Show all grid edges
Do not crop off the outer border or any row and column.
Free tool / Für Papier-Sudokus
Sudoku-Löser
The Sudoku-Löser From Photo transcribes printed 9×9 grids (81 cells across nine 3×3 boxes) via on-device OCR and solves them by constraint propagation and exact cover, returning either a single explained Naked/Hidden Single hint or the unique 81-digit solution. Photograph a printed 9×9 Sudoku, review the digits, then ask for one logical move or reveal the complete grid.
Scan puzzle
Take a clear photo of the entire grid
Keep all four corners visible, hold the page flat and avoid shadows across the digits.
Recognized grid
Check every given digit
Bereit
Scan a puzzle or enter digits manually.
Tap any cell and type its given digit.
Paper photo tips
Good scans make trustworthy solutions.
Avoid handwriting noise
Erase candidate notes when possible or confirm every recognized given.
Review before solving
The editable grid prevents one misread newspaper digit from corrupting the answer.
Algorithm, Formula & Worked Example
How does the 9×9 Sudoku solver verify uniqueness and compute the next logical move?
A standard 9×9 Sudoku is an exact-cover constraint satisfaction problem over 81 cells and 324 binary constraints: every cell holds one digit d ∈ {1..9}, and each of the 9 rows, 9 columns, and 9 boxes (3×3) contains each digit 1 through 9 exactly once, summing to 45 per unit (405 across the whole board).
Plain-Text Constraint Satisfaction Equations
Let binary indicator x[r, c, d] ∈ {0, 1} equal 1 when cell (r, c) (with r, c ∈ {1..9}) is assigned digit d ∈ {1..9}, and let Peer(r, c) denote the 20 peer cells sharing row r, column c, or 3×3 box b = 3·⌊(r-1)/3⌋ + ⌊(c-1)/3⌋ + 1:
- Cell occupancy constraint (81 equations):
∀ r, c ∈ {1..9}: ∑_{d=1..9} x[r, c, d] = 1 - Row uniqueness constraint (81 equations):
∀ r, d ∈ {1..9}: ∑_{c=1..9} x[r, c, d] = 1 - Column uniqueness constraint (81 equations):
∀ c, d ∈ {1..9}: ∑_{r=1..9} x[r, c, d] = 1 - 3×3 Box uniqueness constraint (81 equations):
∀ b, d ∈ {1..9}: ∑_{(r,c) ∈ Box(b)} x[r, c, d] = 1 - Candidate domain & Naked Single rule:
Cand(r, c) = {1..9} \ { grid[r', c'] : (r', c') ∈ Peer(r, c) }. Whenever|Cand(r, c)| = 1, that single digit is immediately forced.
Worked 9×9 Deduction Example With Real Numbers & Output
Suppose you scan a newspaper Sudoku where Row 1 has 8 given digits [5, 3, _, 6, 7, 8, 9, 1, 2] and Center Cell R5C5 (Box 5) sees the peer digits {1, 2, 4, 5, 6, 7, 8, 9} across Row 5, Column 5, and Box 5:
- Row 1 Hidden/Naked Single at R1C3: The sum of placed digits in Row 1 is
5 + 3 + 6 + 7 + 8 + 9 + 1 + 2 = 41. Subtracting from the unit total45 - 41 = 4forces R1C3 = 4 (Cand(1, 3) = {4}). - Center Cell R5C5 Naked Single: Excluding the 8 peer digits
{1, 2, 4, 5, 6, 7, 8, 9}from{1..9}leavesCand(5, 5) = {3}. Clicking Einen Hinweis geben highlights Row 5, Column 5 → Place 3 (Naked Single) without spoiling the rest of the 81-cell grid. - Uniqueness verification: Clicking Rätsel prüfen runs minimum-remaining-values (MRV) backtracking up to 2 solutions; a valid newspaper Sudoku requires at least 17 givens (the proven mathematical minimum for a unique 9×9 solution) and returns Verified: 1 unique solution.
9×9 Sudoku Grid Metrics & Solving Technique Reference Table
| Metric / Solving Stage | Grid Scope (Cells / Units) | Mathematical Threshold / Count | Candidate Condition | Solver Output / Action |
|---|---|---|---|---|
| Board Dimensions | 9 rows × 9 columns × 9 boxes (3×3) | 81 cells total; 27 units; 324 constraints | Each cell belongs to 3 units and sees 20 peers | Validates 0 duplicate digits across all 27 units |
| Minimum Givens for Uniqueness | Whole 9×9 grid (81 cells) | ≥ 17 given digits (typical print: 22–36 givens) | Fewer than 17 givens always yields ≥ 2 solutions | Flags under-constrained scans with multiple solutions |
| Unit Sum Invariant (Rule of 45) | Any 1 row, 1 column, or 1 box (9 cells) | ∑_{i=1..9} d_i = 45 (9 × 45 = 405 total) |
8 known digits in a unit fix the 9th digit as 45 - ∑ |
Immediate single-cell completion |
| Naked Single (Stage 1 Hint) | 1 unresolved cell (r, c) |
|Cand(r, c)| = 1 (8 digits eliminated by peers) |
Only 1 legal digit d ∈ {1..9} remains in cell |
Hint names cell coordinate (r, c) and forced digit d |
| Hidden Single (Stage 2 Hint) | 1 row, column, or 3×3 box (9 cells) | ∑_{(r,c) ∈ Unit} [d ∈ Cand(r, c)] = 1 |
Digit d fits in only 1 cell of that unit |
Hint identifies the unit and places digit d |
| MRV Exact-Cover Search (Stage 3) | All remaining unresolved cells | Branches on cell with minimum |Cand(r, c)| ≥ 2 |
Propagates constraints until all 81 cells are filled | Reveals verified 81-digit solution or reports conflict |