How to Solve Sudoku with Pencil Marks

Using pencil marks to solve sudoku puzzles is an extremely common solving technique. 

Especially as you advance to more difficult puzzles where using pencil marks not only helps but is pretty much necessary. 

So, whether you’re new to using pencil marking or are looking for a refresher on the best practices, let’s look at what pencil marking is and how to pencil mark sudoku puzzles. 

What is pencil marking?

In sudoku, pencil marking involves writing small numbers in each single cell of every possible number that cell could contain. Such as in the example below. 

As you can see above, the given digits are larger and placed in the center of the cells. Whereas all of the potential digits for the remainder of the cells (the empty cells) are ‘pencil marked’ in the top left corner and are smaller. 

Any digit that cannot appear in an empty cell (because that digit already appears in the same row, column or block) is not added as a pencil mark. 

How this helps with solving sudoku puzzles 

From looking at the pencil marked grid above, you’ll notice that some cells have more possible candidates than others.

Same cells have as many as six possible candidates whereas many only have two or three and there’s a single cell that only has one possibility. 

This is known as a naked single (as it’s all by itself) and shows just how incredibly helpful pencil marking can be.

By looking at this cell with the naked single (column 5, row 7), you can immediately tell that this cell contains the digit one.

After placing this 1 in the grid, you can then eliminate the 1s from the pencil marks in the cells sharing the same row, column or block. 

This is particularly helpful when you look at the two remaining empty cells in column 5.  

As the cell in column 5, row 3 originally had 1 and 7 and its only two possibilities, and the 1 has now been erased, we know that the only digit this cell can contain is the 7. 

Repeating the process and removing any 7 pencil marks in cells that share a row, column or block with this cell reduces the possibilities for the very center cell of the grid (column 5, row 5) from 1, 4 and 7 to just 4.

Continuing this process of identifying naked singles will go a long way to solving your sudoku puzzles. 

Also look for hidden singles

So far we’ve covered looking for naked singles (cells with only one possible candidate). 

In addition to naked singles, there is another type of ‘single’ you’ll want to look out for: hidden singles. 

To demonstrate what a hidden single is, let’s go back to the very center of the grid again (column 5, row 5). 

We originally determined that this cell must be a 4 through a process of looking at naked singles, eliminating pencil marks and then looking for new naked singles. 

And while this got us to the solution of the center cell containing a four, there’s another, potentially faster, way of getting there with the help of hidden singles. 

A hidden single is where a particular digit is only a possibility within its row, column or block. 

In this case, the digit 4 was only a possibility in column 5 in this one cell. Making the 4 a hidden single as it could not have been placed anywhere else in the column and we know that the column must contain a 4. 

It was also the only cell a 4 could have been placed in the center 3×3 block. 

By identifying these hidden singles, you can place numbers in the puzzle much faster than only looking for naked singles. Especially when you run out of naked singles to fill in. 

For example, in cell row 8, column 2, there’s another hidden single – the 8. This is because 8 cannot appear anywhere else in the 3×3 block or the row. 

Noticing this hidden single means you can place a digit in this cell much faster than you could have by trying to reduce the five pencil marked candidates down to one possibility. 

How to pencil mark a sudoku puzzle efficiency 

One important thing to note in the example above is that every possible candidate for every empty cell has been penciled into the grid. 

Doing this can be a lengthy and time consuming process. Especially if your sudoku puzzle has few givens to start with. 

Because of this, more experienced sudoku solvers will only pencil mark cells that contain a maximum of two or three possibilities. 

They may also only choose to focus on rows, columns or blocks when pencil marking candidates in. 

This is where different players can have their own styles and preferences when it comes to pencil marking. 

To learn about these more advanced pencil marking techniques, check out Cracking the Cryptic’s video tutorial on YouTube here.