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Darmowe narzędzie / Do sudoku na papierze

Solver sudoku

Zrób zdjęcie wydrukowanego 9×9 Sudoku, przeanalizuj liczby, a następnie poproś o jeden logiczny ruch lub odkryj pełną siatkę.

Gazety i książkiEdytowane skanowanieBez rejestracji

Skanowanie puzzle

Zrób jasne zdjęcie całej sieci

Utrzymuj widoczne wszystkie cztery narożniki, trzymaj stronę płaską i unikaj cieni między cyframi.

Rozpoznana sieć

Sprawdź wszystkie dane cyfry

Gotowe

Skanuj zagadkę lub wprowadź cyfry ręcznie.

Dotknij dowolnej komórki i wpisz jej dane cyfry.

Papierowe zdjęcia Tips

Dobre skany tworzą wiarygodne rozwiązania.

01

Pokaż wszystkie grid edges

Nie odcinaj zewnętrznej granicy ani żadnego rzędu i kolumny.

02

unikać hałasu pisania

W miarę możliwości usunąć notatki kandydatów lub potwierdzić każde uznane dane.

03

Przegląd przed rozwiązaniem

Edytowalna siatka zapobiega jednemu błędnie przeczytanemu cyfrowi gazety zniszczeniu odpowiedzi.

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:

  1. 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 total 45 - 41 = 4 forces R1C3 = 4 (Cand(1, 3) = {4}).
  2. Center Cell R5C5 Naked Single: Excluding the 8 peer digits {1, 2, 4, 5, 6, 7, 8, 9} from {1..9} leaves Cand(5, 5) = {3}. Clicking Daj jedną wskazówkę highlights Row 5, Column 5 → Place 3 (Naked Single) without spoiling the rest of the 81-cell grid.
  3. Uniqueness verification: Clicking Sprawdź łamigłówkę 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