퍼즐 스캔

전체 격자의 명확한 사진을 찍으십시오.

네 개의 각도를 모두 볼 수 있도록, 페이지를 평평하게 유지하고 숫자 사이의 그림자를 피하십시오.

＋퍼즐 촬영 또는 업로드신문, 잡지, 책 또는 스크린샷

인식된 격자

모든 숫자을 확인하라

준비됨

퍼즐을 스캔하거나 수동으로 숫자를 입력합니다.

모든 칸을 탭하고 지정된 숫자를 입력합니다.칸을 탭하고 주어진 숫자를 입력하세요.

종이 사진 tips

좋은 스캔은 신뢰할 수있는 정답을 만듭니다.

격자의 모든 테두리 표시

외부 경계나 행과 열을 잘라내지 마십시오.

글쓰기 소음 피하기

가능한 경우 후보 노트를 삭제하거나 각각의 인정된 주어진 것을 확인하십시오.

해결하기 전에 검토

편집 가능한 격자는 잘못 읽은 한 신문 숫자이 답을 손상시키지 못하게 합니다.

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 힌트 하나 보기 highlights Row 5, Column 5 → Place 3 (Naked Single) without spoiling the rest of the 81-cell grid.
Uniqueness verification: Clicking 퍼즐 확인 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