How to Solve Sudoku: From Scanning to Pencil Marks

Updated June 2026

Almost everyone can fill in a few Sudoku cells; the gap between that and finishing a hard board is just a handful of named techniques applied in the right order. This guide builds them up from the fastest, no-writing scan to the pencil-mark patterns that crack the tough grids — all of it pure logic, no guessing required.

The one rule, three ways

A standard Sudoku is a nine-by-nine grid split into nine three-by-three boxes. Every row, every column, and every box must contain the digits 1 to 9 exactly once. That single rule is really three overlapping constraints on each cell, and every technique below is just a smarter way of asking "which digits are still allowed here?"

Start by scanning — no writing needed

Before you pencil in anything, scan. Pick a digit that already appears several times, say 5, and look at each box that is still missing it. A 5 elsewhere in the same row or column "blocks" entire lines from holding the 5 in that box. Often two blocked lines plus an already-filled cell leave only one empty square in the box where the 5 can go. This is called cross-hatching, and on easy and medium boards it places a remarkable number of digits with zero notation. Work through 1 to 9 this way first; it is by far the highest-value habit.

Naked singles and hidden singles

When scanning dries up, switch to the two "singles", which are the backbone of solving.

Naked single

A naked single is a cell that has only one candidate left — its row, column, and box between them have already used eight of the nine digits, so the ninth is forced. These are easy to miss late in a solve when a cell quietly runs out of options; whenever you place a digit, glance at its neighbours for a freshly created naked single.

Hidden single

A hidden single is sneakier and more powerful. Look at a single unit — one row, column, or box — and a single digit. If there is exactly one cell in that unit where the digit can legally go, it goes there, even if that cell still has other candidates. The digit is "hidden" among other possibilities, but the unit needs it somewhere and there is only one home. Hunting hidden singles unit by unit is what keeps a medium board moving.

When to start pencil marks

On harder boards you eventually need to write candidates — the small digits penciled into each empty cell listing what is still possible. Don't do this too early; fill every single you can find first, because each placement erases candidates and saves you writing. Once you are genuinely stuck, pencil in the full candidate list for the remaining cells and the next techniques become visible.

Naked pairs and triples

If two cells in the same unit have exactly the same two candidates — say both can only be 3 or 7 — then those two digits are locked into those two cells, in some order. That means 3 and 7 can be erased from every other cell in that unit. You have not placed a digit, but you have removed candidates, which frequently exposes a hidden single nearby. The same idea extends to three cells sharing three candidates (a naked triple).

Pointing pairs (box–line reduction)

Here is the technique that unlocks most "hard" boards. Look at a single box and a single digit. If every cell where that digit could go inside the box lies on the same row (or the same column), then the digit must end up on that line within this box — which means it cannot appear anywhere else along that line in the neighbouring boxes. Erase it there. This interaction between a box and a line removes candidates that no single-unit technique can see, and it usually cascades into a fresh run of singles.

A solving order that works every time

1. Cross-hatch scan all nine digits and place every obvious one with no writing.
2. Sweep for hidden singles unit by unit, then naked singles cell by cell.
3. Only when stuck, pencil in full candidates for the rest.
4. Apply naked pairs/triples to thin the candidates.
5. Apply pointing pairs to remove candidates across box–line boundaries.
6. Loop back to singles — each elimination tends to reveal one.

A well-formed Sudoku has exactly one solution reachable by logic alone, so if you ever feel the urge to guess, it means there is a technique on the board you have not spotted yet. Slow down, recheck your candidates, and the forced move is there.