PRINTABLE SUDOKU - HARD - Free Printable

Let's solve this Sudoku puzzle step by step.

Step 1: Understand the Puzzle

We are given a 9×9 Sudoku grid with some numbers filled in. The goal is to fill the empty cells so that each row, each column, and each of the nine 3×3 subgrids (also called "boxes") contains all of the digits from 1 to 9 exactly once.

Here’s the initial grid:

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

We'll denote the rows as R1 to R9 and columns as C1 to C9.

---

Step 2: Start Solving Using Logic

We’ll look for cells where only one number can fit based on row, column, and box constraints.

#### Let's begin with R1C5 (Row 1, Column 5)

- Row 1 has: 8, 3, 7, 9 → missing: 1,2,4,5,6
- Column 5: check values in C5:
- R1C5: ?
- R2C5: ?
- R3C5: ?
- R4C5: ?
- R5C5: 2
- R6C5: ?
- R7C5: ?
- R8C5: 3
- R9C5: ?
→ So far: 2,3 → missing: 1,4,5,6,7,8,9
- Box 2 (top-middle): R1–R3, C4–C6
- Current values: R1C4=9, R1C5=?, R1C6=?, R2C4=?, R2C5=?, R2C6=7, R3C4=4, R3C5=?, R3C6=?
→ Already have: 9,7,4 → missing: 1,2,3,5,6,8

So R1C5 must be from {1,2,4,5,6} ∩ {1,4,5,6,7,8,9} ∩ {1,2,3,5,6,8} = {1,4,5,6}

Not helpful yet.

Let’s look for easier cells.

---

#### Look at R2C2

- Row 2: 9, ?, 6, ?, ?, 7 → missing: 1,2,3,4,5,8
- Column 2: R1C2=3, R2C2=?, R3C2=1, R4C2=?, R5C2=9, R6C2=?, R7C2=4, R8C2=9, R9C2=?
→ So far: 3,1,9,4,9 → duplicates: 9 appears twice (R5C2 and R8C2), but that’s fine since it's not in same row/column.
→ Values present: 1,3,4,9 → missing: 2,5,6,7,8
- Box 1 (top-left): R1–R3, C1–C3
- R1: 8,3,7
- R2: 9,?,6
- R3: ?,1,?
→ Present: 8,3,7,9,6,1 → missing: 2,4,5

So R2C2 must be in:
- Row: {1,2,3,4,5,8}
- Col: {2,5,6,7,8}
- Box: {2,4,5}

Intersection: {2,5}

So possible: 2 or 5

Let’s keep that in mind.

---

Now try R4C2

- Row 4: 4, ?, ?, ?, ?, 3 → missing: 1,2,5,6,7,8,9
- Column 2: already has 3,1,9,4,9 → missing: 2,5,6,7,8
- Box 4 (middle-left): R4–R6, C1–C3
- R4: 4, ?, ?
- R5: ?, 9, 1
- R6: 6, ?, ?
→ Present: 4,9,1,6 → missing: 2,3,5,7,8

So R4C2 ∈ {1,2,5,6,7,8,9} ∩ {2,5,6,7,8} ∩ {2,3,5,7,8} = {2,5,7,8}

Still multiple options.

---

Try R3C1

- Row 3: ?, 1, ?, 4, ?, ? → missing: 2,3,5,6,7,8,9
- Column 1: R1C1=8, R2C1=9, R3C1=?, R4C1=4, R5C1=?, R6C1=6, R7C1=5, R8C1=?, R9C1=?
→ Present: 8,9,4,6,5 → missing: 1,2,3,7
- Box 1: missing: 2,4,5 — but 4 is already in R4C1, so box 1 has: 8,3,7,9,6,1 → missing: 2,4,5

Wait, R3C1 is in box 1, which needs 2,4,5

But column 1 missing: 1,2,3,7

So R3C1 ∈ {2,4,5} ∩ {1,2,3,7} = {2}

So R3C1 = 2

Update grid:

```
R3C1 = 2
```

Now update:

- Row 3: 2, 1, ?, 4, ?, ?
- Column 1: now has 8,9,2,4,6,5 → missing: 1,3,7
- Box 1: now has 8,3,7,9,6,1,2 → missing: 4,5

So now we know box 1 missing 4 and 5.

Now R3C3 is in box 1, row 3, col 3.

- Row 3: 2,1,?,4,?,?,? → missing: 3,5,6,7,8,9
- Col 3: R1C3=7, R2C3=6, R3C3=?, R4C3=?, R5C3=1, R6C3=?, R7C3=7, R8C3=2, R9C3=?
→ Present: 7,6,1,7,2 → duplicates: 7 appears twice (R1 and R7), but different rows, so okay.
→ So far: 1,2,6,7 → missing: 3,4,5,8,9
- Box 1: missing 4,5

So R3C3 ∈ {3,5,6,7,8,9} ∩ {3,4,5,8,9} ∩ {4,5} = {4,5}

So R3C3 = 4 or 5

But row 3 already has 2,1,4 → so 4 is already used in row 3 → cannot use 4

So R3C3 = 5

Update:

R3C3 = 5

Now:

- Row 3: 2,1,5,4,?,?
- Box 1: now complete: 8,3,7,9,6,1,2,5,4 → all filled

Now go back to R2C2: earlier we said possible 2 or 5

But R3C3=5, so column 3 has 5 now, but R2C2 is in column 2.

Column 2: R1C2=3, R2C2=?, R3C2=1, R4C2=?, R5C2=9, R6C2=?, R7C2=4, R8C2=9, R9C2=?

So 9 appears twice in column 2? R5C2=9 and R8C2=9 → same column → invalid!

Wait! That's a problem.

R5C2 = 9, R8C2 = 9 → both in column 2 → conflict unless they are same cell.

But R5C2 and R8C2 are different cells → same column → duplicate 9 → invalid.

But wait, let's double-check the original puzzle.

Original grid:

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

So:

- R5C2 = 9
- R8C2 = 9

Same column → two 9s → impossible.

Wait, is that correct?

R5C2 is row 5, column 2 → yes, value is 9

R8C2 is row 8, column 2 → value is 9

Yes → same column → duplicate → invalid.

But that means the puzzle is unsolvable? But probably I made a mistake.

Wait — no, maybe I misread.

Look again:

Row 5: ` | 9 | 1 | | 2 | ` → so R5C2 = 9

Row 8: ` | 9 | 2 | | 3 | ` → R8C2 = 9

Yes → both in column 2 → duplicate 9 → invalid.

But that can't be — unless the puzzle is wrong.

Wait — perhaps I misread the original image.

Let me reconstruct the grid carefully.

From your image description:

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

Yes, R5C2 = 9, R8C2 = 9 → same column → conflict.

Unless... is it possible that the puzzle has a typo?

But let's double-check if I'm reading it right.

Row 5: " | 9 | 1 | | 2 | " → so C2 = 9

Row 8: " | 9 | 2 | | 3 | " → C2 = 9

Yes → column 2 has two 9s → impossible.

Therefore, either the puzzle is invalid, or I made a mistake in interpretation.

Wait — could it be that the second row of the bottom block is not row 8?

No, standard Sudoku numbering:

- Rows: 1 to 9 from top to bottom
- Columns: 1 to 9 from left to right

So:

- R5C2 = 9
- R8C2 = 9

→ Both in column 2 → conflict.

But that means the puzzle is invalid.

However, perhaps I misread the original image.

Maybe the value in R8C2 is not 9?

Let me re-express the full grid with positions.

Let me write the entire grid clearly:

```
Row 1: 8 3 7 9 ? ?
Row 2: 9 ? 6 ? ? 7
Row 3: ? 1 ? 4 ? ?
Row 4: 4 ? ? ? ? 3
Row 5: ? 9 1 ? 2 ?
Row 6: 6 ? ? ? ? 1
Row 7: 5 4 7 8 ? 1
Row 8: ? 9 2 ? 3 ?
Row 9: ? ? ? ? ? ?
```

So column 2:

- R1C2 = 3
- R2C2 = ?
- R3C2 = 1
- R4C2 = ?
- R5C2 = 9
- R6C2 = ?
- R7C2 = 4
- R8C2 = 9 ← here!
- R9C2 = ?

So R5C2 = 9 and R8C2 = 9 → both in column 2 → duplicate → invalid.

But wait — is it possible that the value in R8C2 is not 9?

Looking at your input:

" | 9 | 2 | | 3 | " — yes, first blank is R8C1, then R8C2 = 9, R8C3 = 2, etc.

So yes, R8C2 = 9.

And R5C2 = 9.

So two 9s in column 2 → impossible.

Therefore, the puzzle as given is invalid.

But perhaps there's a typo in the image.

Alternatively, maybe I misread the position.

Let me count the blocks.

The grid is divided into 3x3 boxes.

Top-left box: R1–R3, C1–C3

Given:
- R1: 8,3,7
- R2: 9,?,6
- R3: ?,1,?

That seems correct.

Middle-left: R4–R6, C1–C3
- R4: 4,?,?
- R5: ?,9,1
- R6: 6,?,?

Bottom-left: R7–R9, C1–C3
- R7: 5,4,7
- R8: ?,9,2
- R9: ?,?,?

So R8C2 = 9

R5C2 = 9

Same column → conflict.

So the puzzle is invalid.

But perhaps you meant something else.

Wait — could it be that the value in R5C2 is not 9? Let's check the original text.

You wrote:

" | 9 | 1 | | 2 | " — yes, R5C2 = 9

" | 9 | 2 | | 3 | " — R8C2 = 9

So yes.

Unless the puzzle has a typo, it's unsolvable.

But maybe I should assume it's a typo and proceed.

Perhaps R8C2 is not 9, but another number?

Or maybe R5C2 is not 9?

But according to your input, it is.

Alternatively, maybe the puzzle is valid and I'm missing something.

Wait — is it possible that R8C2 is not 9?

Let me re-read your message:

" | 9 | 2 | | 3 | "

This is the eighth row: so after " | " is C1, then C2=9, C3=2, etc.

Yes.

So unless there's a formatting error, the puzzle has two 9s in column 2.

Therefore, it's invalid.

But perhaps you meant R8C2 to be something else?

Maybe it's a 6 or 8?

But based on what you provided, it's 9.

Alternatively, maybe the puzzle is solvable if we ignore that, but no.

Another possibility: maybe the image shows a different value.

Since I can't see the image, I rely on your text.

But you said: "I uploaded an image", but I don't have access to it.

So my only source is your text representation.

And in that, R5C2 = 9 and R8C2 = 9 → conflict.

Therefore, the puzzle as described is invalid.

But let's double-check if there's a mistake in my logic.

Could it be that R8C2 is not in the same column as R5C2? No, both are column 2.

Unless the grid is transposed, but no.

Perhaps the value in R8C2 is not 9, but the character is misread.

For example, maybe it's a 6 or 8?

But you wrote "9".

Alternatively, maybe it's a 0, but no.

Perhaps it's a typo in the puzzle.

In many Sudoku puzzles, such errors occur.

So likely, one of the 9s is incorrect.

Let’s assume that R8C2 is not 9, but let's say it's 6 or 8.

But without knowing, we can't proceed.

Alternatively, maybe R5C2 is not 9.

But in your text, it is.

Wait — look at the seventh row: "5 | 4 | 7 | 8 | | 1"

So R7C1=5, R7C2=4, R7C3=7, R7C4=8, R7C5=?, R7C6=1

Then row 8: " | 9 | 2 | | 3 | " → R8C1=?, R8C2=9, R8C3=2, R8C4=?, R8C5=3, R8C6=?

Row 5: " | 9 | 1 | | 2 | " → R5C1=?, R5C2=9, R5C3=1, R5C4=?, R5C5=2, R5C6=?

So yes.

So column 2: R5C2=9, R8C2=9 → conflict.

Therefore, the puzzle is invalid.

But perhaps you meant R8C2 to be 8 or 6.

Maybe it's a typo, and it's supposed to be 8.

Let’s assume that R8C2 is not 9, but let's say it's 8.

Then we can continue.

But since I can't change the puzzle, I must conclude that as given, it's unsolvable due to duplicate 9 in column 2.

But let's check if there are other duplicates.

Column 1: R1C1=8, R2C1=9, R3C1=?, R4C1=4, R5C1=?, R6C1=6, R7C1=5, R8C1=?, R9C1=?

So far: 8,9,4,6,5 → no duplicates.

Column 3: R1C3=7, R2C3=6, R3C3=?, R4C3=?, R5C3=1, R6C3=?, R7C3=7, R8C3=2, R9C3=?

R1C3=7, R7C3=7 → both in column 3 → duplicate 7!

Oh! Another conflict.

R1C3 = 7, R7C3 = 7 → same column → duplicate 7.

So two more conflicts.

So the puzzle has at least two errors:

- Column 2: R5C2=9 and R8C2=9
- Column 3: R1C3=7 and R7C3=7

So it's definitely invalid.

Therefore, the puzzle as presented cannot be solved.

But perhaps you made a typo when typing.

Let me suggest a corrected version.

Maybe R8C2 is 8 instead of 9.

And R7C3 is 6 instead of 7.

But without seeing the image, I can't be sure.

Alternatively, maybe the values are correct, but the grid is different.

But based on your text, it's invalid.

So final answer: The puzzle as given contains duplicate numbers in column 2 (two 9s) and column 3 (two 7s), making it unsolvable. Please verify the input.

If you can provide a corrected version or clarify the image, I can help solve it.