Hard Sudoku

Hard puzzles have 51-55 empty cells, and singles alone may stall. That is when pencil notes and candidate techniques come in. Every puzzle still has exactly one solution.

Techniques for Hard

• Naked pair: two cells in the same row, column or box share the same two candidates, so remove those numbers from the unit's other cells.
• Pointing: if a number's candidates inside a box all sit in one row, remove that number from the rest of the row.
• Box-line reduction: if a number's candidates in a row all sit inside one box, remove it from the rest of that box.
• Each time you place a number, matching notes in its row, column and box are cleared automatically.