The Hardest Sudoku Puzzles

The hardest sudoku is the one with the highest rating, not the one with the fewest clues. Difficulty is measured by the hardest logical step a solver must use, on the Sudoku Explainer scale (ER). Ordinary puzzles sit below ER 7. The top of the scale, ER 10 and above, needs nested forcing chains: you test a candidate, follow what it forces, and sometimes test inside that test. This page is for intermediate and advanced players who want to know what "hardest" means in practice and how to face it.

How is "hardest" measured?

Sudoku Explainer, a solver by Nicolas Juillerat, rates a puzzle by simulating a human solver. It always applies the simplest available technique, and the puzzle's rating is the difficulty of the hardest step it was forced to use. Hidden singles are ER 1.0 to 1.5, pairs and triples roughly 2.0 to 4.0, fish and wings roughly 3.2 to 5.2, forcing chains start at 6.5, and nested chains start at 9.0 and reach ER 10 and higher. Our solver is a port of that engine, and our difficulty levels use its bands directly.

Clue count is a poor guide. In our classic catalogue, Legendary puzzles average 25.3 givens, slightly more than Master (24.4), and a Hard puzzle can have 44. Fewer clues usually make a grid more open, but what matters is how much logic each clue unlocks. A grid with 30 clues can need a chain deeper than one with 20.

What do the hardest puzzles demand?

Top-rated puzzles leave singles, pairs, fish and wings with nothing to remove. Progress needs the last rungs of the technique ladder, in this order of cost:

• Almost Locked Sets and Death Blossom, which link groups of cells that are one candidate short of a locked set.
• Forcing Nets, where you follow every candidate of a cell and keep what all branches agree on.
• Bowman's Bingo, the last resort: assume a candidate, follow the forced placements, and look for a contradiction.

Sudoku Explainer names the deepest of these as forcing chains, split into cell, unit, dynamic and nested variants. They are all the same idea at increasing depth: a branch inside a branch. Nothing here is guessing. Every branch is forced by logic, which is why a puzzle needing them still counts as valid.

What about the "world's hardest sudoku" headlines?

Newspapers have run "world's hardest sudoku" stories for years. The best known concern the Finnish mathematician Arto Inkala, whose puzzles were reported in the press in 2010 and again in 2012, when The Daily Telegraph ran one under that headline. Treat these as history. "Hardest" depends on the rating system, and different solvers rank puzzles differently. There is no single agreed hardest puzzle, so we show none of these grids and quote no ratings for them.

How hard are our hardest puzzles?

All figures below are from puzzles generated for this site and rated by our Sudoku Explainer port. A puzzle the rater cannot solve by logic is never published.

Legendary starts at ER 10.0 and our window runs up to 15.0. Most of our Legendary puzzles cluster just above 10, and the tail toward the top is thin.

Killer is the hardest of the three by rating. In our killer Expert, Master and Legendary puzzles there are no given digits at all: the cages alone fix the solution. If you enjoy that kind of pressure, the Legendary Killer level is where the 12.8 lives.

A Legendary classic example

This is our own classic puzzle rated ER 11.1, the highest classic in our catalogue. It has 26 givens (dots are empty).

5 . 3 | . . 4 | . . .
4 8 . | 9 . . | . . .
. 6 . | . 5 . | . 8 .
------+-------+------
6 . . | 8 . . | . . 2
. . . | . . . | . . 7
. . . | . 6 . | 9 . .
------+-------+------
3 . 6 | . . . | 4 . .
9 5 . | . 4 3 | . . .
. . 8 | 6 9 . | . . 5

Its hardest step, as reported by the rater, is Dynamic Contradiction Forcing Chains, with Multiple Forcing Chains also in play. In plain terms: no pair, fish, wing or simple chain finishes it. You must assume a candidate, follow the consequences several placements deep, and find that the assumption breaks a rule. Then the opposite candidate is true.

How do you approach a Legendary puzzle?

1. Exhaust the cheap moves first. Singles, pairs, triples, pointing pairs, then fish and wings. Every candidate removed makes a later branch shorter.
2. Fill in all pencil marks and keep them accurate. A stale candidate ruins a chain, and one wrong mark can make a valid puzzle look contradictory.
3. Look for structure before you branch: almost locked sets sharing a candidate, or bivalue cells forming a loop.
4. When you must branch, pick a cell with two candidates, or a digit with two places in a unit. Two short branches beat four long ones.
5. Trace a branch only through forced moves. If you write a "probably", the branch proves nothing.
6. Undo each branch fully before starting the next, and keep only what every branch shares or the contradiction of a single branch.
7. Take breaks. At this level most errors are bookkeeping, not logic.

Ready to try? Play Legendary classic or the killer level above. Our difficulty levels guide explains every band below it, and the full ladder of techniques is in Forcing Nets and the pages it links to.