Cross-hatching
Cross-hatching is the technique you will use more than all the others put together. It needs no pencil marks and no notation — only the digits already on the board — and it is how the opening of every puzzle, at every level, gets solved.
The idea
Pick one digit and one box that is still missing it. Every row that already contains that digit rules out the three squares where it crosses your box. Every column that contains it does the same. Strike out enough squares and, often, exactly one survives — and the digit has to go there.
Nothing is being deduced about any individual square. The argument is about the box: the digit has to live somewhere inside it, and if there is only one place left, that is the place.
A worked example
Here the top-left box is missing its 7, and there are four 7s elsewhere on the board doing the eliminating.
Work through it one line at a time:
- The 7 in row 1 kills the whole top line of the box — r1c1, r1c2 and r1c3.
- The 7 in row 3 kills the bottom line the same way.
- The 7 in column 1 kills r2c1.
- The 7 in column 3 kills r2c3.
That leaves r2c2 alone in the middle, so the 7 goes there. Note that r2c2 might accept several other digits too — that is irrelevant. The box has run out of room for its 7.
How to scan fast
Work one digit at a time across all nine boxes before moving to the next digit. Almost everyone does the opposite at first — staring at one box and trying to place all nine digits in it — and it is far slower, because you keep rebuilding a different mental picture for every digit.
Sweeping a single digit means you are looking for one shape repeatedly, and the rows and columns you have already checked stay useful as you move across the grid. Start with the digits that appear most often on the board: they do the most eliminating.
Common mistakes
The usual error is stopping at the first survivor. If two squares in the box survive, you have found nothing yet — a digit with two possible homes is not a placement. Move on and come back after the next placement opens the box up.
The second is forgetting to re-scan. Every digit you place invalidates part of your earlier sweep, and the square it opens up is usually adjacent. After each placement, re-check that digit in the neighbouring boxes before doing anything else.
Where to go next
The next technique in the order is naked single — One square, one surviving candidate. The most direct placement in Sudoku.
All nine techniques: the techniques index. New to the rules? Start here. Ready to use it? Open a board.