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Doku Solver

Sudoku Solving Techniques

Master these essential strategies to solve any Sudoku puzzle by hand. From beginner basics to advanced techniques.

Quick Navigation

1
Naked Singles
2
Hidden Singles
3
Full House
4
Cross-Hatching
5
Naked Pairs
6
Hidden Pairs
7
Naked Triples
8
Pointing Pairs
9
X-Wing
10
Swordfish
11
Y-Wing
12
Unique Rectangles
13
Coloring
14
Chains
15
ALS
1-4: Beginner 5-9: Intermediate 10-12: Advanced 13-15: Expert
1

Level 1: Beginner Techniques

Essential basics

1. Naked Singles (Solo Numbers)

The most fundamental technique. When a cell can only contain one possible number, that's a naked single. This happens when 8 out of 9 numbers are already present in the cell's row, column, or 3×3 box.

In this example:

  • • Look at the highlighted cell (R1C1)
  • • Row 1 has: 3,4,6,2,8,1,9,7 - missing 5
  • • Column 1 has: 6,1,2,3,8,7,9,4 - missing 5
  • • Top-left box has: 3,4,6,7,2,1,8,9 - missing 5
  • • Only 5 satisfies all three constraints
5
3
4
6
2
8
1
9
7
6
7
2
59
1
59
34
48
38
1
8
9
3
7
4
56
26
56
2
15
15
4
35
35
9
6
8
3
4
6
7
9
2
8
1
5
8
9
35
2
6
1
7
34
34
7
6
18
18
4
39
5
23
29
9
2
8
5
13
7
36
34
1
4
15
7
19
8
6
2
35
39

Yellow: Naked single (only 5 possible)
Blue: Other candidates in grid

2. Hidden Singles

A number that can only go in one place. Even if a cell has multiple candidate numbers, if a specific number can only fit in one cell within a row, column, or box, it must go there.

In this example:

  • • Looking for number 4 in the top-left box
  • • R1C1: has candidates {4,5,8} - includes 4
  • • R1C2: has candidates {3,5} - no 4 possible
  • • R2C1: has candidates {7,9} - no 4 possible
  • • R1C1: only cell in box that can contain 4
  • • Therefore R1C1 must be 4 (hidden single)
458
35
6
7
2
8
1
9
3
79
1
2
9
4
5
38
6
78
379
8
37
3
1
6
25
25
9
2
7
1
6
5
3
9
8
4
6
3
5
8
9
2
7
1
45
8
2
9
1
7
4
6
3
5
1
6
8
2
3
7
4
59
67
5
9
4
45
6
1
3
7
2
34
45
37
45
8
9
25
24
1

Yellow: Hidden single for 4
Blue: Other candidates shown

3. Full House

When 8 cells in a row, column, or box are filled, the 9th cell is automatically determined. This is the simplest form of elimination.

In this example:

  • • Middle-left box has: 2,1,7,4,6,5,8,3
  • • Only one cell is empty (R5C1)
  • • Numbers 1-8 are present, 9 is missing
  • • Therefore R5C1 must be 9
  • • This completes the "full house"
5
3
4
16
7
16
28
289
89
6
7
2
189
4
189
38
289
389
1
8
9
3
2
6
47
47
5
2
1
7
4
5
3
9
6
8
9
4
6
7
8
2
1
3
5
8
6
5
2
1
4
3
9
7
3
2
1
56
36
7
456
18
169
7
5
3
168
9
18
246
248
124
4
9
8
156
36
7
256
1
123

Yellow: Full house - only 9 missing
Blue: Other candidates in grid

4. Cross-Hatching (Visual Scanning)

A systematic visual technique where you scan rows and columns to find where specific numbers can be placed. Focus on one number at a time across the entire grid.

Cross-hatching for number 7:

  • • Top-right box needs a 7
  • • Row 1: already has 7 in R1C5 (blocks R1C7, R1C8, R1C9)
  • • Row 2: already has 7 in R2C2 (blocks R2C7, R2C8, R2C9)
  • • Row 3: no 7 yet, so R3C7, R3C8, R3C9 are possible
  • • Column 7: has 7 in R6C7 (blocks R3C7)
  • • Column 8: has 7 in R8C8 (blocks R3C8)
  • • Therefore: only R3C9 can contain 7 in this box
5
3
4
6
7
8
X
X
X
6
7
2
1
4
5
X
X
X
1
8
9
3
2
6
X
X
7
2
1
378
4
5
3
9
6
8
4
6
378
8
9
2
1
3
5
8
2
36
59
1
4
7
9
26
3
59
1
2
6
79
4
5
8
79
59
5
79
3
1
6
7
4
79
59
68
57
8
79
2
1
3

Blue: Existing 7s guide placement
Yellow: Only valid position for 7
Red: Eliminated by cross-hatching

2

Level 2: Intermediate Techniques

Requires pencil marks

5. Naked Pairs

When two cells in the same row, column, or box contain exactly the same two candidate numbers, those numbers can be eliminated from all other cells in that unit.

In this example:

  • • R7C1 and R9C1 both have exactly candidates {4,9}
  • • This forms a naked pair in column 1
  • • Since these cells must be 4 and 9 (in some order)
  • • 4 and 9 can be eliminated from all other cells in column 1
  • • R1C1: eliminate 4 and 9, leaving only {2,3}
  • • R4C1: eliminate 4, leaving only {3}
234
7
8
6
5
1
369
239
29
5
6
1
78
2
78
4
3
9
7
3
9
4
1
6
5
8
2
34
8
2
5
7
9
1
6
34
1
5
346
2
8
34
7
9
36
6
2
7
1
9
5
8
4
3
49
1
5
3
6
2
9
7
8
8
9
3
7
4
28
6
1
5
49
4
6
89
3
78
2
5
1

Blue: Naked Pair {4,9} in column 1
Red: Eliminate 4,9 from other cells

6. Hidden Pairs

Two numbers that can only appear in the same two cells within a row, column, or box. Other candidates in those cells can be eliminated.

In this example:

  • • In row 2, numbers 1 and 8 appear only in R2C4 and R2C6
  • • R2C4 has candidates {1,8,9}, R2C6 has {1,7,8}
  • • Since 1 and 8 can only go in these two cells
  • • They form a hidden pair {1,8}
  • • We can eliminate 9 from R2C4 and 7 from R2C6
  • • Leaving R2C4: {1,8} and R2C6: {1,8}
5
3
4
6
7
2
9
1
8
6
7
2
189
4
178
3
5
9
189
189
189
3
29
9
6
2
7
2
1
7
4
5
3
8
9
6
4
6
5
79
29
89
1
3
25
3
29
689
579
129
4
25
678
125
7
4
3
259
6
15
25
8
125
189
5
689
279
3
78
4
67
12
19
29
169
257
29
57
25
4
3

Yellow: Hidden pair {1,8} in row 2
Blue: Other candidates in grid

7. Naked Triples

Three cells that collectively can only contain the same three numbers. These three numbers can be eliminated from all other cells in that unit.

In this example:

  • • R5C2: {1,5}, R5C3: {1,6}, R5C6: {5,6}
  • • Together they contain only numbers {1,5,6}
  • • This forms a naked triple in row 5
  • • 1, 5, and 6 can be eliminated from other cells in row 5
  • • R5C4: eliminate 5,6 leaving {3,4}
  • • R5C9: eliminate 1,5 leaving {3,4}
5
3
4
6
7
2
9
1
8
6
7
2
1
4
5
3
8
9
1
8
9
3
2
7
6
4
5
2
1
7
4
5
3
8
9
6
4
15
16
3456
9
56
2
7
135
8
2
3
7
1
4
5
6
9
7
4
5
2
8
9
1
3
6
9
6
8
5
3
1
4
2
7
3
9
1
8
6
7
7
5
2

Blue: Naked triple {1,5,6} in row 5
Red: Eliminate 1,5,6 from other cells

8. Pointing Pairs (Box-Line Reduction)

When a candidate number appears in only one row or column within a 3×3 box, it can be eliminated from the same row or column in other boxes.

In this example:

  • • In top-middle box, number 3 appears only in R2C4 and R2C6
  • • 3 is confined to row 2 within this box
  • • All 3s in row 2 must be within this box
  • • Therefore 3 can be eliminated from R2C1, R2C2 (left box)
  • • And from R2C7, R2C8, R2C9 (right box)
5
2
4
6
7
8
1
9
3
13
139
8
39
1
39
26
26
7
6
7
9
2
4
5
8
1
3
2
1
7
4
5
3
9
8
6
4
6
5
8
9
2
3
7
1
8
3
2
7
6
1
4
5
9
7
4
1
5
8
9
6
3
2
9
8
6
3
2
7
5
4
1
1
5
3
9
3
4
7
6
8

Blue: Pointing pair {3,9} in row 2
Red: Eliminate 3 from rest of row 2

9. X-Wing

A rectangular pattern where a candidate appears in exactly two cells in two different rows, and those cells are aligned in the same two columns (or vice versa).

In this X-Wing example:

  • • Number 4 appears in exactly 2 cells in row 3: R3C2 and R3C9
  • • Number 4 appears in exactly 2 cells in row 7: R7C2 and R7C9
  • • Forms rectangle: R3C2, R3C9, R7C2, R7C9
  • • Logic: If R3C2=4 then R7C9=4, if R3C9=4 then R7C2=4
  • • All other 4s in columns 2 and 9 can be eliminated
5
3
6
7
9
8
1
2
4
2
1457
7
1
6
5
3
8
49
1
4
8
3
2
7
6
5
4
6
1
2
4
5
3
9
7
8
8
6
9
2
7
1
5
3
6
7
245
3
8
1
6
4
9
25
9
4
5
6
3
2
8
1
4
3
278
1
9
8
4
7
6
235
4
789
4
5
4
9
2
4
137

Blue: X-Wing corners for 4
Red: Eliminate 4s from columns 2,9

3

Level 3: Advanced Techniques

Complex patterns

10. Swordfish

Extension of X-Wing to three rows and three columns. A candidate appears in at most three cells in three rows, and these cells are constrained to exactly three columns (or vice versa).

In this Swordfish example:

  • • Number 5 in Row 2: appears in R2C3 and R2C7 (2 cells)
  • • Number 5 in Row 4: appears in R4C3, R4C6, R4C7 (3 cells)
  • • Number 5 in Row 8: appears in R8C6 and R8C7 (2 cells)
  • • All constrained to columns 3, 6, and 7
  • • Forms Swordfish: eliminate all other 5s in columns 3, 6, 7
  • • Logic: 5s must fill exactly these positions across the 3 rows
4
3
15
6
7
15
28
9
2
6
7
5
1
4
2
5
8
3
1
8
2
3
9
5
6
4
7
2
1
5
4
8
5
5
6
9
8
6
9
2
1
7
3
5
4
7
4
35
8
6
139
19
2
15
3
9
67
7
2
4
18
1
6
9
2
1
57
3
5
5
7
8
5
5
68
9
5
168
47
3
15

Blue: Swordfish pattern with 5
Red: Eliminate 5s in columns 3,6,7

11. Y-Wing (XY-Wing)

Three bi-value cells forming a Y shape: one pivot cell with candidates XY, connected to two wing cells with XZ and YZ. This eliminates Z from cells that can see both wings.

In this Y-Wing example:

  • • Pivot (R5C5): {2,9} connects to both wings
  • • Wing1 (R5C2): {2,7} shares 2 with pivot
  • • Wing2 (R3C5): {9,7} shares 9 with pivot
  • • R3C2 sees both wings: eliminate 7
  • • Logic: If pivot=2→Wing2=7, if pivot=9→Wing1=7
  • • Therefore R3C2 cannot be 7, must be 1
5
3
4
6
8
1
9
2
7
6
8
2
3
4
5
1
7
9
1
17
9
2
97
7
6
8
3
2
1
7
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5
3
8
9
6
4
27
6
8
29
9
3
1
5
8
9
3
7
1
2
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6
4
7
4
1
5
6
8
2
3
9
9
6
8
1
3
4
7
5
2
3
5
5
9
2
6
4
7
1

Yellow: Pivot {2,9}
Blue: Wings {2,7} & {9,7}
Red: Eliminate 7 from R3C2

12. Unique Rectangles

Four cells forming a rectangle that would create multiple solutions if filled with the same two numbers. This violates the uniqueness principle, so it can be avoided.

In this Unique Rectangle example:

  • • Rectangle corners: R2C1, R2C9, R8C1, R8C9
  • • Three corners have candidates {2,7}
  • • R8C9 has {2,7,6} - extra candidate 6
  • • If all corners = {2,7}, deadly pattern creates multiple solutions
  • • Therefore R8C9 cannot be 2 or 7, must be 6
  • • This breaks the deadly rectangle and preserves uniqueness
5
3
8
6
9
4
1
5
3
27
6
4
1
8
3
5
9
27
1
9
3
2
5
7
8
4
6
4
1
7
3
2
8
6
9
5
8
5
6
9
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1
4
7
2
9
2
5
7
4
6
3
1
8
3
7
1
8
6
9
2
3
4
27
8
9
5
1
2
7
6
276
6
4
2
4
7
5
9
8
1

Blue: Rectangle corners {2,7}
Yellow: Extra candidate 6, breaks deadly pattern

4

Level 4: Expert Techniques

Master level

13. Simple Coloring

Uses two colors to track conjugate pairs (strong links) for a single number. When coloring leads to contradictions or forces, eliminations can be made.

In this Simple Coloring example:

  • • Tracking number 3 with two alternating colors
  • • Blue chain: R2C2, R7C2, R7C8 (strong links)
  • • Yellow chain: R2C8, R6C8 (strong links)
  • • R7C2 and R7C8 are both blue - same row!
  • • Contradiction: blue chain is invalid
  • • Therefore yellow chain must be true: R2C8=3, R6C8=3
5
6
4
8
7
9
1
2
4
2
39
1
4
5
6
7
39
8
7
8
3
1
2
4
9
5
6
1
4
7
2
9
8
6
3
5
6
5
9
3
1
7
8
4
2
8
2
6
5
4
1
2
39
7
4
39
2
7
6
5
4
39
1
9
7
5
6
8
2
3
1
4
3
1
8
9
3
4
5
6
9

Blue: Color A chain (contradicted)
Yellow: Color B chain (valid)
Elimination using conjugate pairs

14. Forcing Chains

Follow logical implications from a starting assumption. If all possible paths lead to the same conclusion, that conclusion must be true.

In this Forcing Chain example:

  • • Starting bi-value cell R1C3: {4,8} - test both values
  • • Chain A: If R1C3=4 → R1C6=8 → R3C6=5 → R3C9=6
  • • Chain B: If R1C3=8 → R4C3=4 → R6C3=6 → R3C9=6
  • • Both chains force R3C9=6
  • • Therefore R3C9 must be 6, eliminate 5 and 9
  • • Forcing chains prove conclusions regardless of starting value
5
3
48
7
2
58
1
9
2
6
7
9
1
4
3
8
2
5
1
8
2
9
6
5
3
4
569
2
1
4
3
5
6
7
8
9
8
5
3
2
9
7
4
1
6
7
9
6
4
8
1
2
5
3
3
2
1
5
7
9
6
3
4
9
6
7
8
3
4
5
6
1
4
4
5
6
1
2
9
7
8

Blue: Bi-value starting cell
Yellow: Forced conclusion (must be 6)
Both chains lead to same result

15. Almost Locked Sets (ALS)

Groups of N cells in a unit that contain exactly N+1 candidate numbers. These create powerful elimination patterns when combined correctly.

In this ALS-XZ example:

  • • ALS1 (Box 1): R1C1, R2C1, R3C1 = 3 cells, 4 candidates {1,5,7,9}
  • • ALS2 (Box 3): R1C7, R2C7, R3C7 = 3 cells, 4 candidates {2,4,6,9}
  • • Common restricted candidate X = 9 (in both ALS)
  • • Z candidates: 1,5,7 from ALS1 and 2,4,6 from ALS2
  • • R2C4 sees both ALS groups: eliminate Z candidates
  • • R2C4 cannot be 1,2,4,5,6,7 → must be 3 or 8
579
3
4
6
8
2
269
1
5
19
6
8
38
5
1
49
7
2
157
2
1
7
9
3
246
8
4
8
1
7
2
4
6
3
9
1
6
4
5
1
3
9
8
2
7
2
9
3
5
7
8
1
4
6
3
5
2
9
1
4
7
6
8
7
8
6
4
2
5
9
3
4
4
7
9
8
6
3
5
5
3

Blue: ALS1 cells {1,5,7,9}
Yellow: ALS2 cells {2,4,6,9}
Red: Eliminated by ALS-XZ rule

General Solving Tips

Solving Order

  • 1. Start with naked singles
  • 2. Look for hidden singles
  • 3. Use pencil marks for complex puzzles
  • 4. Apply intermediate techniques when stuck
  • 5. Try advanced techniques as last resort

Best Practices

  • • Work systematically, don't guess
  • • Use pencil marks for possibilities
  • • Check your work frequently
  • • Focus on areas with fewer possibilities
  • • Take breaks when stuck

Practice makes perfect!

Apply these techniques to real puzzles, or let our AI solver handle the hard work for you.

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