When you first learn to play Sudoku, you usually progress by finding a cell, determining the exact number that belongs there, and writing it in boldly. For "Easy" difficulty puzzles, this method of direct scanning is often enough to complete the entire grid. However, once you graduate to "Medium," "Hard," or "Expert" levels, you will inevitably hit a wall where you cannot find a single cell with a definitive answer.
This is exactly where the most powerful tool in a Sudoku player's arsenal comes into play: Candidate Digit Analysis (also known as Pencil Marks). This technique involves writing small numbers in the corners of empty cells to track all legal possibilities, turning the grid into a visual database. If you ever get stuck and need help analyzing specific regions, our Sudoku Solver, Difficulty & Candidate Analyzer can instantly calculate these missing candidates for you.
What Are Candidate Digits?
Candidate digits are the numbers that are legally allowed to be placed in a specific empty cell, based on the current state of the board.
The mathematical definition of candidates for any given cell is:Candidates = {1..9} \ (Digits in Row ∪ Digits in Column ∪ Digits in Block)
In other words, you take the full set of numbers from 1 to 9, and you subtract all the numbers that already exist in the cell's intersecting row, column, and 3x3 block. The remaining numbers are your "candidates."
Strategies for Noting Candidates
Globally, players use two main methods for taking notes on candidates: Snyder Notation and Full Notation.
1. Snyder Notation (Selective Noting)
This method is incredibly popular among speed-solvers because it prevents the grid from becoming cluttered and visually overwhelming.
- The Rule: Within a 3x3 block, you only write down a candidate number if it can be placed in exactly two cells (sometimes extended to three if they are in a straight line). If a number has 4 or 5 possible spots in a block, you do not write it down at all.
- The Benefit: Snyder Notation makes finding "Hidden Singles" and "Naked Pairs" incredibly easy. If another number is later placed in one of those two cells, you immediately know that the other cell must be your candidate, without needing to re-scan the board.
2. Full Notation
When Snyder Notation stops yielding results (which usually happens in the mid-to-late stages of Hard or Expert puzzles), you must switch to Full Notation.
- The Rule: You write down every single possible candidate for every empty cell on the board.
- The Benefit: Advanced techniques like X-Wings, Swordfish, and complex logical chains (Forcing Chains) require complete data. If you miss even one candidate, you might execute an advanced technique incorrectly and break the puzzle.
The Process of Elimination
Solving advanced Sudoku is not really about finding the right numbers; it is about eliminating the wrong numbers (candidates). Once you eliminate all impossible candidates, whatever remains—no matter how unlikely—must be the correct answer.
After taking full notation, you apply elimination techniques in stages:
Stage 1: Basic Eliminations
- Naked Single: If you look at your notes and see that a cell has only one candidate digit written in it, that digit is confirmed. Write it in large, and then erase that digit from the candidate lists of all other cells in the same row, column, and block. This erasure often triggers a chain reaction of new Naked Singles.
- Hidden Single: Look at a 3x3 block. If the digit "5" is written as a candidate in only one cell within that block, then that cell must be the 5, regardless of how many other candidates (e.g., 2, 4, 8) are also written in that same cell.
Stage 2: Pairs and Triples
- Naked Pairs: If two cells in the same region (row, column, or block) contain the exact same two candidates (e.g., only 3 and 7), those two numbers are locked into those two cells. You can safely erase 3 and 7 from the notes of every other cell in that region.
- Hidden Pairs: If, within a region, two specific numbers appear as candidates in only two specific cells (even if those cells have other candidates written in them too), those two cells are reserved for those two numbers. You must erase all the other (extra) candidates from those two cells.
Stage 3: Regional Reductions
- Box-Line Reduction / Pointing Pairs: If all the candidate notes for a specific digit within a 3x3 block are aligned in a single row or column, that digit must appear in that line within the block. Therefore, you can erase that digit as a candidate from the rest of that row/column outside the block.
Using Tools for Candidate Analysis
If writing down candidates across the whole board feels tedious, or if you are afraid of making a mistake (forgetting to write one candidate can cause a fatal error later), software tools can be highly beneficial.
Our Sudoku Solver, Difficulty & Candidate Analyzer allows you to analyze specific lines or blocks instantly.
For example, if you input a row containing 1, 4, 7, 9, the tool processes the math:Present Digits = {1, 4, 7, 9}Candidates = {1,2,3,4,5,6,7,8,9} \ {1,4,7,9} = {2, 3, 5, 6, 8}
It then provides you with the exact missing candidate list for that region, eliminating human error from your note-taking.
Common Mistakes in Candidate Analysis
- Forgetting to Update Notes: When you solve a cell and write a large "6", you must immediately erase "6" from the candidates of the intersecting row, column, and block. If you forget, you might later use that obsolete "6" to falsely trigger an X-Wing.
- Going to Full Notation Too Early: Writing all candidates for 60 empty cells at the start of a puzzle is a waste of time and causes severe eye strain. Always start with Snyder Notation and basic scanning, and only switch to Full Notation when you are truly stuck.
- The 'Congratulations' Effect: After finding a difficult Naked Pair and erasing the candidates, players often pause, feeling accomplished. However, every erasure changes the board state. Immediately after making an elimination, you must re-scan the affected regions for new Naked Singles that may have just been created.
Frequently Asked Questions
Can't I just guess instead of writing all these tiny numbers?
You might get lucky on Easy or Medium puzzles. But on Hard puzzles, guessing a 50/50 shot and realizing 15 moves later that you broke the board (by causing a collision of two identical numbers) is incredibly frustrating. Candidate analysis is the guaranteed path; guessing is a gamble.
How long does it take to write full notes for a board?
It depends on practice. Digital apps have "auto-candidate" buttons, but on paper, it might take 5 to 7 minutes. However, this small time investment will save you hours of staring blankly at an unsolved board.
Should I note candidates that appear 3 times in Snyder Notation?
Strict Snyder Notation only tracks pairs (2 times). However, intermediate players often adapt the rule to include 3 instances, especially if those 3 cells are arranged in a straight line (which sets up Box-Line Reductions). It ultimately depends on your personal solving style.
