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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 total 45 - 41 = 4 forces 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} leaves Cand(5, 5) = {3}. Clicking Dammi un indizio highlights Row 5, Column 5 → Place 3 (Naked Single) without spoiling the rest of the 81-cell grid.
Uniqueness verification: Clicking Controlla il puzzle 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