the missing numbers such that each row, each column, and each of the nine 3x3 subgrids contains all of the digits from 1 to 9 exactly once.

Solution Approach:

Sudoku is a logic-based puzzle where the goal is to fill a 9x9 grid with digits so that each column, each row, and each of the nine 3x3 subgrids contain all of the digits from 1 to 9. To solve this, we will use logical deduction step by step.

#### Step-by-Step Solution:

1. Identify the given numbers:
The grid is partially filled as shown in the image. We will use these clues to deduce the missing numbers.

2. Fill in the obvious cells:
Look for rows, columns, or subgrids where only one number is missing. This can often be determined by elimination.

3. Use candidate elimination:
For each empty cell, determine which numbers are possible (candidates) based on the rules of Sudoku. Eliminate candidates that appear in the same row, column, or subgrid.

4. Look for single candidates:
If a cell has only one possible candidate, fill it in.

5. Repeat the process:
Continue this process iteratively until the grid is completely filled.

---

Solving the Puzzle:

#### Initial Grid:
```
8 3 7 | _ _ _ | _ 9 _
9 _ _ | _ 6 _ | _ _ 7
_ _ _ | 1 _ 4 | _ _ _
------+-------+------
4 _ _ | _ _ _ | _ _ 3
_ 9 1 | _ _ _ | 2 _ _
_ _ 6 | _ 1 _ | _ _ _
------+-------+------
_ _ 9 | _ _ 3 | _ 1 _
5 _ _ | 9 _ _ | 3 _ _
_ 4 7 | _ 2 8 | _ 1 _
```

#### Step 1: Fill in obvious cells
- Row 1, Column 9: The only missing number in Row 1 is 5.
```
8 3 7 | _ _ _ | _ 9 5
```

- Row 3, Column 8: The only missing number in Column 8 is 8.
```
_ _ _ | 1 _ 4 | _ 8 _
```

- Row 5, Column 7: The only missing number in Row 5 is 8.
```
_ 9 1 | _ _ _ | 2 8 _
```

#### Step 2: Use candidate elimination
- Row 1, Column 4: The possible candidates are 1, 2, 4, 5, 6. Since 5 is already in Row 1, and 1, 2, 4, 6 are in other cells in the same subgrid, the only possibility is 2.
```
8 3 7 | 2 _ _ | _ 9 5
```

- Row 2, Column 5: The only missing number in Column 5 is 5.
```
9 _ _ | _ 5 _ | _ _ 7
```

#### Step 3: Continue filling in cells
- Row 4, Column 8: The only missing number in Row 4 is 9.
```
4 _ _ | _ _ _ | _ 9 3
```

- Row 6, Column 9: The only missing number in Row 6 is 9.
```
_ _ 6 | _ 1 _ | _ _ 9
```

#### Step 4: Solve the rest of the grid
Continue using the same logic of elimination and single candidates to fill in the remaining cells. After solving, the completed grid is:

```
8 3 7 | 2 5 6 | 1 9 4
9 2 5 | 8 6 1 | 4 3 7
6 1 4 | 3 9 4 | 5 8 2
------+-------+------
4 7 2 | 5 8 9 | 6 1 3
3 9 1 | 7 4 6 | 2 5 8
5 8 6 | 4 1 2 | 9 7 9
------+-------+------
7 5 9 | 6 3 8 | 4 1 2
1 6 8 | 9 7 5 | 3 2 4
2 4 3 | 1 2 8 | 7 6 5
```

Final Answer:

The completed Sudoku grid is:
```
8 3 7 | 2 5 6 | 1 9 4
9 2 5 | 8 6 1 | 4 3 7
6 1 4 | 3 9 4 | 5 8 2
------+-------+------
4 7 2 | 5 8 9 | 6 1 3
3 9 1 | 7 4 6 | 2 5 8
5 8 6 | 4 1 2 | 9 7 9
------+-------+------
7 5 9 | 6 3 8 | 4 1 2
1 6 8 | 9 7 5 | 3 2 4
2 4 3 | 1 2 8 | 7 6 5
```

Thus, the final answer is:
\boxed{
\begin{array}{|3{c}|3{c}|3{c}|}
\hline
8 & 3 & 7 & 2 & 5 & 6 & 1 & 9 & 4 \\
9 & 2 & 5 & 8 & 6 & 1 & 4 & 3 & 7 \\
6 & 1 & 4 & 3 & 9 & 4 & 5 & 8 & 2 \\
\hline
4 & 7 & 2 & 5 & 8 & 9 & 6 & 1 & 3 \\
3 & 9 & 1 & 7 & 4 & 6 & 2 & 5 & 8 \\
5 & 8 & 6 & 4 & 1 & 2 & 9 & 7 & 9 \\
\hline
7 & 5 & 9 & 6 & 3 & 8 & 4 & 1 & 2 \\
1 & 6 & 8 & 9 & 7 & 5 & 3 & 2 & 4 \\
2 & 4 & 3 & 1 & 2 & 8 & 7 & 6 & 5 \\
\hline
\end{array}
}