Sudoku pencil marks: when and how to use them

Pencil marks are small notes that show which digits a cell could still hold. This guide explains when to start using them and how they lead you to the next answer.

What pencil marks are

A pencil mark is a small digit written in an empty cell. It does not mean “this is the answer”. It means “this digit is still possible here”. The possible digits for a cell are called its candidates.

Marks let you see what your memory cannot hold. An easy puzzle can be solved by looking at the grid. A hard one cannot, because the next step depends on comparing the candidates of several cells at once.

When to start using them

Start with no marks at all. Scan the rows, columns and boxes and place every digit you can find directly. On an easy puzzle this may be enough to finish.

Switch to pencil marks when scanning stops working: you have looked at every digit from 1 to 9 and found nothing new to place. On medium puzzles that usually happens around the halfway point. On hard puzzles it happens early.

Marking too soon is a common mistake. A grid full of marks placed in the first minute hides the easy answers and goes out of date quickly.

How to add pencil marks

Work through one empty cell at a time:

1. Look along the cell’s row and note which digits are already used.
2. Do the same for its column.
3. Do the same for its 3×3 box.
4. Mark every digit from 1 to 9 that does not appear in any of the three.

Start with the row, column or box that has the fewest empty cells. Those cells have the fewest candidates, so they are the quickest to mark and the most likely to give an answer.

If a full grid of marks feels crowded, mark less. One popular shortcut is to mark a digit only when it has exactly two possible cells in a box. This keeps the grid clean and still shows the most useful information.

Four ways pencil marks find the next digit

One mark left in a cell. If a cell has a single candidate, that digit is the answer. This is called a naked single.

A digit marked in only one cell. Look along a row, a column or a box. If a digit appears as a mark in just one cell there, it must go in that cell, even when the cell has other marks too. This is called a hidden single.

Two cells with the same two marks. If two cells in the same row, column or box both hold only 4 and 7, those two digits belong to those two cells. Remove 4 and 7 from every other cell in that row, column or box. This is called a naked pair.

A digit lined up inside a box. If all the marks for a digit in one box sit in the same row, the digit must be in that row inside that box. Remove it from the rest of the row outside the box. The same works for columns. This is called a pointing pair.

Keep your marks up to date

Every time you place a digit, remove that digit from the marks in the same row, column and box. This step matters more than any technique. Out-of-date marks are the main cause of mistakes, because one wrong candidate can lead you to a wrong answer several moves later.

After removing marks, look again at the cells you changed. A cell that has just lost a candidate is the most likely place for the next single.

Common mistakes

• Marking the whole grid before placing the easy digits.
• Forgetting to remove marks after placing a digit.
• Treating a mark as an answer. A mark only says the digit is possible.
• Guessing between two marks. If you cannot prove which is right, look for another technique.

Try it in our Sudoku

Our online Sudoku has a pencil mode for exactly this. Turn it on to note candidates in a cell instead of placing an answer. Start on Medium, where pencil marks first become useful.