Naked single

What is a naked single?

A naked single is a cell that has only one candidate left. Every digit from 1 to 9 except one already appears somewhere in the cell’s row, column or box, so the missing digit is the only thing that can go there.

It is the most basic deduction in Sudoku and the one every other technique eventually feeds. The advanced patterns rarely place a digit themselves; they remove candidates until a naked single appears.

How to spot a naked single

Pick an empty cell in a crowded part of the grid — the more digits around it, the fewer possibilities remain. Read the digits in its row, then its column, then its box, ticking each one off as you go. If eight different digits turn up, the ninth is your answer.

Checking every cell like that is slow, so look for the shortcuts: a row or column with only one or two gaps, or a cell where a nearly full row crosses a nearly full box. If you keep pencil marks, a naked single shows itself as a cell with one mark left in it.

A worked example

Cells are named by row and column: r4c1 is row 4, column 1. Rows count from the top, columns from the left, and boxes are numbered 1 to 9 reading like a book, so box 1 is top-left and box 9 is bottom-right.

The diagram is a Medium puzzle exactly as our game dealt it, before a single digit has been added. It has 47 empty cells, and only one of them is a naked single. Look at r1c2, the second cell of the top row. Row 1 already holds 1, 2, 3, 4, 5 and 8. Column 2 adds a 9; its other digits repeat ones the row already had. Box 1, the top-left box, adds a 6. That makes eight different digits, and the only one missing is 7, so r1c2 is 7.

No single unit gives the answer on its own: the row rules out six digits, and the column and the box add one each. That is what makes this a naked single rather than a hidden one. You are asking what this cell can hold, not where a particular digit can go.

Why it works

Every row, column and box must contain each digit exactly once. If a digit already appears in any of the three units a cell belongs to, putting it in the cell would repeat it. When eight digits are ruled out that way, the cell still needs a digit, and the ninth is the only one that breaks no rule.

Which difficulty levels need it

Every level. Our Easy puzzles are generated so that naked singles alone solve them from start to finish — the game’s own solver checks this before an Easy grid is dealt, and in 40 Easy puzzles we sampled it never needed anything else.

In Medium, Hard and Expert you will still use it constantly. Each eliminating technique in this series ends the same way, by leaving some cell with one candidate, and that is a naked single.