Sudoku solving techniques: from naked singles to hidden pairs

Every Sudoku puzzle with a unique solution can be solved through logic alone, no guessing required. The trick is knowing what to look for. These three techniques cover most of what you’ll need for easy and medium puzzles, and they build on each other: each one looks at the candidate digits, the numbers that could legally go in a cell, from a slightly different angle.

Candidates: the starting point for everything

Before any of these techniques apply, it helps to think in candidates rather than final answers. For any empty cell, its candidates are the digits 1–9 that aren’t already used by that cell’s row, column, or 3×3 box (its three “peer groups”). A cell in a row that already contains 1, 4, and 7, a column that already contains 2 and 8, and a box that already contains 5, has the candidates {3, 6, 9}, the digits left over once everything already placed is ruled out.

Naked singles: only one candidate left

A naked single is a cell whose row, column, and box have, between them, already ruled out eight of the nine digits, leaving exactly one candidate. If a cell’s row has 1, 2, 3, 4, its column has 5, 6, and its box has 7, 8, then 9 is the only digit not eliminated anywhere, so that cell must be 9. This is the easiest technique to spot: work through a cell’s candidates and if only one survives, it’s the answer.

Hidden singles: only one cell left

A hidden single looks at the same information from the opposite direction. Instead of asking “which digits can go in this cell,” it asks “which cells can this digit go in, within one row, column, or box.” A cell can have several candidates and still be a hidden single, if one of those candidate digits doesn’t fit anywhere else in that unit. For example, suppose a box has three empty cells with candidates {2, 4, 6}, {2, 4}, and {4, 6}. Digit 2 appears as a candidate in two of those cells, and digit 6 also appears in two, but digit 4 appears in all three, so checking 4 alone doesn’t narrow anything down. Now suppose instead the three cells have candidates {2, 4, 6}, {4, 6}, and {4, 6}: digit 2 now appears in only one of the three cells, the first one, even though that cell also allows 4 and 6. Since 2 has to go somewhere in the box and this is its only legal home, that cell must be 2, ignoring its other candidates entirely.

Naked pairs: two cells, two digits, nowhere else to go

A naked pair is two cells in the same row, column, or box whose candidate lists are identical and contain exactly two digits. If cell A and cell B, both in the same row, each have candidates {2, 7} and nothing else, then between the two of them, one must be 2 and the other must be 7. It doesn’t matter which is which yet, but it does mean no other cell in that row can be 2 or 7, since both digits are already spoken for by A and B. You can safely erase 2 and 7 from every other cell’s candidates in that row.

This often cascades into new naked or hidden singles. Take a row with five empty cells and these candidates:

cell 1: {2, 7}
cell 2: {2, 7}
cell 3: {1, 2, 7, 9}
cell 4: {2, 5, 7}
cell 5: {3, 8}

Cells 1 and 2 are a naked pair on {2, 7}. Removing 2 and 7 as candidates from the rest of the row leaves cell 3 with {1, 9} and cell 4 with just {5}, a naked single. Cell 4 can be filled in immediately, and cell 3 is narrowed down to a two-way choice it wouldn’t have had otherwise. Cell 5 is untouched, since it never had 2 or 7 as a candidate to begin with.

The same logic extends to naked triples (three cells sharing exactly three candidates between them) and to columns and boxes, not just rows. Once naked singles and hidden singles stop finding anything new, scanning for naked pairs is usually the next place to look before reaching for a harder technique.