Hard Sudoku
Play it here or print a sheet. Where the interaction between a box and a line starts to matter — and still no guessing, because the puzzle is checked first.
Print a sheet of 6
Six tough puzzles and their answers, on separate sheets so a puzzle cannot be spoiled by the page it is on.
The puzzles
Puzzle 1 — Tough 8 2 1 7 2 2 4 6 5 6 4 9 9 3 6 2 4 9 7 8 3 1 4 6 5 3 7 3 Puzzle 2 — Tough 4 7 3 5 1 3 8 5 1 3 9 7 6 4 9 9 2 1 8 4 2 7 5 5 9 1 5 3 Puzzle 3 — Tough 1 4 9 5 3 8 2 7 4 5 8 6 6 5 2 1 3 4 6 3 6 9 2 8 5 4 2 Puzzle 4 — Tough 8 3 2 5 2 1 4 9 6 3 9 1 7 9 1 3 4 6 4 6 9 7 3 7 4 6 3 7 8 6 Puzzle 5 — Tough 4 5 3 1 2 3 9 6 4 6 5 7 2 8 9 1 7 2 7 2 1 3 9 8 2 6 5 Puzzle 6 — Tough 2 6 9 8 4 2 1 3 2 3 6 7 3 6 4 7 9 4 8 1 2 7 4 4 6 2 7
The answers
Puzzle 1 — Tough (answer) 4 9 6 8 3 5 1 2 7 5 1 7 6 2 9 4 8 3 2 3 8 7 1 4 6 5 9 7 6 1 2 4 8 3 9 5 8 5 9 3 7 6 2 4 1 3 4 2 5 9 1 8 7 6 9 8 3 1 5 2 7 6 4 1 2 4 9 6 7 5 3 8 6 7 5 4 8 3 9 1 2 Puzzle 2 — Tough (answer) 1 9 4 2 6 5 7 3 8 5 8 2 7 1 3 9 6 4 3 6 7 8 4 9 2 5 1 8 5 3 9 7 4 6 1 2 2 4 1 6 3 8 5 9 7 6 7 9 5 2 1 8 4 3 4 2 6 3 9 7 1 8 5 7 3 8 1 5 6 4 2 9 9 1 5 4 8 2 3 7 6 Puzzle 3 — Tough (answer) 7 5 6 8 1 3 2 4 9 2 9 4 5 6 7 8 3 1 3 1 8 4 9 2 7 6 5 4 3 5 2 8 6 1 9 7 9 6 1 7 5 4 3 2 8 8 7 2 1 3 9 4 5 6 1 8 3 6 4 5 9 7 2 6 2 9 3 7 8 5 1 4 5 4 7 9 2 1 6 8 3 Puzzle 4 — Tough (answer) 4 7 9 8 3 5 2 6 1 5 3 2 6 1 9 7 8 4 1 6 8 2 7 4 5 3 9 6 5 3 4 9 7 1 2 8 7 9 1 5 2 8 3 4 6 2 8 4 1 6 3 9 5 7 3 4 6 7 5 1 8 9 2 8 1 5 9 4 2 6 7 3 9 2 7 3 8 6 4 1 5 Puzzle 5 — Tough (answer) 4 7 8 5 6 3 9 1 2 9 1 6 4 2 8 5 3 7 2 5 3 7 1 9 6 8 4 8 3 4 6 5 1 7 2 9 6 2 7 9 8 4 1 5 3 5 9 1 3 7 2 8 4 6 7 6 2 1 4 5 3 9 8 3 4 5 8 9 7 2 6 1 1 8 9 2 3 6 4 7 5 Puzzle 6 — Tough (answer) 2 3 6 9 8 7 1 4 5 9 8 1 3 5 4 7 2 6 4 5 7 6 2 1 8 9 3 5 2 4 8 9 3 6 7 1 8 1 3 7 6 2 4 5 9 6 7 9 4 1 5 3 8 2 1 6 5 2 7 8 9 3 4 7 4 2 1 3 9 5 6 8 3 9 8 5 4 6 2 1 7
Pointing pairs: when a box constrains a line
Take one box and one digit. If every square in that box which could hold the digit lies in the same row, then the digit is definitely somewhere in that row inside the box — so it cannot be anywhere else in that row outside the box.
The same argument works for a column. It is a small step from the naked pair and it feels like a large one, because for the first time you are reasoning across two units at once rather than inside a single one.
Nothing gets placed, again. What you get is a set of removals along a line, often a long way from the box that justified them, and those removals are what makes some distant square a single.
Why hard puzzles feel unfair before they feel fair
The technique is invisible unless candidates are pencilled in, and the removals it produces are non-local — you reason about a box and act on a row two boxes away. That combination is what makes hard puzzles feel like guesswork the first few times.
It is not guesswork, and it is worth being certain of that: a solver on this site checked, before the puzzle was printed, that reasoning alone finishes it. Anything that needed trial and error was discarded rather than labelled "expert".
That distinction matters more on paper than on a screen. A wrong guess on a screen is one undo; a wrong guess in pen halfway down a printed grid is a ruined sheet.
A working order for hard puzzles
Fill in candidates for the whole grid before starting. At this level the partial-annotation approach that works for medium puzzles stops paying, because the useful deductions depend on squares you would not have chosen to annotate.
Then work in the same loop as before, with one addition: singles, naked pairs, and only then pointing pairs. Reaching for the newest technique first is the commonest way to spend ten minutes finding something a simpler rule would have given you.
Rewrite candidates onto a fresh sheet if the grid gets messy. It sounds wasteful and it is faster — a misread pencil mark is the source of most wrong answers at this level, and they are very expensive to unwind.
Common questions
What is a pointing pair?
A digit whose possible squares inside one box all lie in a single row or column. The digit must be in that line within the box, so it can be removed from the rest of that line outside the box.
Do hard Sudoku puzzles need guessing?
Not the ones here. Every puzzle is run through a solver that only uses human techniques, and anything it could not finish was discarded rather than published as harder.
Should I write in all the candidates?
At this level, yes. Pointing pairs depend on squares you would not have picked out to annotate, so partial pencil marks hide exactly the deductions you need.