How to Use Pencil Marks Effectively in Sudoku

Pencil marks are small candidate numbers written in sudoku cells to track which digits are still possible for that cell. They are not a crutch — they are a core solving tool used by every serious solver, from intermediate players to world championship competitors. Without pencil marks, techniques like naked pairs, pointing pairs, and X-Wing are essentially impossible to apply. This guide covers when to use pencil marks, the two main notation systems, strategies for keeping them manageable, and how digital apps handle notation differently from paper.

When to Start Using Pencil Marks

You do not need pencil marks for every puzzle or every phase of solving. Here is a practical framework:

• Easy puzzles: Usually solvable with naked singles, hidden singles, and basic scanning — no pencil marks needed. If you want to build speed, solving easy puzzles without pencil marks is good practice.
• Medium puzzles: You will typically reach a point where singles are exhausted and you need to find pairs or other patterns. This is when pencil marks become valuable. Start adding them when your first scan stalls.
• Hard and expert puzzles: Pencil marks are essential from the start. These puzzles require candidate analysis for almost every placement, so building complete notation early saves time.

The general rule: if you have scanned the puzzle twice and cannot find any more naked or hidden singles, it is time for pencil marks.

Snyder Notation: The Speed Solver's Choice

Snyder notation, named after puzzle author Thomas Snyder, is a minimalist pencil marking system designed for speed. The rule is simple: only write a candidate in a cell when that number has exactly two possible positions within the cell's box.

How It Works

For each 3x3 box, go through each missing number. If that number can go in exactly two cells within the box, write it as a small notation in both cells. If it can go in three or more cells, do not write anything. If it can go in only one cell, place it — that is a hidden single.

Why Snyder Works

Snyder notation captures the most useful information with the least clutter:

• Naked pairs become immediately visible. If two cells in a box have identical Snyder marks, you have a naked pair.
• Pointing pairs jump out. If a candidate's two Snyder positions are in the same row or column, it is a pointing pair.
• Minimal writing time. You only mark candidates with exactly two positions per box, which is far fewer marks than full notation.
• Clean grid. With fewer marks, the grid stays readable and patterns stay visually clear.

Limitations

Snyder notation does not capture all candidate information. Numbers with three or more positions in a box are not marked. This means some techniques — particularly those that depend on knowing all candidates in a cell — may not be directly visible. For hard puzzles, you may need to supplement Snyder notation with additional marks or switch to full notation for specific areas.

Full Notation: Complete Candidate Tracking

Full notation means writing every possible candidate in every empty cell. This gives you complete information but produces a dense grid.

How to Do Full Notation

For each empty cell, check its row, column, and box. Any number from 1 to 9 that does not already appear in any of these three groups is a candidate for this cell. Write all candidates as small numbers in the cell.

Most solvers arrange candidates in a consistent 3x3 grid within the cell:

  1 2 3
  4 5 6
  7 8 9

Each candidate occupies a fixed position. When you eliminate 5, you erase the center spot. When you eliminate 7, you erase the bottom-left. This positional consistency lets you scan candidates quickly without reading individual numbers.

When to Use Full Notation

Full notation is most valuable for:

• Hard and expert puzzles where advanced techniques require complete candidate lists.
• X-Wing and Swordfish detection, which requires knowing exactly where each candidate appears across the grid.
• Advanced strategies like XY-Wing, simple coloring, and unique rectangles.
• Situations where Snyder notation is insufficient and you need more information.

Keeping Full Notation Manageable

The biggest challenge with full notation is visual clutter. Here are strategies for keeping it manageable:

• Apply singles first. Before adding full notation, solve every naked and hidden single you can find. Fewer empty cells means fewer marks to write.
• Update immediately. Every time you place a number, immediately erase it as a candidate from all cells in the same row, column, and box. Stale pencil marks lead to errors.
• Use consistent positioning. Always put each number in the same position within the cell. This lets you scan for a specific candidate across multiple cells without reading each one individually.
• Focus on regions of interest. You do not always need full notation across the entire grid. If you are looking for an X-Wing on candidate 7, you only need to know where 7 appears — not the full candidate lists for every cell.

Hybrid Approach: Best of Both

Many experienced solvers use a hybrid approach:

1. Start with no pencil marks and solve using scanning.
2. When scanning stalls, add Snyder notation.
3. If Snyder is insufficient, upgrade to full notation in specific areas where you need more information.

This layered approach minimizes writing time while providing the information you need when you need it. It is particularly effective for speed solving, where writing time directly impacts your total solve time.

Pencil Marks in Digital Sudoku

Digital sudoku apps handle pencil marks differently from paper, and these differences affect your strategy:

Advantages of Digital Pencil Marks

• Auto-elimination: Many apps automatically remove pencil marks when you place a number. This eliminates the most error-prone aspect of manual notation — forgetting to update marks.
• Speed of entry: Tapping to add or remove a candidate is faster than writing with a pencil. This makes full notation more practical on digital than on paper.
• Visual clarity: Digital marks are uniform and legible. No squinting at smudged handwriting.
• Auto-fill options: Some apps can fill in all candidates automatically, giving you instant full notation. This is controversial among purists but undeniably practical.

Disadvantages of Digital Pencil Marks

• Smaller cells: Mobile screens make cells small, and fitting nine potential candidates in a cell can be visually cramped.
• Tap precision: Adding pencil marks on a small screen can be finicky. Good input design matters — Sudoku Royale's slide-to-select input helps minimize errant taps.
• Less spatial memory: On paper, the physical act of writing helps encode the information in memory. On digital, candidates can feel more abstract.

Paper Pencil Marks: Tips for Clean Notation

If you prefer paper sudoku, these tips keep your pencil marks clean and useful:

• Use a mechanical pencil with fine lead (0.5mm). Thick lead makes marks too large and messy. Fine lead keeps candidates small and legible.
• Write lightly. You will be erasing frequently. Light marks erase cleanly; heavy marks leave ghosts that clutter the grid.
• Use the positional system. Place each number in its fixed position within the cell (1 top-left, 5 center, 9 bottom-right). This lets you check for a specific candidate without reading every mark.
• Erase with a quality eraser. A clean erase is essential. Smudged marks are worse than no marks at all because they create ambiguity about which candidates remain.

Common Pencil Mark Mistakes

Over-Marking

Writing candidates in every cell from the start buries you in information. On an easy puzzle, this wastes time. On any puzzle, the initial time spent writing marks delays actual solving. Use the layered approach: start without marks, add Snyder when needed, upgrade to full notation only where necessary.

Under-Updating

The most dangerous mistake is failing to update pencil marks after placing a number. If you place a 5 in a cell but forget to remove 5 from pencil marks in the same row, column, and box, your subsequent deductions will be based on incorrect information. This is the number one cause of errors in intermediate solving.

Trusting Marks Over Logic

Pencil marks are a record of your analysis, and that analysis can be wrong. If you missed a constraint when adding marks, a candidate might be present that should not be. Always verify important deductions — especially when a technique like naked pairs leads to a placement. Double-check the elimination against the actual grid, not just the pencil marks.

Pencil Marks and Puzzle Difficulty

Different difficulty levels have different notation needs:

• Easy: No pencil marks needed. Solve with scanning.
• Medium: Snyder notation after initial scanning stalls. Enough for pairs and pointing pairs.
• Hard: Full notation in key areas. Needed for X-Wing and other line-based techniques.
• Expert/Evil: Full notation across the grid. Required for advanced strategies like XY-Wing and coloring.

Understanding this progression helps you calibrate your notation effort. Using full notation on an easy puzzle wastes time; using no notation on a hard puzzle wastes even more time in failed deductions and missed patterns.

For practical advice on applying pencil marks in a speed-solving context, see our speed solving guide. For a complete beginner introduction to sudoku, check out how to play sudoku.