If you have ever stared at a medium or hard Sudoku grid and felt completely stuck, there is a good chance that pencil marks are the missing tool in your solving kit. Pencil marks — also called candidate notes, possibility lists, or simply candidates — are the small numbers you write inside a cell to record every digit that could legally go there. Used well, they transform a confusing jumble of empty squares into a structured map that your eye and brain can navigate with confidence. This guide explains exactly what pencil marks are, when to start using them, how to write them efficiently, and how to use them to unlock techniques that would otherwise be nearly impossible to apply.
What Are Pencil Marks and Why Do They Matter?
In standard Sudoku, each of the 81 cells in the 9×9 grid must contain a digit from 1 to 9. The rule is simple: no digit can repeat in any row, column, or 3×3 box. When a puzzle is easy, many cells have only one possible digit, and you can fill them in directly. But as difficulty increases, cells become ambiguous — two, three, or even six different digits might all be legal at first glance. This is where pencil marks earn their name.
A pencil mark is a tiny digit written lightly inside a cell to indicate that this digit is still a candidate — a possible answer — for that cell. When you later prove that a digit cannot go in that cell, you erase (or cross out) the mark. Eventually, if you work methodically, each cell will have exactly one candidate left, and that becomes the cell’s answer.
Without pencil marks, your working memory has to track dozens of possibilities simultaneously. That is cognitively exhausting and error-prone. With pencil marks, the grid itself does the remembering for you, freeing your mind to spot patterns. This is why every serious Sudoku solver — from competitive players to enthusiastic hobbyists — relies on candidate notation for anything beyond easy puzzles.
When Should You Start Writing Pencil Marks?
One of the most common mistakes new solvers make is adding pencil marks too early or too late. Here is a practical approach that balances speed with thoroughness.
Step 1: Do a quick “naked singles” pass first. Before touching pencil marks, scan the grid for any cells where only one digit is possible without needing notes. These are called naked singles. Fill them in immediately. Similarly, look for hidden singles — rows, columns, or boxes where a particular digit can only fit in one cell. Work through these easy wins first. On many easy and some medium puzzles, this alone will solve the entire grid.
Step 2: Switch to pencil marks when you get stuck. The moment you complete a full scan of the grid and cannot place any more digits with certainty, it is time to annotate. Do not wait until frustration sets in; switch modes as soon as the straightforward logic runs out.
Step 3: Keep marks updated as you fill in answers. Every time you place a confirmed digit, immediately erase that digit from the pencil marks of every other cell in the same row, column, and box. Failing to do this is the number one cause of mistakes when using candidate notes.
How to Write Pencil Marks Efficiently
There are two popular layouts for writing candidates inside a cell, and choosing the right one makes a real difference to readability.
The fixed grid layout (recommended): Divide the cell mentally into a 3×3 mini-grid and always place each digit in the same position — 1 in the top-left, 2 in the top-centre, 3 in the top-right, 4 in the middle-left, and so on up to 9 in the bottom-right. This method means you can glance at any cell and instantly see which candidates are present and which are absent, because the position is fixed. Spotting that a cell contains only 1 and 5, for example, is immediate — the top-left and middle-right spots are filled, the rest are blank.
The sequential list layout: Some solvers simply write candidates in a small line across the top of the cell (e.g., “1 4 7”). This works fine when there are only two or three candidates but becomes hard to read when there are five or six.
For digital Sudoku apps — including those on playsudoku.org — most platforms have a dedicated “notes” or “pencil” mode. Tap the notes button, then tap the digits you want to add as candidates. The app typically displays them in the fixed grid layout automatically, saving you the effort entirely.
Practical tips for physical puzzles:
- Use a mechanical pencil with a fine tip (0.5 mm or 0.3 mm) so marks stay small and neat.
- Write lightly so erasing is clean and does not smudge.
- If you are solving in pen for a challenge, use a different colour for candidates versus confirmed answers.
A Worked Example: Using Pencil Marks to Find a Naked Pair
Let us walk through a concrete illustration of how pencil marks reveal a pattern that is invisible to the naked eye.
Imagine a row that already contains the digits 3, 5, 6, 7, and 9. The remaining four empty cells in that row still need the digits 1, 2, 4, and 8 placed between them. After adding pencil marks and eliminating candidates based on the column and box rules, suppose you find the following situation:
- Cell A: candidates 1, 4
- Cell B: candidates 1, 2, 4
- Cell C: candidates 1, 4
- Cell D: candidates 2, 8
Look at cells A and C. Both contain exactly the same two candidates: 1 and 4. This is called a naked pair. It means that, whatever happens, digits 1 and 4 must go into cells A and C (in some order). We do not know which way around yet, but we know for certain that neither 1 nor 4 can go into any other cell in the row. Therefore, you can eliminate 1 and 4 from cell B’s candidate list. Cell B then becomes a naked single containing only 2, so you fill it in immediately. Cell D already only has 2 and 8; removing 2 (now placed in cell B) leaves cell D with only 8, so that is filled in too. Two cells solved in seconds — purely because pencil marks made the naked pair visible.
Without candidate notes, spotting a naked pair like this would require holding multiple possibilities in your head at once and comparing them mentally. With pencil marks on the page, the pattern jumps out visually.
Advanced Techniques That Rely on Pencil Marks
Once you are comfortable with basic candidate notation, pencil marks become the foundation for a whole family of powerful solving strategies. Understanding the names of these techniques helps you search for tutorials and build your skills progressively.
Naked pairs, triples, and quads: As shown in the example above, when two (or three, or four) cells in a unit share the exact same set of candidates, those digits can be eliminated from all other cells in that unit.
Hidden pairs and triples: The mirror image of the above. If two digits only appear as candidates in exactly two cells within a row, column, or box, those two cells must contain those two digits — even if they have other candidates listed. You can eliminate the extra candidates from those cells.
Pointing pairs and box-line reduction: When a candidate digit appears in only two or three cells within a box, and all those cells share the same row or column, the digit can be eliminated from the rest of that row or column outside the box. This technique is called a pointing pair (or pointing triple) and is a staple of medium-difficulty solving.
X-Wing: One of the first “fish” patterns in Sudoku. If a candidate digit appears in exactly two cells in each of two different rows, and those cells share the same two columns, you can eliminate that digit from the rest of those two columns. Spotting an X-Wing without pencil marks is essentially impossible.
Swordfish and Jellyfish: Extensions of the X-Wing idea across three or four rows and columns respectively. These are advanced techniques used on the hardest expert-level puzzles.
Chains and forcing chains: These involve following a chain of candidate eliminations — “if cell A is 3, then cell B cannot be 3, which means cell B must be 7, which means…” — until a contradiction or a forced answer emerges. Candidate notes are absolutely essential here because chains can run through ten or more cells.
The jump in puzzle-solving ability you experience when you add these techniques to your toolkit is dramatic. Most people find they can comfortably tackle hard and even expert Sudoku puzzles after mastering pencil-mark-based strategies, whereas before they could only complete easy grids reliably.
Common Pencil Mark Mistakes to Avoid
Even experienced solvers fall into certain traps with candidate notation. Being aware of them saves a lot of backtracking.
- Not updating after placing a digit. This is the most common error. Always eliminate the placed digit from all candidates in the same row, column, and box immediately.
- Adding pencil marks before simpler techniques are exhausted. Cluttering the grid with marks when straightforward logic would work makes the grid harder to read, not easier.
- Forgetting to check box eliminations. When updating marks after placing a digit, solvers often remember the row and column but forget to remove the candidate from the relevant 3×3 box.
- Treating pencil marks as confirmed answers. A candidate note means “this might be the answer,” never “this is the answer.” Stay flexible.
- Messy handwriting. Illegible pencil marks cause misreads that lead to wrong answers. Neatness pays off.
Key Takeaways
- Pencil marks (candidate notes) list every possible digit for each empty cell, giving you a visual map of the puzzle’s remaining possibilities.
- Always exhaust straightforward naked singles and hidden singles before switching to pencil-mark mode.
- Use a fixed-position layout so candidates are always in the same spot, making patterns easy to spot at a glance.
- Update your marks every single time you place a confirmed digit — in the row, column, and box.
- Pencil marks unlock advanced techniques like naked pairs, pointing pairs, X-Wings, and chains that are otherwise practically impossible to use.
- Both physical and digital solvers benefit equally; most Sudoku apps include a built-in notes mode.
Pencil marks might look like extra work at first, but they are actually the key to solving Sudoku faster, more accurately, and with far more satisfaction. The moment a hidden pattern leaps off the page because you have your candidates written clearly in front of you is genuinely exciting — and that feeling never really gets old. Give candidate notation a serious try on your next medium or hard puzzle at playsudoku.org, and you may be surprised just how quickly those once-impossible grids start to yield their secrets.