Easy puzzles dissolve in your head. Medium puzzles ask you to keep track of a few candidates. Hard puzzles and up demand pencil marks. Once you start using them properly, the techniques above singles finally become visible.

How it works

A pencil mark is a small candidate digit written inside an empty cell. Each cell carries the list of digits that could legally go there given the current state of its row, column and 3x3 box. As you place digits, those candidates get crossed off the peers that share a unit with the new placement.

Pencil marks turn sudoku from a single-decision puzzle into a constraint-elimination puzzle. Instead of asking "what fits here?" you ask "which candidate can I remove based on the latest move?" Most intermediate and advanced techniques operate purely on candidate sets, not on the placed digits.

Two notation styles for the same candidate set {1, 4, 8}. Positional reads faster once you are used to it.

When to start using them

Not too early. Easy sudoku is full of naked singles and hidden singles; marking candidates there is busywork. The right time to switch is the moment a full scan of every digit and every cell produces no new placement.

On medium puzzles that moment usually arrives a third of the way through. On hard and above, you may need pencil marks from the start. Two rules: either mark every empty cell, or mark none. Half-marked grids trick you into trusting bad information.

Step-by-step example

Look at an empty cell. List the digits 1 to 9, then strike out anything already present in its row, column or box. What remains is its candidate set.
Write those candidates small inside the cell. Pick a notation style (positional or list) and stick to it.
Repeat for every empty cell. The first pass is slow. Budget two or three minutes for a fresh hard puzzle.
When you place a digit, immediately scan its row, column and box and erase that digit from every peer's candidate set.
After each sweep, look for cells that now hold one candidate. Those are naked singles, ready to place.

The two notation styles

	Positional	List
Layout	Each digit in its own 1-9 slot inside the cell	Candidates written in a row
Reading speed	Fast once memorised	Slower
Writing speed	Slower (precise placement)	Faster
Best for	Paper or digital pad	Scratch paper, quick solves

Pick one and stay with it. Switching mid-puzzle creates errors.

Tips for spotting it

Sweep peers immediately after every placement. Stale marks are the biggest cause of getting stuck.
Use a digital pad if your eraser is wearing through the paper.
Pause once the grid is fully marked and look for naked pairs before touching anything else. Pairs are everywhere right after the first marking pass.
If a cell drops to one candidate, place the digit and erase it from peers before scanning anywhere else.

Common mistakes

Pencil-marking too early. Easy and most medium puzzles do not need them.
Leaving stale candidates after placements. The mark you trust may already be eliminated.
Mixing positional and list styles in the same solve. Pick one.
Marking only some cells. Half-marked grids hide real candidate sets and produce wrong eliminations.

Practise it

Switch to medium sudoku and mark the entire grid before you place anything else. The discipline of full marks is what separates faster solvers from slower ones. Once the habit is in your hands, every technique above singles becomes reachable.

Pencil Marks