How to use pencil marks in sudoku
When to start marking, why partial marking beats full marking on paper, the maintenance rule that keeps notes trustworthy, and what changes when the app does the bookkeeping.
Pencil marks — candidates, notes, whatever your app calls them — are the small digits you write in an empty cell to record what could still go there. Every technique past the beginner level assumes you have them.
They also ruin more games than any other habit, and always for the same reason: a pencil mark that was true ten moves ago and is not true now is worse than no pencil mark at all. It is not just clutter, it is a false statement you will reason from.
Do not start too early
The instinct is to fill in every candidate in every empty cell before doing anything. On a fresh grid this is bad practice. Half those cells are about to be resolved by simple scanning, and every one you resolve invalidates candidates in up to twenty other cells that you then have to go and fix.
Work the cheap techniques first. Place every digit you can find by looking — naked singles, hidden singles, pointing and claiming, all of which can be done from the placed digits alone. Only when nothing more falls out should you start writing candidates, and by then the grid is smaller and the marks are more stable.
Mark by digit, not by cell
When you do start, most people go cell by cell: pick an empty cell, work out everything it could be, write those in, move on. That is thorough and exhausting, and it makes it easy to miss something.
Going digit by digit is faster and less error-prone. Take the digit 1. Sweep the whole grid and mark 1 in every cell where a 1 is still possible. Then do the same for 2. You are holding one rule in your head at a time instead of nine, and you go through the grid nine times cheaply rather than 50 times expensively.
A useful side effect: while sweeping a single digit you will spot hidden singles, pointing patterns and X-Wings almost for free, because those are all single-digit patterns and you are looking at exactly one digit.
Partial marking, when you are on paper
Full marking means every candidate in every empty cell. Partial marking means writing only what you actually need. On paper, partial marking is usually better, and the version most solvers converge on is: mark only the cells that have exactly two candidates.
Two-candidate cells are the raw material of nearly every mid-level technique. Naked pairs are made of them. XY-Wings are made of nothing else. A grid with the two-candidate cells marked and nothing else is readable at a glance and carries most of the information you will use.
The cost is that hidden pairs and hidden singles become invisible, since those depend on knowing where a digit can go across the whole unit. If a partially marked grid stalls, that is the moment to fill in properly — but for one unit at a time, not the whole board.
The one maintenance rule
When you place a digit in a cell, immediately erase that digit from the candidates of every cell sharing its row, its column or its box. That is twenty cells to check, and it must happen at the moment of placement, not later.
Every stale-mark disaster comes from skipping this once. The failure mode is nasty: nothing goes wrong immediately, you build ten correct-looking deductions on top of a mark that should not exist, and the contradiction surfaces far away from its cause with no way to find it. The usual outcome is starting the grid over.
If you are on paper and dislike the erasing, that is a strong argument for partial marking. Fewer marks means fewer things that can go stale.
What changes on a phone
The clerical half of this problem is exactly what software should absorb, and in Sudoku - Calm Number Puzzle it does. Notes are stored as a nine-bit mask per cell, so a cell can hold any combination of candidates with no bookkeeping. Auto-fill writes every legal candidate across the board in one action. And when you enter a digit, that digit is stripped automatically from the notes of every cell that shares its row, column or box — the maintenance rule, applied without you remembering it.
Undo restores the notes too, including the peer cleanups, so backing out of a move does not leave the candidate grid in a state that no longer matches the board.
What this does not do is think for you. Auto-filled candidates tell you what is legal; they do not tell you which pattern is sitting in front of you. The techniques still have to be run by hand — the app has just removed the part of paper sudoku that is pure clerical work, and the part where paper players actually lose games.
A workable routine
- Scan for placements with no marks at all, until nothing more falls out.
- Run pointing and claiming, which also need no marks.
- Now mark — digit by digit, either fully or restricted to two-candidate cells.
- On every placement, clear that digit from all twenty peers before doing anything else.
- When stuck, fill in candidates completely for the one unit you are working on, rather than the whole grid.
Frequently asked questions
Should I write every candidate in every cell?
On a phone, yes — auto-fill costs nothing and stays correct by itself. On paper, usually not. Full marking is a lot of writing and a lot of erasing, and most solvers do better marking only the cells with exactly two candidates, filling in a unit completely only when they get stuck.
When should I start pencilling in candidates?
After the placements you can find by scanning have run out. Marking a fresh grid means writing candidates for cells that are about to be resolved anyway, and then having to correct them.
Why do my pencil marks lead me to a contradiction?
Almost always a stale mark: a digit was placed and not cleared from a peer cell. Occasionally it is an invalid pattern applied earlier, such as treating a three-candidate cell as half of a naked pair. Both produce a contradiction far away from the cause.
Is using pencil marks cheating?
No. They are a record of deductions you already made, not information from outside the puzzle. Every technique above the beginner level was designed with them in mind.
Try it on a real puzzle
Sudoku - Calm Number Puzzle grades every puzzle by the hardest technique it actually needs, verifies a single solution before you ever see the grid, and its hints name the technique instead of just filling in a digit. Free, offline, no ads and no account.