Hard Sudoku

Sudoku is played on a grid of 9 rows and 9 columns, split into nine 3×3 boxes. Some cells start with a digit (the clues). You fill in the rest.

On hard puzzles you will usually need notes to see what can still go where.

• Each row holds the digits 1 to 9, each exactly once.
• Each column holds the digits 1 to 9, each exactly once.
• Each 3×3 box holds the digits 1 to 9, each exactly once.
• The clues can't be changed. The puzzle is solved when every cell is filled and no digit repeats in a row, column or box.

Every hard puzzle here reaches a point where no single is left. To go on you have to rule out candidates with one of four techniques: pointing, box-line reduction, a naked pair or a hidden pair. Our grader checks that at least one of them is needed and that nothing harder is.

These steps don't place a digit by themselves. They remove notes, and the singles they open up do the rest.

Pointing: when a digit's possible cells inside a box all lie on one row (or column), that digit will be in this box on that line. So it can't be anywhere else on the line, outside the box.

Box-line reduction is the reverse: when a digit's possible cells on a row or column all sit in one box, the digit can't be in the rest of that box.

Naked pair: two cells of a unit that can each only be the same two digits, say 4 and 7. One gets the 4 and the other the 7, so no other cell of that unit can be 4 or 7.

Hidden pair: two digits that, within a unit, fit only in the same two cells. Those cells must hold those digits, so any other notes in them can go.

Tap or click a cell, then a number below the grid. On a keyboard, the arrow keys move between cells, 1 to 9 fill a cell, and Backspace or Delete clears it.

Notes turns the number buttons into pencil marks: small digits that remind you what can still go in a cell. Press N to switch on a keyboard, or hold Shift while you type a digit. When you place a digit, it is taken out of the notes in its row, column and box.

Undo takes back as many steps as you like. Hint first points out any wrong digit, then shows the next step a person could find, with the technique and why it works; press Hint again to fill it in. "Show mistakes" marks a wrong digit in red as soon as you enter it; switch it off to check your own work. Your game is saved in this browser, so you can close the page and come back.

These techniques only work when your notes are complete. Before you look for pairs, fill in every possible digit in the area you are working on, then remove digits as clues and placed digits rule them out.

Placing a digit here clears it from the notes of its row, column and box for you, so the notes stay correct as long as each digit you place is right. With "Show mistakes" on, a wrong digit shows at once.