The hardest sudoku puzzles, and what makes them hard

What makes a sudoku hard

Not the number of givens. Seventeen is the fewest numbers a sudoku with one solution can start with, and some such puzzles are easy, while one that starts with 25 can call for a technique few players know.

What counts is the hardest step: the technique you need at the point where nothing simpler finds a number or removes a note. Our game grades every puzzle that way. Its first four levels need singles alone, and each level after them adds techniques of its own.

Some puzzles need more than any level allows: a technique such as the W-Wing or the Finned X-Wing, or, where no pattern is left, the contradiction: suppose a number, follow the singles it forces, and remove it if the board breaks. The puzzles on this page are of that kind.

Every technique, level by level

The puzzle called the world's hardest sudoku

In June 2012 British newspapers printed a new puzzle by the Finnish mathematician Arto Inkala as the world's hardest sudoku. It starts with 21 numbers and has one solution. It is the first puzzle on the sheet below, as SudokuWiki.org gives it, and its answer is among the others.

Our game's hint cannot find its first step. None of the techniques it knows applies, and its contradiction, which follows one supposition through singles alone, breaks nothing. The answer below comes from a search by trial and error, not from steps a person could follow.

Whether it is the hardest depends on how hardness is measured. SudokuWiki, whose solver knows many more strategies than our hint, ranks it third by its own measure, after two puzzles by David Filmer.

It cannot open in our game: a code there names a puzzle our generator makes, and no code names this one.

Open it in our solver and see where the techniques stop

Six of ours, harder than Extreme

Each starts with about as many numbers as our Extreme puzzles do, yet the grader places it above every level of the game. Out of the puzzles our generator made from 100,000 seeds, these six needed the contradiction most often: each at least 76 times.

Scan a puzzle's QR code, or tap the line beside it, and it opens in Telegram as an Extreme puzzle, the hardest level a code can name.

Contradiction: how it works

Print Extreme sudoku

The puzzles to print