Sudoku Solving Strategies & Techniques

1. Naked Singles Beginner

What is it?

A naked single is a cell that has only one possible candidate number remaining. This is the most fundamental Sudoku technique and the first thing you should look for in any puzzle.

How to identify:

• Look at an empty cell
• Check which numbers already exist in its row
• Check which numbers already exist in its column
• Check which numbers already exist in its 3x3 box
• If only one number (1-9) is missing from all three units, that's your answer

Example:

If a cell's row contains {1,2,3,4,5,6,7,8}, its column contains {1,2,3,5,6,7,8,9}, and its box contains {2,3,4,5,6,7,8,9}, then the only number that doesn't appear in any of these units is 1. Therefore, this cell must be 1.

2. Hidden Singles Beginner

What is it?

A hidden single occurs when a specific number can only be placed in one position within a row, column, or box, even if that cell has multiple candidates.

How to identify:

• Pick a number (e.g., 5)
• Look at a specific row, column, or box
• Check all empty cells in that unit to see where 5 could go
• If only one cell can contain 5, place it there
• Repeat for all numbers 1-9 in all rows, columns, and boxes

Example:

In a particular box, the number 7 is missing. Looking at all empty cells in this box, you notice that six cells already have 7 eliminated (because 7 appears in their rows or columns). Only one cell remains where 7 could possibly go - that's a hidden single.

3. Cross-Hatching / Scanning Beginner

What is it?

Cross-hatching is a visual technique where you scan rows and columns to determine where a specific number must be placed in a box.

How to use it:

• Choose a number (e.g., 4)
• Find boxes where 4 has been placed
• Draw imaginary lines through the rows and columns containing those 4s
• Look at boxes where 4 hasn't been placed yet
• The intersecting lines eliminate certain cells, often leaving only one possible position

This technique works best with numbers that appear frequently in the given clues.

4. Naked Pairs Intermediate

What is it?

When two cells in the same row, column, or box both contain exactly the same two candidates (and only those two), they form a naked pair. These two numbers must occupy these two cells.

How to use it:

• Find two cells in the same unit (row/column/box) with identical candidate pairs, like {3,7}
• Even though you don't know which cell gets 3 and which gets 7, you know these cells must contain 3 and 7
• Remove 3 and 7 from all other cells in that same unit

Example:

In row 5, cells A and B both have candidates {2,8}. This means 2 and 8 must go in these two cells. You can now eliminate 2 and 8 from all other cells in row 5.

5. Hidden Pairs Intermediate

What is it?

When two candidates can only appear in two specific cells within a row, column, or box (and nowhere else in that unit), those cells must contain those two numbers, even if they have other candidates.

How to identify:

• Look at a row, column, or box
• Find two numbers that can only go in the same two cells within that unit
• Remove all other candidates from those two cells

Example:

In box 3, the numbers 1 and 6 can only appear in cells X and Y (and nowhere else in that box). Cell X might have candidates {1,4,6,9} and cell Y might have {1,2,6,8}, but since 1 and 6 must go in these cells, you can remove the other candidates, leaving X={1,6} and Y={1,6}.

6. Pointing Pairs/Triples Intermediate

What is it?

When a candidate in a box is limited to a single row or column within that box, that candidate can be eliminated from the rest of that row or column outside the box.

How to identify:

• Look at a candidate number within a box
• If that candidate only appears in one row or column within the box (in 2 or 3 cells)
• Eliminate that candidate from the rest of that row or column outside the box

Example:

In box 1 (top-left), the candidate 9 only appears in the top row of the box (in 2 cells). This means 9 must be in one of those two cells. Therefore, you can eliminate 9 from all other cells in the top row that are outside box 1.

7. Box/Line Reduction Intermediate

What is it?

The opposite of pointing pairs. When all candidates for a number in a row or column are confined to a single box, that candidate can be eliminated from other cells in that box.

How to identify:

• Look at a row or column
• Find a candidate that only appears in one box within that row/column
• Eliminate that candidate from other cells in that box (outside the row/column)

Example:

In row 4, all the 5s are confined to box 5 (middle box). This means 5 in box 5 must be in row 4. You can eliminate 5 from all other cells in box 5 that aren't in row 4.

8. Naked Triples/Quads Advanced

What is it?

An extension of naked pairs. When three cells in the same unit contain only three specific candidates (distributed among them), or four cells contain only four candidates, those candidates belong to those cells.

How to identify:

• Find three cells in the same unit where the combined candidates contain exactly three numbers
• The cells might have {1,2}, {2,3}, {1,3} or {1,2,3}, {2,3}, {1,2} etc.
• These three numbers must go in these three cells
• Remove these three candidates from all other cells in the unit

Example:

Three cells in column 7 have candidates: {4,9}, {4,6}, {6,9}. These three cells must contain 4, 6, and 9. Remove 4, 6, and 9 from all other cells in column 7.

9. X-Wing Advanced

What is it?

X-Wing is a powerful technique that works across two rows (or columns). When a candidate appears in exactly two positions in two different rows, and these positions are in the same columns, eliminations can be made in those columns.

How to identify:

• Pick a candidate number (e.g., 3)
• Find two rows where 3 appears in exactly two cells each
• Check if these four cells align to form a rectangle (same column positions)
• If yes, eliminate 3 from all other cells in those two columns

Why it works:

If 3 must appear once in each of the two rows, and there are only two possible positions (the corners of the rectangle), then 3 must occupy two diagonally opposite corners. Either way, 3 is locked into those two columns, so it can be eliminated elsewhere in those columns.

Example:

Row 2 has candidate 7 in columns 3 and 8. Row 6 has candidate 7 in columns 3 and 8. This forms an X-Wing. You can eliminate 7 from all other cells in columns 3 and 8.

10. Y-Wing Advanced

What is it?

Y-Wing involves three cells forming a Y-shape: one "pivot" cell and two "wing" cells. The pivot has two candidates, and each wing shares one candidate with the pivot and has one common candidate with the other wing.

Structure:

• Pivot cell: {A,B}
• Wing 1: {A,C} (shares A with pivot)
• Wing 2: {B,C} (shares B with pivot)
• Both wings share candidate C

How to use it:

Any cell that can "see" both wing cells (is in the same row, column, or box as both wings) cannot be C. Why? If the pivot is A, then Wing 2 must be C. If the pivot is B, then Wing 1 must be C. Either way, C must be in one of the wings.

Example:

Pivot at R5C5: {3,7}. Wing 1 at R5C2: {3,8}. Wing 2 at R8C5: {7,8}. Any cell that sees both R5C2 and R8C5 cannot be 8.

11. Swordfish Advanced

What is it?

Swordfish is an extension of X-Wing involving three rows and three columns instead of two. It's a rare but powerful technique for difficult puzzles.

How to identify:

• Pick a candidate number
• Find three rows where this candidate appears in 2 or 3 positions
• If these positions align to three columns, you have a Swordfish
• Eliminate the candidate from all other cells in those three columns

Example:

Candidate 5 appears: Row 1 in columns 2,5,9; Row 4 in columns 2,9; Row 8 in columns 5,9. These align to columns 2, 5, and 9, forming a Swordfish. Eliminate 5 from all other cells in columns 2, 5, and 9.

12. XYZ-Wing Expert

What is it?

Similar to Y-Wing but involves three candidates instead of two. The pivot cell has three candidates {X,Y,Z}, and the wing cells each have two of these three candidates.

Pattern:

• Pivot: {X,Y,Z}
• Wing 1: {X,Y} or {X,Z} or {Y,Z}
• Wing 2: {X,Y} or {X,Z} or {Y,Z}

Any cell that sees all three cells (pivot and both wings) cannot contain the common candidate that appears in all three cells.

13. Unique Rectangles Expert

What is it?

A technique based on the assumption that a well-formed Sudoku has only one solution. When you find four cells forming a rectangle with the same two candidates, you can make eliminations to prevent multiple solutions.

Requirements:

• Four cells form a rectangle (two rows, two columns, two boxes)
• All four cells contain the same two candidates {A,B}
• This would create a "deadly pattern" with multiple solutions
• To prevent this, eliminate candidates to avoid the pattern

Note: This technique assumes the puzzle has a unique solution, which is standard for published Sudoku puzzles.

14. Jellyfish Expert

What is it?

Jellyfish is an extension of X-Wing and Swordfish, involving four rows and four columns. It's extremely rare but useful for the hardest puzzles.

How it works:

When a candidate appears in 2-4 positions in four different rows, and these positions align with four columns, eliminations can be made in those columns (and vice versa).

Due to its rarity and complexity, Jellyfish is typically only needed for extremely difficult puzzles.