Hard Sudoku
A hard board here is one that stops yielding to single cells. At some point you will be looking at a row where nothing can be placed, and the only way forward is to prove that certain digits are impossible somewhere else. That is a different kind of move: it fills nothing in, it only narrows what is left, and a cell somewhere across the grid becomes readable two steps later.
What makes a sudoku hard is elimination, not placement
Easy and medium boards are solved by placing digits. From this grade on, most of your moves place nothing at all – they remove possibilities, and the placement arrives later as a consequence. That changes how the grid feels: long stretches where you are working and the board looks unchanged, then three or four cells falling at once. If you are used to steady progress, this reads as being stuck when it is nothing of the kind.
The naked pair
Two cells in the same unit, each with exactly the same two candidates left – say 4 and 9. You cannot tell which is which, and it does not matter. Between them those two cells will consume both the 4 and the 9 of that unit, so no other cell in the row, column or box the pair shares can be either. Cross the 4 and the 9 off every one of them. Often that is enough to leave some third cell with a single candidate, and the board opens up again.
The pointing pair
This one runs the other way, from a box out into a line. Inside one box, find every cell where a 3 is still possible. If all of them happen to sit in the same row, then whichever cell finally takes the 3, it will be in that row and inside that box. So the 3 can be struck from the rest of that row – the six cells belonging to the two neighbouring boxes. Nothing has been placed; a whole line has just got narrower. The same argument run down a column is what the board calls box line when it names a hint.
Twelve attempts, then the closest one
Building a puzzle to a stated grade is not deterministic. The generator makes a complete grid, digs holes while uniqueness holds, grades what is left, and if that does not come out at the grade this page asked for it throws the whole thing away and starts again – up to twelve times. If none of the twelve lands exactly, it hands over whichever attempt came closest rather than looping forever. In practice that means the occasional board on this page that a medium solver walks straight through. A hard sudoku that falls apart in four minutes is that, rather than a mis-labelled page. Press New puzzle and you get a fresh set of twelve attempts.
The hint takes one of two routes
Ask for a hint and the board first sweeps the grid for a cell with a single candidate, because that is cheap and it is the move you were most likely to have missed. If there is no such cell, it runs the full human solver over the position instead, takes the first cell that solver managed to fill, and tells you the sort of deduction it had reached for. What it will never do is copy a digit out of the answer. A position that no technique on the list can advance gets told so plainly, and you are left with the board you built – which is itself information, because on this grade an unadvanceable position nearly always means an earlier elimination was wrong.
Pencil marks start earning their keep
Naked pairs are invisible unless the candidates are written down, so this is the grade where the Notes button stops being optional. Turn it on and the digits you press are added to the cell as small marks instead of being entered as an answer; press the same digit again to remove it. Erase clears a cell completely, digit and marks together, whichever mode you are in.