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June 3, 2026 · 10 min read · By

Sudoku Strategies, Explained with Diagrams

Most people learn Sudoku by scanning for obvious numbers, then hit a wall on harder boards and assume they need to guess. They don't. Every standard Sudoku is solvable by pure logic — you just need a bigger toolbox. This guide walks through the eight techniques that carry you from beginner to confident solver, each one explained with a worked diagram you can read at a glance.

The strategies build on each other. The first three place digits directly and need no notes; they are covered in full on this page. The next five work on pencil marks — the small candidate numbers you jot into empty cells — and clear those candidates away until placements appear. Those are covered in two companion guides, linked below. Learn them in order and each new technique will feel like a natural extension of the last.

If you want a board to practise on while you read, open Sudoku Zen in another tab. Its notes mode lets you pencil-mark candidates exactly as the diagrams below show. Brand new to the game? Start with how to solve Sudoku for beginners first, then come back here.

The eight techniques

How the techniques fit together

Beginner techniques — last remaining cell, hidden singles, and cross-hatching — are about placing digits. They finish easy boards on their own and get you most of the way through a medium one. None of them require notes; you do them by looking. All three are explained below.

Intermediate techniques are about eliminating candidates so that a placement becomes possible. Naked and hidden pairs, naked triples, pointing pairs, and box/line reduction never put a number on the board by themselves. Instead they shave possibilities off other cells until a hidden single or last remaining cell appears. That two-step rhythm — eliminate, then place — is the whole game at the intermediate level.

Once these are reflexive, the advanced patterns (X-Wing, Swordfish, XY-Wing) are just the same idea stretched across several rows and columns at once. For now, master the eight here and you'll solve the vast majority of "hard"-rated puzzles without ever guessing.

The last remaining cell (full house)

The last remaining cell — sometimes called a "full house" — is the simplest move in Sudoku. When a row, column, or 3×3 box already contains eight of its nine digits, the empty cell can only be the one that's missing. It's the first technique every solver learns, and it's worth doing deliberately rather than by accident.

Sudoku's one rule is that every row, every column, and every 3×3 box must contain the digits 1 through 9 exactly once. The last remaining cell is that rule at its most direct: count what's there, name what's missing, write it in.

Row with eight cells filled; the ninth must be 2534678912672195348198342567859761423426853791713924856961537284287419635345286179
Row 1 already holds eight digits. Only 2 is missing, so it goes in the empty cell — a last remaining cell (full house).

In the row above, eight cells are already filled. Reading along, the digits present are 5, 3, 4, 6, 7, 8, 9, and 1 — every digit except 2. Since the row must contain a 2 somewhere and there is only one empty cell left, the 2 goes there. No candidates, no notes, no doubt.

The same logic applies to columns and boxes. Whenever a unit is down to its last empty cell, you can fill it instantly. These moments cascade: filling one last cell often completes a neighbouring unit, which completes another, and a corner of the board falls in seconds.

Where to look for them

Last remaining cells appear most often near the end of a solve, but they also show up early in any unit that started with lots of clues. Scan the board for the rows, columns, and boxes that look most crowded — the ones with seven or eight digits already placed. Those are where a single missing number is easiest to spot.

A practical habit: every time you place a digit anywhere, glance at the three units that cell belongs to (its row, its column, its box). If placing your digit left any of them with exactly one empty cell, fill that cell immediately before moving on. This keeps the cascade going and stops you from re-scanning the same regions.

Last remaining cell vs. naked single

These two sound alike but come from opposite directions. A last remaining cell looks at a whole unit and asks "which digit is missing?" A naked single looks at one cell and asks "which digits are still allowed here?" — checking its row, column, and box together. Both leave exactly one answer, but they're found by scanning differently. The last remaining cell is usually the faster of the two to spot because you only have to read one line.

It's tempting to dismiss the last remaining cell as too obvious to name. But solvers who treat it as a deliberate step — actively hunting for almost-full units — finish easy and medium boards far faster than those who only notice these cells by luck. It costs nothing, and it's the foundation every harder technique is built on.

Hidden singles

A hidden single is a digit that can legally go in only one cell of a row, column, or box — even though that cell looks like it has several options. It's the most useful beginner technique after basic scanning, and it's the one that quietly solves most of a medium Sudoku. Learning to see hidden singles is the moment Sudoku starts to click.

The word "hidden" is the key. With a last remaining cell or a naked single, the cell itself obviously has one answer. A hidden single is disguised: the cell carries two or three candidates, so nothing about it stands out. The single only appears when you look at the whole unit and notice that one particular digit has nowhere else to go.

In this box, 5 fits only one cell534678912721953468198324546785976123426853791713924856961537284287419635345286179
Pencil marks for the top-right box. The digit 5 appears as a candidate in just one cell, even though that cell also allows 4.
5 placed53467891272195346819832546785976123426853791713924856961537284287419635345286179
Because 5 has nowhere else to go in the box, it is placed — a hidden single.

Look only at the top-right box. Three cells are empty, with candidates {4, 6}, {4, 5}, and {4, 6}. If you judged the middle cell on its own, you couldn't decide between 4 and 5. But scan the box for the digit 5 specifically: it appears in just one of the three cells. The box must contain a 5 somewhere, and there is only one cell that accepts it — so 5 is placed, even though that cell also listed a 4.

That's the whole technique. Instead of asking "what can go in this cell?", you ask "where in this unit can this digit go?" When the answer is "only here," you have a hidden single.

How to hunt for them

Work one digit at a time within one unit. Pick a box, then run through the digits it's still missing. For each missing digit, count how many of the box's empty cells could legally hold it — remembering that a digit is blocked if it already appears in that cell's row or column. If exactly one cell survives, place the digit.

Boxes are the easiest place to start because the 3×3 shape makes the elimination visual, but hidden singles live in rows and columns too. After you've swept the boxes, run the same check along any row or column that still has several gaps.

Why they're easy to miss

Beginners overlook hidden singles because they scan cell by cell, and a hidden single's cell never looks special — it has the same two or three candidates as its neighbours. The fix is to switch your unit of attention from the cell to the digit. Once you train yourself to ask "where can the 5 go in this box?" the hidden singles light up.

This is also why pencil marks help. With every cell's candidates written in, a hidden single is visible the moment you notice a digit that appears only once across a unit. On Sudoku Zen, turn on notes mode and the pattern becomes much easier to spot.

Cross-hatching (slicing and dicing)

Cross-hatching — also called slicing and dicing — is the scanning method that places a digit inside a 3×3 box by reading the rows and columns that pass through it. It's how experienced solvers fill in numbers quickly without writing a single pencil mark, and it's the most efficient way to find hidden singles in a box.

The idea is simple. A box needs each digit exactly once. If the rows and columns crossing that box already contain a particular digit, they block most of the box's cells. Often they block all but one — and that surviving cell is where the digit must go.

Cross-hatching places a 1 in the bottom-left box53467891267219534819834256785761423685379171924856961537284287419635345286179
Rows 4 and 5 already contain a 1 (amber lines), and so do columns 1 and 3. Inside the bottom-left box that rules out every cell except one — the 1 must sit there.

We want to place a 1 in the bottom-left box. Look at the rows passing through it: the top two rows of the box already contain a 1 elsewhere on the board (the amber lines), so a 1 can't go in either of those rows inside the box. That leaves only the bottom row. Now look at the columns: the left and right columns of the box already have a 1, leaving only the middle column. The one cell that sits on both the open row and the open column is forced — the 1 goes there.

"Slicing" refers to the horizontal scan across rows; "dicing" is the vertical scan down columns. Doing both narrows a box from nine cells to one without any notes.

How to do it efficiently

Pick a digit that already appears several times on the board — a digit with five or six placements gives you the most blocking lines to work with. Then check each box that's still missing it. For every such box, mentally extend the rows and columns that already contain the digit; if they cut the box down to a single open cell, place it.

Cycle through the digits this way, 1 to 9, and you'll place a surprising number of cells. When a digit stops yielding placements, move to the next. This systematic sweep is the engine of fast solving on easy and medium boards.

Cross-hatching vs. hidden singles

These are two views of the same logic. A hidden single describes the result — a digit with only one legal cell in a unit. Cross-hatching is the technique you use to find that result inside a box, by scanning the crossing lines. If you prefer working from pencil marks, you'll spot the hidden single in the notes; if you prefer scanning a clean board, you'll cross-hatch your way to the same cell.

When scanning runs out

Cross-hatching solves easy boards almost entirely and gets you deep into medium ones. When no box can be sliced down to a single cell and no unit is one digit from full, scanning has done its job. That's the moment to start pencil-marking and move to candidate elimination — the techniques that don't place digits directly, but remove possibilities until scanning works again.

They come in two families, one per guide:

A note on guessing

You may have seen "guess and check" listed as a Sudoku method. It works, but it isn't solving — it's brute force, and on a contaminated board it's hard to undo. Every technique in this guide is deductive: each move is forced by the rules, so you can always explain why a digit goes where it does. That's the difference between finishing a puzzle and actually getting better at them.

Play Sudoku Zen — six difficulties

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