Techniques · 2 of 10

The hidden single

The same situation viewed from the digit's side of the desk — and the technique that finishes most easy and medium puzzles.

A hidden single is a digit which, within some row, column, or box, has only one legal cell to live in. The cell itself may have many other candidates — that is what makes the single hidden — but every other empty cell in the unit is excluded for that particular digit. The technique flips the viewpoint of the naked single: instead of asking, what can go in this cell?, you ask, where can this digit go in this row?

Worked example

Below is a Sudoku grid with the middle box highlighted in our mind, and four scattered 5s placed elsewhere. The middle box has nine empty cells. The question: in how many of them can the digit 5 legally appear?

Four 5s already placed. In the middle box, only the highlighted cell can hold a fifth.

Trace it. The 5 in the top band lives in column 4, so no cell of the middle box that shares column 4 can be a 5 — but the top three cells of the middle box span columns 4, 5, 6, and the column-4 cell is already ruled out. Look more carefully: the 5 at the top of the grid sits in column 4, which strikes through the upper-middle, middle-middle, and lower-middle boxes. The 5 in the left column sits in row 4, eliminating the entire middle row of the middle box from holding another 5. The 5 in the right column sits in row 6, eliminating the bottom row. The 5 in the bottom row sits in column 6, eliminating column 6 of the middle box.

Walk the nine cells of the middle box and you will find that eight of them lie in a row or column already containing a 5. Only one — the highlighted cell at the centre — survives. The 5 must go there. The digit is forced even though the cell itself has many other candidates.

Why this is called hidden

If you looked only at the highlighted cell, you would not see a single. The cell is empty, and most of its row and column are empty too; for all the cell knows, it could hold a 1, 2, 3, 5, 6, 7, or 9. The single is hidden inside that list. It only becomes visible when you change your unit of analysis from the cell to the digit — asking where 5 can go, rather than what can go in this cell.

That viewpoint shift is the heart of the technique. Beginners scan cells; intermediate solvers scan digits. After the easy naked singles have all been placed, most of the next wave of placements are hidden singles.

How to find them

1. Pick a digit and a unit. "Where can 5 go in this box?" Walk the nine cells of the box and cross out any whose row or column already contains a 5.
2. One survivor wins. If exactly one cell remains, place the digit there. If two or more remain, move on; the digit isn't a hidden single in this unit yet.
3. Repeat for every digit in every unit. This sounds exhausting but in practice it is fast. You skip the easy ones (digit already placed) and the impossible ones (no candidates in the unit), and focus on the borderline cases.

Most experienced solvers automate this with their eyes. They notice that a 5 has been placed in two of the three columns crossing a band, and immediately look at the third column inside the band to see whether one box is forced.

Hidden singles versus naked singles

The two techniques are reflections of one another:

• A naked single is a cell whose set of candidates has collapsed to one digit.
• A hidden single is a unit (row, column, or box) where one digit's set of legal cells has collapsed to one cell.

Every hidden single, once placed, removes candidates from peers, frequently triggering naked singles elsewhere. The two patterns play off each other — a cascade of hidden singles often resolves a quiet middle stretch of a puzzle.

A note on box-hidden singles

Hidden singles in boxes are by far the most common and the easiest to spot, because the nine cells of a box are spatially close. The Sudoku Explainer difficulty rater scores them at 1.2 — the lowest difficulty rating in its catalogue. Hidden singles in rows and columns score slightly higher (1.5) because the nine cells of a row span the full width of the board and require a longer scan. If you are training your eye, start with boxes: pick one, pick a digit, find a hidden single. Then graduate to rows and columns.