A pencil mark is a small digit written inside an empty cell to record that the digit is still legally possible there. A cell with three pencil marks, say 2, 5 and 8, means those are the only digits left once its row, column and box are taken into account. Nothing else could go there. Once you have marks on a good stretch of the board, singles start turning up on their own, without you re-checking every row and column from scratch.
The habit is simple but the payoff compounds. Every time you place a digit, it should disappear from the pencil marks of every cell in that digit's row, column and box. Do that consistently and a cell that had two candidates yesterday can drop to one candidate today, purely because a neighbor got filled in. You do not need to re-scan the cell; the marks already tell you.
On this site, press the Notes button to switch into pencil-mark mode. Placing a digit automatically clears that digit from the notes of every peer in its row, column and box, so the bookkeeping described above happens for you.
Worked example
Watch four cells carry pencil marks, then see what happens to three of them the moment a neighboring cell gets filled in.
Writing marks that you can trust
A pencil mark is only useful if it is complete. When you first mark a cell, check all nine digits against its row, column and box, and write down every one that survives, not just the ones you happen to notice. A cell with a single mark you wrote in a hurry is not the same as a real naked single; if you skipped checking a digit, the mark is unreliable.
Most players mark an entire empty region at once rather than one cell at a time, since it is faster to sweep a row or box in one pass than to revisit it later.
Keeping marks updated as you place digits
The habit that makes pencil marks worth the effort is removing a digit from a peer's marks the instant you place it. Skip this step even once and a cell can look like it has two or three options when it really has one, which is how naked singles get missed entirely. On this site the Notes system does this automatically: placing a digit clears it from the marks of every cell sharing its row, column or box, so a cell that was down to two candidates can drop to one right in front of you.
Common mistakes
- Marking some cells and skipping others. Pencil marks only pay off once a decent stretch of the board has them. A few marked cells scattered around will not reveal much; mark a full row, column or box at a time.
- Writing incomplete lists. If you jot down "2, 5" for a cell but never checked whether 8 was also possible, your marks are lying to you. Always run through all nine digits, at least the first time you mark a cell.
- Forgetting to erase after a placement. A stray mark for a digit that is no longer possible can hide a real single sitting one candidate away. Clear a digit from every peer as soon as you place it, or let the Notes feature do it for you.
- Re-marking a cell instead of updating it. If a cell already has marks and you place a digit nearby, do not leave the old list untouched and hope you remember the change. Cross the digit out or delete it, so the marks always match what is actually still possible.
When pencil marks matter most
On an Easy board (40 or more givens) you can often solve without writing a single mark; scanning and obvious singles carry most of the puzzle. Once you reach Medium boards (33 to 39 givens) and beyond, cells start holding onto three or four live candidates for longer, and tracking that in your head becomes unreliable. That is where pencil marks earn their keep, and where techniques like naked pairs become possible in the first place, since they depend on comparing candidate lists between cells.
Paper solvers usually write marks in a tiny grid inside the cell, three digits per line. On screen it does not matter how the marks are laid out, only that they are complete and current. A cell that shows the wrong marks is worse than a cell with no marks at all, since it looks solved when it is not, or looks open when it has actually narrowed down to one digit.
Turn on Notes before you start and keep it on. Watch how many cells collapse to one candidate purely from placements you make elsewhere.
Frequently asked questions
Pencil marks are small candidate numbers written inside a cell, listing every digit that is still legally possible there once its row, column and box are taken into account.
Yes. Turn on the Notes button, and placing a digit clears that digit from the pencil marks of every cell in its row, column and box.
Mark a full row, column or box at once rather than a handful of scattered cells. Partial marking rarely reveals much, since singles usually depend on comparing several neighboring cells at the same time.
Either the marks were written before all nine digits were checked, or a digit was placed nearby and never cleared from that cell's notes. Both problems disappear if you mark a whole unit at once and keep the marks updated after every placement.
Easy puzzles can usually be solved without them. Once you move to Medium boards and beyond, where cells hold three or four candidates for longer, pencil marks become far more reliable than tracking candidates in your head.
Pencil marks are the record; a naked single is what you find once that record shows only one digit left for a cell. Keeping accurate marks is what makes naked singles easy to spot as the board fills in.