Candidates / Pencil marks

What are candidates?

Candidates are, generally speaking, possible numbers for a specific cell or region within the Sudoku grid.

They are those little hints that you write onto the Sudoku grid to be able to solve it quicker or better (or at all).

Since they are oftentimes done with pencil directly in the Sudoku cells, most people and several Sudoku apps just generalize them as "pencil marks", although "pencil marks" can be used for much more than just candidates.

Why Candidates?

With many Sudoku positions, you can directly deduce a cell's solution.

Take this position for example.

Here, it's easy to see, that the solution to the orange cell is 9, because 1-8 are already used in that row. Though this is an easy example, most parts of most Sudokus can be solved like this. But as soon as the Sudoku gets harder, you will need to deal with candidates (be it in your head, on paper, or within the app.)

By the way, most Sudoku techniques don't directly give you fixed solutions. Most just let you eliminate candidates. The two easiest techniques "Hidden Single" and "Naked Single" are among the few that allow us to write a big number into a cell.

For all other techniques, you will not get a fixed solution, but simply a hint.

They will only yield the info, what numbers cannot be in that cell, or region. So at that point you have started an elimination process.

From all possible cell candidates (1-9) you keep eliminating candidates until there is only one candidate left, which then is the cell's solution.

Types of candidates

The two main types of candidates are cell candidates and box candidates.

What candidate types you track mostly depends on how cluttered you want your grid to be. The more different candidate types you use, the more chaos you will have on your Sudoku grid, and some candidate types are easier to handle than others.

That's why most people just stick to one or two.

Be aware that with the webapp, you can notate both kinds of candidates. However, the solver and the hints system only takes cell candidates into consideration. It completely ignores box candidates that you have notated.

Cell candidates

Cell candidates are the remaining possible solutions for a specific cell.

Each cell can be one of the numbers 1-9. So we start with candidates 1-9 for each cell. (This example only shows it for a single cell.)

When we can logically exclude a certain number, we "eliminate that candidate".

  • The fixed 1 sees the center cell via column, so we know the center cell is not a 1.
  • The fixed 2 sees the center cell via box, so we know the center cell is not a 2.
  • The fixed 3 sees the center cell via row, so we know the center cell is not a 3.

1, 2, 3 are eliminated from the center cell.

The candidates 4, 5, 6, 7, 8 and 9 remain.

Box candidates

The second most often used type of candidate is the box candidate.

Box candidates are somehow the opposite of cell candidates:

  • With cell candidates you ask "For this cell, what numbers are possible".
  • With box candidates you ask "For this number, what cells are possible (within the 3x3 box)".

You can see how the 1 can only be in one of two cells within that box.

Other candidates types

The problem with candidate handling is that you somehow need to notate candidates, i.e. write them into the grid.

A Sudoku grid has only so much space for pencil marks, and the more you fill it, the less clear it becomes.

For that reason other candidate types are not used very often.

Line candidates (i.e. column candidates or row candidates) are possible for example.

Line candidates are basically like box candidates, in that you ask "For this number, what cells are possible (within the column or row)".

You can easily see that box, column and row candidates don't go well hand in hand.

Take this position for example. Imagine you figured out these candidates.

After a couple steps you come back to these candidates. How do you know to what region those candidates belong? Is the 1 on the left a box candidate and can therefore be entered directly, or is it a line candidate?

You can, of course, color code them, but bringing colored pencils to the table might be a little too much.

Why prefer box candidates over line candidates?

There are two reasons why box candidates are better.

The first one is the ease of perception. Our eyes and brains work with a relatively small visual focus area. The farther things are away from that area, the more blurry they become. A Sudoku 3x3 box is visually denser then a 1x9 line, so its information is easier to perceive.

What number is missing in the top left box? What number is missing in the line? With the line, your eyes need to move a lot more.

The second reason is how the regions (box, columns, rows) actually interact with each other. While the intersection of a row and a column is a single cell, the intersection of any line with a box is three cells, which can lead to faster candidate elimination. E.g. the technique "Locked Candidate" makes use of lines and boxes intersecting.

Candidate Notation

There are several notations for Sudoku solving. We will discuss two of them on the next pages: