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Bezplatný nástroj / pro papírové sudoku
Řešitel sudoku
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Skenování hlavolamu
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Rozpoznaná mřížka
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Připraveno
Skenujte hlavolam nebo zadávejte číslice ručně.
Klepněte na kteroukoli buňku a zadejte zadanou číslici.
Tipy pro fotografování papírového sudoku
Kvalitní snímek umožní spolehlivě vyřešit hlavolam.
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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 Dej mi jednu nápovědu highlights Row 5, Column 5 → Place 3 (Naked Single) without spoiling the rest of the 81-cell grid. - Uniqueness verification: Clicking Zkontrolovat hlavolam 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 |