The Elimination Method in Sudoku
Elimination is the foundational logic behind every sudoku technique. Whether you are solving an easy puzzle or a championship-level expert grid, you are always doing the same thing at the core: ruling out digits that cannot go in a cell until only one remains. This article explains the elimination method in full — what it is, how to apply it, and how it scales from beginner puzzles to advanced ones.
The Core Idea
Every empty cell in a sudoku grid has a set of candidate digits — the digits from 1 to 9 that could theoretically go there without immediately breaking a rule. Elimination is the process of reducing that candidate set until only one digit remains. That digit is the answer.
The three sudoku rules give you three sources of eliminations:
- Any digit already in the cell's row cannot go in that cell.
- Any digit already in the cell's column cannot go in that cell.
- Any digit already in the cell's 3×3 box cannot go in that cell.
Combine all three, and you have the complete list of eliminations for that cell. Whatever is left from 1–9 after those eliminations are your candidates.
Naked Singles: Elimination Down to One
When elimination reduces a cell's candidates to exactly one digit, that digit is called a naked single. It's the simplest outcome of elimination, and easy puzzles are full of them.
Example: An empty cell belongs to a row containing 1, 2, 4, 6, 7, 8, 9 — eliminating seven digits. Its column contains 3, eliminating one more. Its 3×3 box adds no new digits. After all eliminations, only 5 remains. That cell must be 5.
How to Apply Elimination Systematically
Random cell-by-cell checking is slow and error-prone. A systematic approach is faster and more reliable:
- Work row by row. For each empty cell in a row, note which digits are already in that row. Those are eliminated for every empty cell in the row simultaneously.
- Add column eliminations. For each empty cell, also note which digits are in its column. Add those to the elimination list.
- Add box eliminations. Finally, check the 3×3 box. Add any new digits to the elimination list.
- Check what remains. If one digit is left, place it. If more than one remains, move on and come back later.
The key insight is that you don't need to find the answer for every cell on the first pass. Some cells will yield answers immediately; others require more of the grid to be filled before elimination narrows them down. Make one pass, place everything you can, then make another pass. The puzzle solves itself iteratively.
Pencil Marks: Tracking Candidates on Paper
For harder puzzles where elimination doesn't immediately produce a naked single, experienced solvers write small candidate digits inside each empty cell. These are called pencil marks.
The process:
- For each empty cell, write all remaining candidates (after elimination) in small digits.
- When you place a digit elsewhere on the grid, erase it as a candidate from all cells in the same row, column, and box.
- When a cell's pencil marks are reduced to one digit, place that digit.
Pencil marks transform elimination from a mental exercise into a visual one. You can see the candidates shrinking in real time as you fill in the grid. For easy and medium puzzles, mental elimination is usually sufficient. For hard and expert puzzles, pencil marks are essentially required.
Hidden Singles: A More Subtle Form of Elimination
Sometimes a digit doesn't have only one candidate cell across the whole grid — but within a specific row, column, or box, it can only go in one place. This is called a hidden single.
Example: The digit 4 needs to be placed in the middle-left 3×3 box. Three of the empty cells in that box are in rows that already contain a 4, eliminating them. The fourth empty cell is in a column that already contains a 4, eliminating it too. Only one empty cell in the box has no elimination reason — that cell must be 4.
Hidden singles are slightly harder to spot than naked singles because you're thinking about one digit across multiple cells, rather than one cell across multiple digits. But the underlying logic is identical: elimination.
Why Elimination Never Requires Guessing
A common frustration for beginners: "I've eliminated everything I can and I'm still stuck. Do I have to guess?"
On a properly constructed puzzle — one with a unique solution — the answer is no. If elimination isn't yielding results, it means one of two things:
- You haven't tried all three groups. Double-check that you've eliminated based on the row, the column, AND the box for every cell you examined.
- You need a more advanced technique. Medium and hard puzzles require techniques that go beyond basic elimination — naked pairs, pointing pairs, X-Wings, and others. These are all still elimination at heart, just applied across multiple cells at once.
Guessing is never the right answer on a valid sudoku puzzle. If you feel forced to guess, treat it as a signal to slow down and look more carefully.
Elimination as the Foundation of All Techniques
Every advanced sudoku technique — no matter how complex it sounds — is ultimately a way to perform eliminations that basic cell-by-cell checking can't find. Naked pairs eliminate candidates from neighboring cells. X-Wings eliminate candidates across rows and columns. Swordfish extends that logic to three rows and columns.
Understanding elimination deeply means you can learn any technique quickly, because you already understand what it's trying to accomplish. The technique is just the method; elimination is the goal.
Practice Elimination on a Real Puzzle
The best way to internalize elimination is to apply it deliberately on a real puzzle. Pick any empty cell, write out its row digits, column digits, and box digits, and see what remains. Do this for ten cells in a row and you'll start to do it automatically.
Start with easy puzzles where naked singles are abundant, then progress to medium puzzles where hidden singles become necessary.
Play a free Easy Sudoku puzzle →
Next, see elimination in action with named techniques: How to Use Scanning →