How to Write Sudoku Pencil Marks

The puzzle below is one I made. It has 27 clues, and I checked with a solver that it has exactly one solution. There are 54 empty cells. If you write every digit that could go in each of those 54 cells, you end up with 209 candidates.

・ ・ ・ | ・ 2 ・ | ・ ・ 1
9 5 ・ | 1 3 8 | ・ ・ ・
・ ・ 6 | ・ ・ 9 | ・ ・ 4

・ ・ ・ | ・ ・ ・ | 3 ・ ・
8 ・ ・ | 4 ・ 3 | ・ ・ ・
・ ・ 3 | 2 ・ ・ | ・ 8 5

1 ・ ・ | 8 ・ ・ | ・ 6 ・
6 ・ ・ | 3 9 ・ | 5 4 ・
・ ・ ・ | ・ ・ ・ | ・ 1 ・

Writing all 209 from the start is a lot of work on paper. On the other hand, this puzzle cannot be finished with the techniques that need no pencil marks. Using only those, you can fill 51 of the 81 cells, and then you stop. At that point the remaining 30 cells hold 82 candidates in total.

So when should you write pencil marks, where, and how? This post goes through it step by step. The words are the same as in my earlier posts: a row runs across, a column runs down, and a box is one of the 3 × 3 squares with a thick border.

What pencil marks are

Pencil marks are small digits written in an empty cell to show which digits could still go there. They are also called candidates or notes. They are not answers. They are a way of recording what is still undecided, as undecided.

A cell’s candidates are the digits from 1 to 9 that do not already appear in its row, its column or its box.

When to start

I think it is best to go as far as you can without pencil marks and start writing them where you get stuck. In the puzzle above, if you start once 51 cells are filled, you write 82 candidates instead of 209.

The techniques that need no pencil marks are 1 to 3 in How to Solve Sudoku: The First Techniques Beginners Should Learn. An easy puzzle can sometimes be finished without writing a single mark.

Every cell, or only where needed

There are two broad approaches.

Mark every cell. Write all the candidates in every empty cell. It takes time, but once it is done, everything about the grid is visible. For hard puzzles that need techniques based on how candidates relate to each other, I think this approach is the strongest. The risk is that one wrong mark breaks every conclusion that follows from it.

Mark only where needed. Write candidates only where something is close to being decided, such as boxes or rows with few empty cells. You write less, so you make fewer mistakes. An even lighter version is to write a digit only when it is limited to two cells in a box, and then only in those two cells.

My suggestion is to start, when you get stuck, by marking only where needed in the boxes with the fewest empty cells, and to widen it to every cell only if that stops working.

Trying it on one box

Take the top middle box of the puzzle above. Four cells are empty.

 567  |  2   | 4567
  1   |  3   |  8
  57  |  57  |  9

For row 1, column 4, remove the 1, 2, 3, 8 and 9 in the box and the 4 in the same column, and 5, 6 and 7 are left. Counting the same way, row 1, column 6 gets 4, 5, 6 and 7, and row 3, columns 4 and 5 both get 5 and 7.

Once written, two things are visible at a glance. First, 4 appears only in row 1, column 6 within this box, so 4 goes there. Second, the two cells in row 3 both hold only 5 and 7. Those two digits are used up by those two cells, so 5 and 7 drop out of row 1, column 4, and 6 is left.

You could settle this box without marks if you looked carefully. But with marks, what you need to look at is right there on the paper. In my view, the job of pencil marks is to reduce what you have to keep in your head.

A fixed position for each digit

On paper, it helps to give each digit its own spot inside the cell. Treat the cell as a small 3 × 3 grid and lay the digits out like the buttons on a phone: 1 at the top left, 2 at the top middle, 3 at the top right, down to 9 at the bottom right. Then, when you want to know where a digit’s marks are, you can find them across the whole grid more easily.

Write them in pencil, clearly smaller than your answers. If marks and answers are the same size, you lose track of which is which.

Erasing

Most pencil-mark mistakes happen when erasing.

When you place a digit, remove it from the marks in the same row, the same column and the same box. Checking those three directions in the same order every time cuts down on missed erasures.

Watch out for erasing too much as well. If you remove a digit that actually belongs in a cell, the right answer vanishes from the grid. Only erase when you can state the reason clearly. I wrote about finding mistakes in your marks in Stuck on a Sudoku? How to Find the Next Move.

Pencil marks in Sudoku365

Sudoku365 is a sudoku app I make for iPhone, iPad and Android. Its notes work like this.

Writing. Tap NOTES OFF and it changes to NOTES ON. While it is on, tapping a digit button writes that digit as a mark in the selected cell. Tapping the same digit again removes it. The marks sit in a 3 × 3 layout inside the cell, with 1 at the top left and 9 at the bottom right, the same as described above.

Automatic erasing. With Remove numbers from notes on in settings (the default), placing a correct digit automatically removes the same digit from the marks in its row, column and box. You cannot forget to erase, which makes it easier than paper. Separately from that setting, placing a digit always clears the marks in that cell.

No automatic marking. There is no feature that fills in every candidate for you. You write the marks yourself. I think deciding where to write them is itself the practice.

Undo. Writing a mark can be undone one step at a time with UNDO. Marks that were cleared automatically when you placed a digit come back with UNDO as well.

In Solve together, part of Play together, two people fill in one grid. Placed digits and pencil marks are both shared, and Remove numbers from notes is always on during a Solve together game.

The app has four difficulties, Easy, Medium, Hard and Expert, with 300 puzzles each, 1,200 in total. All 300 Easy puzzles can be finished with the techniques that need no pencil marks. If you want to practise pencil marks, Medium is a good place to start.

The app is free with ads, and a one-time Remove Ads purchase turns them off. For your first puzzle, see Getting Started with Sudoku365.

Pencil marks are not where you write answers. They are where you sort out what you do not know yet. Even if you write only a few, write them correctly and erase them correctly. Sometimes that alone is enough to get a stuck grid moving again.