Sudoku solving techniques for hard and expert grids

Scanning gets you through easy sudoku and most of the way through medium. Past that it runs out, and the grid goes quiet. What follows is not harder logic — it is the same logic applied to candidate marks instead of to digits. These are the techniques that unstick a stalled board, roughly in the order you should reach for them.

All of them assume you have pencilled candidates into the empty cells. If you have not, do that first; none of what follows is visible without it. And if candidate marking itself is new, start with how to solve sudoku and come back.

1. Naked pairs, triples and quads

A naked pair is two cells in the same region — a row, a column or a box — that hold exactly the same two candidates. If two cells in row 4 both read {2, 8}, then one of them is the 2 and the other is the 8. Which is which does not matter. What matters is that between them they have used up row 4’s 2 and its 8, so every other cell in row 4 can have 2 and 8 struck out.

The same idea scales. Three cells sharing exactly three candidates between them form a naked triple, and four cells sharing four form a quad. The cells do not each need all the candidates — {2,8}, {2,5} and {5,8} across three cells is a valid triple on 2, 5 and 8, because those three digits are locked into those three cells. Triples are worth hunting for; quads are rare enough that you will usually find something else first.

2. Hidden pairs

A hidden pair is the same situation wearing a disguise. Here, two digits can only appear in two cells of a region — but those cells carry other candidates too, so the pair is not obvious. If in one box the 4 and the 9 both appear only in cells A and B, then A and B must be the 4 and the 9 in some order, and every other candidate in A and B can be deleted.

Hidden pairs are the technique most solvers under-use, because a naked pair announces itself visually and a hidden one does not. The way to find them is to work digit by digit through a region, noting which cells each digit could occupy, rather than reading the cells. Two digits with the same two-cell footprint is your pair.

3. Pointing pairs and box-line reduction

These two work on the interaction between a box and the rows or columns crossing it, and together they are the workhorses of hard sudoku.

A pointing pair works outward from the box. If, inside one box, every possible position for the 6 lies in the same row, then the 6 for that box is somewhere in that row — you do not know where, but you know it is in there. Which means the 6 cannot be anywhere else in that row outside the box. Strike it out of those cells.

Box-line reduction is the same argument in reverse. If every possible position for the 6 in a given row lies inside a single box, then that box’s 6 is in that row, and the 6 can be struck from the box’s other six cells.

Both are easy to apply once you are looking for them, and both tend to produce a cascade — the eliminations they generate usually create a naked single or a hidden single within a move or two.

4. The X-wing

The X-wing is the first technique that spans the whole board, and it is the point where most solvers feel sudoku becomes a different game.

Find a digit — say 7 — that in two different rows can only go in two cells each. Now check the columns: if in both rows those two cells sit in the same pair of columns, you have an X-wing. The four cells form a rectangle. Whichever way the 7s fall, they occupy one cell from each row and one from each column, taking a diagonal of that rectangle. Either way, both of those columns have their 7 accounted for inside the rectangle, so the 7 can be eliminated from every other cell in both columns.

The same works with the roles swapped: two columns with only two positions each, aligned on the same two rows, eliminates from those rows. An X-wing rarely solves a cell directly. What it does is remove candidates that were blocking a simpler technique.

5. Swordfish and beyond

A swordfish is an X-wing widened to three rows and three columns: a digit confined to at most three cells in each of three rows, all falling within the same three columns, eliminates that digit from the rest of those columns. It is genuinely useful on expert grids and genuinely hard to spot without a systematic sweep.

Beyond that lie XY-wings, unique rectangles and forcing chains. They are worth learning if you enjoy the hunt, but they come with diminishing returns: most expert puzzles yield to candidate marking, pairs, pointing pairs and an X-wing applied patiently. Reaching for a forcing chain is usually a sign that something simpler was missed.

A working order

When a grid stalls, run down this list rather than staring:

1. Re-scan for naked and hidden singles — you have probably created some since your last pass.
2. Sweep every region for naked pairs and triples.
3. Go digit by digit through each region looking for hidden pairs.
4. Check every box for pointing pairs, then every row and column for box-line reduction.
5. Only then start hunting for an X-wing.

Each step feeds the ones above it, so after any successful elimination, go back to the top. The board you are looking at is not the board you were looking at a move ago.

Practise on paper

These techniques are far easier to learn on a printed grid than on a screen, because you can see every candidate mark at once and annotate freely. Print a batch of hard puzzles one or two per page so there is room to write, and keep the answer key to check yourself when you finish. Every puzzle from the generator is verified to have exactly one solution, so if a technique leads you into a contradiction, the fault is in the pencil marks and not in the puzzle.