Pencil Marks: How to Take Notes Without Drowning in Them

What a pencil mark is

A candidate is a digit that has not yet been ruled out for a square. A pencil mark is just a candidate written down. If a square's row, column and box already contain 1, 2, 4, 5, 6 and 9, its candidates are 3, 7 and 8, and those are what you pencil in.

Candidates are not guesses. Every one is a digit that might still go there, and the job of every technique beyond the basic singles is to cross candidates off until one is left.

Notes in our game

• Press Notes (or the N key) to switch between writing digits and writing notes. Then tap a digit to add it to the selected square's notes, and tap it again to remove it.
• Each digit has a fixed spot in the square: 1 top left, 5 in the middle, 9 bottom right. After a while you can read a square's notes by their pattern without reading the numbers.
• When you place a digit, the game clears that square's notes and removes that digit from the notes of every square in the same row, column and box. You never have to tidy up after a placement yourself.
• Undo brings back notes as well as digits.

What the game does not do is fill in notes for you. That is deliberate: writing candidates is where you notice things.

When to start writing notes

Not straight away. Early in a puzzle most squares have five or six candidates, and writing them all is slow and tells you little. Work the singles first, using the scans in our singles guide, until they dry up. By then the grid is fuller, squares have fewer candidates, and notes start paying for themselves.

On our easy and medium levels you may never need notes at all. On hard and expert you almost certainly will, because those puzzles need techniques that work on candidates rather than on placed digits.

Two ways to mark

Full marking

Write every candidate in every empty square. It takes a few minutes, but once done, every pattern in the grid is visible: pairs, locked candidates, X-wings and the rest. This is the reliable choice for hard puzzles, and the way the diagrams in our technique guides are drawn.

The cost is accuracy. One missing candidate can lead you to a wrong conclusion much later, so fill notes one unit at a time and double-check as you go. A good order: for each digit 1 to 9, go box by box and pencil it into every square where it is still possible. Working by digit rather than by square makes it much harder to skip one.

Partial marking

Many paper solvers only note a digit in a box when it has exactly two possible squares there. These "two-spot" marks are quick to make and are exactly what you need to spot hidden singles and locked candidates. You can do the same in our game by only entering those notes.

The drawback is that partial marks can't show you a square's full set of options, so they miss pairs and the wing patterns. A common habit is partial marks first, then a switch to full marks when the partial ones stop giving results.

What notes let you see

Here is a real position from a puzzle on its way to being solved: 5 digits have been added to the starting grid since the beginning, and the 52 remaining empty squares carry full notes. Every simpler step has already been used.

Highlighted: 7 the pattern · struck through: 7 removed

Look at row 9. Two squares, row 9, column 4 and row 9, column 5, have exactly the same two candidates: 4 and 9. Between them they must hold those two digits, one each, so no other square in row 9 can be 4 or 9. The two squares also share box 8, so the same goes there. That rules out 4 from row 9, column 8 (in row 9) and 4 from row 7, column 5 and row 8, column 4 (in box 8). This pattern is a naked pair, and without notes it is almost impossible to see. With notes, it jumps out.

In this position 7 of the 52 empty squares are down to two candidates. Squares like that are where most advanced patterns start, so they are worth a second look whenever you scan the grid.

Keeping notes trustworthy

• Update after every placement. Our game does this for you. On paper, cross the digit off everywhere in its row, column and box straight away, not later.
• Remove, don't add. Once your notes are complete, they should only ever shrink. If you find yourself adding a candidate back, something earlier went wrong. Find it before carrying on.
• Write down eliminations at once. When a technique tells you a digit can't go somewhere, remove the note immediately. Remembering it for later is how notes drift out of date.
• Check a suspicious square. If a square ever ends up with no candidates, or a unit has nowhere left for a digit, an earlier step was wrong. Undo back to where it last made sense.

Reading notes for the next move

With notes in place, scan for these, in this order:

1. A square with a single candidate: a naked single.
2. A digit with only one spot left in a row, column or box: a hidden single.
3. A digit whose spots in a box all sit in one row or column: locked candidates.
4. Two squares in a unit with the same two candidates: a naked pair.
5. One digit forming a rectangle across two rows and two columns: an X-wing.

Each of these has its own guide with a worked example. The techniques page has the short version of all of them.