Hidden single

What is a hidden single?

A hidden single is a digit that has only one possible place in a row, column or box. The cell itself may still have several candidates — that is why it is called hidden — but if every other cell in the unit is ruled out for that digit, the digit must go in the one cell that is left.

How to spot a hidden single

Work digit by digit instead of cell by cell. Choose a digit and a box that does not contain it yet. Every copy of that digit already on the board rules it out along its own row and column, so any cell in the box crossed by one of those lines cannot take it. Filled cells cannot either. If exactly one empty cell survives, the digit goes there.

This is usually called cross-hatching, and the same scan works on a row or a column. Boxes are the quickest place to start, because the lines from two or three nearby copies of a digit often cover a whole box at once. It is the main tool for Medium puzzles.

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.

This is a Medium puzzle from our game at the moment its solver runs out of naked singles: no empty cell on the board is down to one candidate. Look at box 4, the middle-left box, and ask where its 1 can go. It has five empty cells.

The 1 in r4c5 sits on row 4, which rules out r4c2 and r4c3. The 1 in r9c2 sits on column 2, which rules out r5c2 and r6c2. That leaves r6c3 as the only cell in box 4 that can hold a 1, so r6c3 is 1.

On its own, r6c3 still has three candidates — 1, 4 and 9 — so staring at the cell gets you nowhere. Looking at the box from the digit’s point of view solves it at once.

Why it works

Box 4 must contain a 1 somewhere, because every box holds each digit once. Each cell you crossed out would put a second 1 into a row or column that already has one. When only one cell is left that breaks no rule, the 1 has to go there, whatever else that cell could otherwise have held.

Which difficulty levels need it

Medium and above. Our Medium puzzles are graded so that naked singles alone are not enough: at some point no cell has a single candidate left, and a hidden single is the next step. The solver used hidden singles in 80 of the 80 Medium puzzles we sampled, and in 199 of 199 Hard ones.

Easy puzzles never need it, though it is often the quicker way to find a digit there too.