Three strategies to look for Naked Singles

What is Naked Single

Let's begin with the idea of Naked Single technique. According to Sudoku rules, digits in rows, columns, and 3x3 blocks cannot repeat. This means a cell cannot contain any digit that is already present in the same row, column, or block. If an empty cell can contain only one possible digit, you can fill that cell with this digit.

For more details and examples, refer to Naked Single technique.

How to find it

If you use pencil notes when solving Sudoku, finding Naked Singles is easy. Just look for cells with only one pencil note left.

However, if you prefer to solve Sudoku without pencil notes, finding a Naked Single might take quite a lot of time. Let's explore different strategies to do that and hopefully find a way to speed up the search.

Strategy #1: scan all cells one by one

This is the most straightforward approach: choose any simple order to scan empty cells and check how many digits are allowed in each cell. For instance, you can check all cells in row 1, then in row 2, and so on, until you have checked all cells in all rows. Alternatively, check cells column by column or block by block.

If your Sudoku puzzle doesn't have any Naked Singles, you'll end up checking all empty cells. However, if there is a Naked Single somewhere on the grid, you'll have to check only some of the cells. On average, you can expect to check about half of all empty cells before finding a Naked Single.

In the first example below, scanning the grid row by row works well. In the second example, it does not - almost all empty cells have to be checked before finding a Naked Single.

Example 1.

Example 2.

Strategy #2: scan all cells in smarter order

For brevity, let's agree to say that two cells see each other if they share the same row, column, or block. Now, imagine that there is a cell on Sudoku grid that sees ten cells already filled with digits, and another cell sees fifteen filled cells. Which of them is more likely to contain a Naked Single? Statistically, it's the second cell, because it sees more filled cells.

This leads us to an idea. Instead of scanning rows from first to last as in strategy #1, let's first scan rows with only two empty cells, then rows with three empty cells, and so on. This simple trick usually works quite well and allows us to find Naked Singles much faster compared to strategy #1. Of course, you can do the same with columns or blocks instead of rows.

Examples 3 and 4 show how many cells need to be checked to find a Naked Single using strategy #2. The puzzles are the same as in examples 1 and 2.

Example 3.

Example 4.

Strategy #3: scan only some cells

Let's focus on two digits, say, 1 and 2. If there is a Naked Single in some cell, then we have three possibilities:

• The Naked Single is digit 1. This means the cell cannot contain digit 2 and, thus must see at least one cell filled with 2.
• Conversely, the Naked Single is digit 2. This means the cell cannot contain digit 1 and, thus must see at least one cell filled with 1.
• It's neither digit 1 nor digit 2. This means the cell cannot contain either digit 1 or 2, so it must see at least one cell filled with 1 and at least one cell filled with 2.

What does all this mean? If there is a cell with a Naked Single somewhere on Sudoku grid, this cell sees at least one digit 1 or digit 2. This leads us to the conclusion that we don't need to check all empty cells. It's enough to check only those cells that see either digit 1 or digit 2.

We can choose any other pair of digits instead of 1 and 2. The best approach is to take the two digits that occur the least number of times on the grid. This way, we'll significantly reduce the number of cells that need to be checked.

Examples 5 and 6 show how many cells need to be checked to find a Naked Single using strategy #3. Again, the puzzles are the same as in examples 1 and 2. In example 5, we only check cells that see digits 5 and 9. In example 6, we check cells that see digits 2 and 8. Please note, that in examples 1-4, we show all cells that will be checked before you find a Naked Single. Whereas in examples 5 and 6, we show all cells that might be checked. Depending on the order you choose to check these cells in, you'll end up checking only part of them. On average, it's going to be half of them.

Example 5.

Example 6.

Our recommendation

Strategies #2 and #3 certainly work better than strategy #1, so our recommendation is to use one of these strategies. After you gain some experience with solving Sudoku and finding Naked Singles, you'll be able to look at the frid and assess what strategy is going to work best. So, keep playing Sudoku and improve your skills.