Sudoku solving techniques for hard and expert grids

Scanning gets you through easy sudoku and most of medium. Past that, you stop finding cells. The techniques below apply the same logic to candidate marks instead of digits, and they get a stalled board moving again. They appear roughly in the order you should reach for them.

Each one assumes you have pencilled candidates into the empty cells. If you have not, do that first, because you cannot see any of these patterns without them. If candidate marking is new to you, 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 the same two candidates. If two cells in row 4 both read {2, 8}, one of them holds the 2 and the other the 8. You do not need to know which. Between them they use up row 4’s 2 and 8, so you can strike 2 and 8 from the other cells in row 4.

The same idea scales. Three cells that share three candidates between them form a naked triple, and four cells sharing four form a quad. Each cell does not need all the candidates: {2,8}, {2,5} and {5,8} across three cells make 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 in disguise. Two digits can appear in only two cells of a region, but those cells carry other candidates too, so the pair is hard to see. If in one box the 4 and the 9 both appear only in cells A and B, then A and B hold the 4 and the 9 in some order, and you can delete the other candidates from A and B.

Solvers under-use hidden pairs, because a naked pair stands out on the page and a hidden one does not. To find them, go through a region digit by digit and note which cells each digit could occupy, instead of reading cell by cell. Two digits with the same two-cell footprint form your pair.

3. Pointing pairs and box-line reduction

These two use the overlap between a box and the rows or columns crossing it, and together they do most of the work on hard sudoku.

A pointing pair works outward from the box. If each possible position for the 6 inside one box lies in the same row, the box’s 6 sits somewhere in that row. You do not know where, but you know the 6 cannot sit anywhere else in that row outside the box, so strike it from those cells.

Box-line reduction runs the same argument in reverse. If each possible position for the 6 in a row lies inside a single box, that box’s 6 is in that row, and you can strike the 6 from the box’s other six cells.

Both are easy to apply once you look for them, and the eliminations they give you usually lead to a naked or hidden single within a move or two.

4. The X-wing

The X-wing is the first technique that spans the whole board, and many solvers find it changes how they look at a grid.

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

The same works with the roles swapped: two columns with two positions each, aligned on the same two rows, let you eliminate from those rows. An X-wing rarely solves a cell by itself. It removes candidates that were blocking a simpler technique.

5. Swordfish and beyond

A swordfish widens the X-wing to three rows and three columns. If a digit is confined to at most three cells in each of three rows, all within the same three columns, you can eliminate that digit from the rest of those columns. Swordfish patterns turn up on expert grids, and you will rarely spot one without a systematic sweep.

Beyond that lie XY-wings, unique rectangles and forcing chains. Learn them if you enjoy the hunt, but each adds less than the last: most expert puzzles yield to candidate marking, pairs, pointing pairs and a patient X-wing. If you find yourself reaching for a forcing chain, you have probably missed something simpler.

A working order

When a grid stalls, work down this list:

1. Re-scan for naked and hidden singles; your last pass has probably created some.
2. Sweep each region for naked pairs and triples.
3. Go digit by digit through each region looking for hidden pairs.
4. Check each box for pointing pairs, then each row and column for box-line reduction.
5. Then hunt for an X-wing.

Each step feeds the ones above it, so after any successful elimination, go back to the top of the list.

Practise on paper

You will learn these techniques faster on a printed grid than on a screen, because you can see all your candidate marks at once and annotate freely. Print a batch of hard puzzles one or two per page so you have room to write, and keep the answer key to check yourself at the end. The solver confirms each puzzle from the generator has exactly one solution, so if a technique leads you into a contradiction, look for the mistake in your pencil marks.