Sudoku Elimination Method

What Is the Elimination Method?

The elimination method (also called candidate elimination or process of elimination) is the core logic principle underlying all Sudoku solving. The idea is simple: for any empty cell, you determine which numbers are impossible by checking what already appears in the same row, column, and 3x3 box. Whatever remains after elimination must include the answer.

If elimination leaves only one candidate in a cell, you have found the answer — this is a Naked Single. If a candidate can only go in one cell within a row, column, or box, that is a Hidden Single. Every named Sudoku technique is ultimately a more sophisticated application of this same elimination principle.

How Basic Elimination Works

Consider an empty cell at the intersection of row 3, column 5, inside box 2. To determine what can go in this cell:

1. Start with all possibilities: The cell could theoretically contain any digit from 1 to 9.
2. Eliminate by row: Check row 3. If it already contains 1, 4, 7, and 9, eliminate those — leaving {2, 3, 5, 6, 8}.
3. Eliminate by column: Check column 5. If it contains 2, 5, and 8, eliminate those — leaving {3, 6}.
4. Eliminate by box: Check box 2. If it contains 3, eliminate that — leaving {6}.
5. Result: Only 6 remains. The cell must contain 6.

This is the most basic form of elimination, and it is exactly what you do when you write pencil marks (candidates) in each cell. The power comes from doing this systematically across the entire grid and then looking for patterns among the remaining candidates.

Types of Elimination in Sudoku

Elimination techniques can be grouped by complexity. Each level builds on the one before it:

1. Direct Elimination (Singles)

The simplest form. After eliminating all impossible candidates, only one remains.

• Naked Single — A cell with only one candidate after row, column, and box elimination.
• Hidden Single — A candidate appears in only one cell within a unit (row, column, or box).
• Last Free Cell — The simplest case: a unit has 8 numbers filled, so the 9th is forced.

2. Pair and Triple Elimination

When two or three cells in a unit share exactly two or three candidates, those candidates can be eliminated from all other cells in that unit.

• Naked Pair — Two cells in a unit with the same two candidates. Those two numbers are eliminated from all other cells in the unit.
• Naked Triple — Three cells sharing three candidates. The same elimination principle applies.
• Hidden Pairs and Triples — The inverse: two or three candidates confined to two or three cells in a unit. All other candidates in those cells can be eliminated.

3. Intersection Elimination

When a candidate within a box is restricted to a single row or column, it can be eliminated from the rest of that row or column outside the box (and vice versa).

• Pointing Pair — A candidate in a box confined to one row/column, eliminating it from the rest of that row/column.
• Box-Line Reduction — A candidate in a row/column confined to one box, eliminating it from the rest of that box.

4. Pattern-Based Elimination (Fish)

These advanced techniques find patterns across multiple rows and columns simultaneously:

• X-Wing — A candidate appears in exactly two positions in two rows, forming a rectangle. The candidate is eliminated from those two columns (or vice versa).
• Swordfish — The same principle extended to three rows and three columns.
• Y-Wing — A chain of three cells with paired candidates that eliminates a candidate from cells seeing both endpoints.

Step-by-Step Elimination Strategy

For most puzzles, follow this sequence of elimination techniques from simplest to most complex:

1. Scan for Last Free Cells — Check each row, column, and box for units with only one empty cell.
2. Crosshatch for Hidden Singles — Use crosshatching to find where each number must go within boxes.
3. Write pencil marks — Fill in candidates for all remaining empty cells.
4. Find Naked Singles — Look for cells with only one candidate.
5. Check for Pointing Pairs and Box-Line Reduction — Eliminate candidates using intersection logic.
6. Look for Naked Pairs and Triples — Find cells sharing candidates within a unit.
7. Apply Fish patterns — Search for X-Wings and Swordfish if simpler techniques stall.
8. Repeat — Each elimination may create new singles. Always restart from step 1 after making progress.

Common Elimination Mistakes

• Forgetting the box constraint: New players often check rows and columns but forget that the 3x3 box also eliminates candidates. Always check all three constraints.
• Not updating candidates: After placing a number, you must remove that number from all candidates in the same row, column, and box. Stale candidates lead to errors.
• Skipping simple techniques: Do not jump to X-Wings before exhausting all singles and pairs. Simpler techniques are faster and less error-prone.
• Eliminating from the wrong scope: A Naked Pair in a row only eliminates candidates from that row — not from the box, unless the pair also happens to share a box.

Practice Elimination with Our Solver

The best way to master elimination is to see it in action. Our AI-powered Sudoku Solver shows you exactly which technique it applies at each step, including which candidates are eliminated and why. You can also try puzzles at different difficulty levels:

• Easy puzzles — Solvable with direct elimination (singles only)
• Medium puzzles — Require pair and intersection elimination
• Hard puzzles — May require pattern-based elimination (X-Wing, Y-Wing)
• Expert puzzles — Demand mastery of all elimination types

For a daily challenge, try the Daily Sudoku. And for more tips on using pencil marks effectively, read our Pencil Marks Guide.