Hard Sudoku

What makes a puzzle hard

Our grader solves each puzzle with human techniques, simplest first. A puzzle is hard when singles alone run out before the grid is full, and the grader has to use locked candidates or naked pairs to move on, but nothing harder. Hard puzzles start with at least 26 digits, and like every puzzle here they have exactly one solution.

The difference from medium is bigger than it looks. Locked candidates and naked pairs don't place a digit. They rule candidates out, and the digit appears a move or two later. That means you need notes.

Why notes matter now

At this level you will reach points where no single is visible. The way forward is to write candidates in each open square (use Notes, or N on a keyboard) and then remove the ones that logic rules out. When you place a digit, the game removes it from the notes in the same row, column and box for you. Our pencil marks guide explains how to keep notes useful rather than cluttered.

Two techniques do most of the work:

• Locked candidates: if, inside a box, a digit can only sit in one row, it cannot sit anywhere else in that row outside the box. The same works for columns. It also works the other way round: if a digit in a row can only sit inside one box, it can't go anywhere else in that box.
• Naked pairs: two squares in the same row, column or box that can only be the same two digits. Those two digits are used up, so remove them from every other square in that unit.

A routine for hard puzzles

1. Place every single you can find, both naked and hidden.
2. Fill in candidates for the remaining squares. Check each square's row, column and box carefully; a missing candidate causes more trouble than an extra one.
3. Look for locked candidates, box by box and digit by digit. Remove what they rule out.
4. Look for naked pairs: scan each unit for two squares with the same two candidates.
5. Go back to step 1. New singles almost always appear after a removal.

Common traps at this level

• Incomplete notes. If a square's notes leave out a digit that is really possible, a "pair" can look real when it isn't. When notes lead you to a contradiction, check them before anything else.
• Forgetting to remove. Spotting a pair is only half the job. The eliminations are what unlock the next digit.
• Hints with no single. When no single is left on the board, a hint fills a square directly and tells you the next step needs pairs or locked candidates. Before carrying on, try to find the step yourself; it's the best way to learn it.

More help: common sudoku mistakes and how to get unstuck.

What comes next

When hard stops being hard, try expert sudoku, which needs patterns such as hidden pairs, the X-Wing and the XY-Wing. There's also a daily hard puzzle, the same for everyone on the same date.