How to play Sudoku

The board

The grid is 81 cells, arranged three ways at once. Nine rows run across, nine columns run down, and nine boxes of 3×3 sit inside the thick lines. Each of those 27 groups is a unit, and each unit needs the numbers 1 to 9 exactly once.

The numbers printed at the start are givens. They never change. Everything else is yours to work out.

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Two cells see each other when they share a row, a column or a box. That single idea is the engine of the whole game: a number placed in one cell is banned from the twenty cells that see it.

Move one: finish what is nearly finished

Look for a unit with exactly one empty cell. Name the numbers already in it, find the one that is missing, and write it in. No candidates, no notes, no thinking beyond counting.

In the grid above, the top-left box already holds 9, 4, 6, 7, 1, 3, 8 and 5. One cell is empty and one number is missing, so the answer writes itself.

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Every number you place makes three units, its row, its column and its box, one step closer to being finished, so check those three immediately before looking anywhere else. Done repeatedly, this move alone clears a large part of an Easy grid. It is called the Last Remaining Cell.

Move two: ask where a number goes

When no unit has a single blank left, change the question. Instead of looking at an empty cell and asking what could fit, take one number and ask where it can still go inside one box.

Pick a number that already appears several times on the board, because each appearance blocks a whole row and a whole column. Trace those lines through the box you are looking at and cross off every cell they touch. If exactly one cell survives, the number belongs there, even if that cell could also have held other numbers.

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In the bottom-left box, the 3s already on the board rule out every cell but one. That cell could hold a 3, a 7 or an 8 as far as its own row and column are concerned, which is exactly why this is called a Hidden Single. It is hidden by the other candidates sitting alongside it.

This is the move that separates people who finish Medium puzzles from people who stall on them, and it costs nothing but a change of question.

When scanning stops working: take notes

Sooner or later, no unit is nearly full and no number is pinned to one cell. That is the signal to write candidates: small pencil marks listing the numbers each empty cell could still take.

Notes turn an invisible problem into a visible one. A cell that ends up with a single candidate is solved on the spot: that is a Naked Single. Two cells in one unit holding the same two candidates reserve both numbers, so every other cell in that unit can drop them: that is a Naked Pair. From there the whole ladder of techniques opens up.

Keeping notes accurate by hand is tedious, which is why the app can fill and maintain them for you while you concentrate on the reasoning.

Five habits that prevent most mistakes

1. Check all three units before you write. Row, column, box. Nearly every wrong number in Sudoku is one of the three going unchecked.
2. Never guess. A proper puzzle has exactly one solution and never requires a guess. If you cannot see the next move, you are missing a technique, not a lucky break.
3. Re-scan after every placement. The number you just wrote changed twenty cells. The next move is usually right next to the last one.
4. Update your notes as you go. A stale pencil mark is worse than no pencil mark, because you will trust it.
5. When stuck, change the question. If "what fits here" has stopped paying, switch to "where does this number go", and the other way round.

What to learn next

Once Last Remaining Cell, Hidden Single and Naked Single feel automatic, the next useful step is not a harder pattern but a different kind of move: techniques that remove candidates rather than place numbers. Pointing Pairs and Box/Line Reduction are the two to learn, in that order, and they are what Hard puzzles are built on.

The full ladder lists all 37 in learning order, and the difficulty guide explains which ones each level will actually ask of you.