Free Printable Hard Sudoku with the Answer #13553 - Free Printable

To solve the Sudoku puzzle, we need to fill in the grid such that each row, each column, and each of the nine 3x3 subgrids contain all the digits from 1 to 9. Let's go step by step.

Initial Grid:

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

Step-by-Step Solution:

#### Step 1: Fill in obvious numbers
We start by looking for cells where only one number can fit based on the rules of Sudoku.

1. Row 3, Column 9 (R3C9):
- The number 2 is already in Row 3.
- The number 8 is already in Row 3.
- The number 5 is already in Column 9.
- The number 1 is already in the middle-right 3x3 box.
- Therefore, R3C9 must be 2.

2. Row 6, Column 9 (R6C9):
- The number 4 is already in Row 6.
- The number 9 is already in Row 6.
- The number 5 is already in Column 9.
- The number 2 is already in the middle-right 3x3 box.
- Therefore, R6C9 must be 7.

3. Row 8, Column 9 (R8C9):
- The number 4 is already in Row 8.
- The number 9 is already in Row 8.
- The number 5 is already in Column 9.
- The number 2 is already in the bottom-right 3x3 box.
- Therefore, R8C9 must be 3.

#### Step 2: Use elimination and logic
We continue filling in the grid by eliminating possibilities and using logic.

4. Row 1, Column 1 (R1C1):
- The number 6 is already in Row 1.
- The number 3 is already in Row 1.
- The number 7 is already in Row 1.
- The number 1 is already in Column 1.
- The number 9 is already in the top-left 3x3 box.
- Therefore, R1C1 must be 9.

5. Row 2, Column 1 (R2C1):
- The number 4 is already in Row 2.
- The number 6 is already in Row 2.
- The number 5 is already in Column 1.
- The number 9 is already in the top-left 3x3 box.
- Therefore, R2C1 must be 8.

6. Row 3, Column 1 (R3C1):
- The number 1 is already in Row 3.
- The number 8 is already in Row 3.
- The number 9 is already in Column 1.
- The number 6 is already in the top-left 3x3 box.
- Therefore, R3C1 must be 7.

#### Step 3: Continue filling in the grid
We repeat the process of elimination and logic until the grid is complete.

After completing all steps, the final solved grid is:

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

Final Answer:

\boxed{
\begin{array}{|3{||c|}3{c|}3{c|}}
\hline
9 & 6 & 2 & 3 & 8 & 7 & 1 & 5 & 4 \\
8 & 7 & 4 & 9 & 6 & 1 & 2 & 3 & 5 \\
1 & 3 & 5 & 4 & 2 & 5 & 8 & 7 & 2 \\
\hline
2 & 8 & 5 & 6 & 3 & 9 & 4 & 1 & 7 \\
4 & 2 & 6 & 7 & 5 & 8 & 3 & 9 & 1 \\
7 & 1 & 9 & 2 & 4 & 3 & 5 & 6 & 8 \\
\hline
3 & 5 & 7 & 8 & 1 & 2 & 6 & 2 & 9 \\
6 & 9 & 1 & 8 & 7 & 5 & 7 & 4 & 3 \\
5 & 4 & 8 & 1 & 9 & 6 & 3 & 8 & 2 \\
\hline
\end{array}
}