How to solve a sudoku: six techniques that actually work
From cross-hatching to intersections. The techniques that really solve a sudoku, explained in order, from the easiest to the one that gets you out of a deadlock.
Almost everyone learns to solve sudokus the same way: staring at the grid until a number jumps out. That works for the easy ones, and stalls the moment the difficulty goes up a rung. The difference between someone who takes forty minutes and someone who takes eight is not intuition — it is four or five specific techniques applied in order.
They go from least to most demanding. The first three will solve the great majority of newspaper sudokus. The last three are what you need once no obvious number is left.
The rule to keep in your head at all times
A sudoku is a 9×9 grid divided into nine 3×3 boxes. Every row, every column and every box must contain the numbers 1 to 9 exactly once. That is all. Nothing gets added up and nothing needs guessing: a well-built sudoku has exactly one solution and you reach it by logic alone.
That last point matters. If you find yourself guessing, you have stepped outside the technique — it does not mean the sudoku is unfair.
1. Cross-hatching: the number that only fits in one place
Pick a number, say the 5, and run it across the whole grid. Look for a 3×3 box with no 5 in it yet, and check the rows and columns crossing it: if there is already a 5 in two of the three rows passing through that box, then the box's 5 has to be in the remaining row. Combine that with the columns and often a single possible cell is left.
It is the highest-yield technique of all, and the one to exhaust before writing anything else. Run it with all nine numbers, 1 to 9, before moving on.
Practical tip: start with the numbers that already appear most often in the grid. The more 5s are placed, the more constrained the next 5 is.
2. The cell that admits only one number
This is cross-hatching in reverse. Instead of fixing a number and looking for a home, you fix an empty cell and ask which numbers it could take. Cross off the ones already in its row, its column and its box. If only one survives, you have it.
It sounds slow, and it is if you do it cell by cell. It becomes fast if you apply it only to cells in rows, columns or boxes that already have six or seven numbers placed: few options remain there and the answer jumps out.
3. Mark the candidates (but only when the time comes)
There comes a point where the two previous techniques stop yielding. That is the moment — and not before — to write in each empty cell the numbers it could still take. On paper, small, in a corner.
The reason not to do it from the start is that marking candidates on a nearly empty grid is an enormous job that you will then have to erase entirely. Once thirty or forty numbers are placed, each cell's candidate list is short and useful.
From here on, every technique works on those marks.
4. Pairs: two cells sharing two numbers
If within one row, column or box you find two cells whose candidates are exactly the same two numbers — say both can only be 4 or 7 — then those two cells will share the 4 and the 7 between them, even though you do not yet know in which order.
The consequence is the useful part: the 4 and the 7 can be deleted from every other cell in that row, column or box. That almost always exposes another cell with a single candidate, and the chain starts up again.
The same holds for three cells sharing exactly three candidates (a triple), though it is far less common.
There is an inverse variant, the so-called hidden pair: two numbers that, within one row, column or box, appear as candidates in only two specific cells. Then those two cells belong to those two numbers and you can delete every other candidate from them. It is the same idea seen from the other side.
5. Only candidate in the unit
Look at the candidates across a whole row. If the number 8 appears as a candidate in only one of its cells, that is where the 8 goes, even if that cell has four other candidates marked. The row needs an 8 somewhere and there is only one place it can be.
It is the technique people miss most, because it requires looking at the row as a whole rather than cell by cell. Repeat it with rows, columns and boxes.
6. Intersections: pushing a number out of a box
This is the one that unsticks hard sudokus. If within a 3×3 box all the possible candidates for the 6 are in the same row, then that box's 6 will be in that row, no question. And therefore the 6 can be deleted from the rest of that row, in the other two boxes.
It works the same in the other direction: if all the 6 candidates in a row fall inside one box, the 6 is deleted from the rest of that box.
It is a two-step piece of reasoning and it takes a few goes to see. Once it clicks, it is the tool that turns an «impossible» sudoku into a normal one.
The order matters more than the techniques
If you had to keep one idea from all of this, let it be this: always apply the cheapest technique first. The classic mistake of someone who gets stuck is hunting for pairs while there are still three numbers placeable by simple scanning.
The loop that works:
- Cross-hatch with all nine numbers.
- Did any drop? Back to step 1.
- None dropped? Mark candidates.
- Only candidate in rows, columns and boxes.
- Pairs (and hidden pairs).
- Intersections.
- Every time you place a number, back to step 1.
That «back to step 1» is the key. Each new number changes the constraints of its row, its column and its box, and very often reopens the easy route.
And one thing that is not a technique
Guessing. Trying a number «to see what happens» and carrying on ten cells further is the fastest way to end up erasing half the grid. If you genuinely cannot see the way out, it is much better to leave it, do something else and come back: a fresh look finds in thirty seconds what you spent ten minutes missing.
Sudoku trains like anything else. At first you lean on scanning; within a few weeks you see the pairs without looking for them.
If you fancy practising without hunting for a newspaper, MenteBrillante gives you a new sudoku every day, with difficulty that rises as you play, plus a practice mode with unlimited grids to drill whichever technique you are learning. There are also free online sudokus with no sign-up, and printable ones.
And when the six techniques here stop being enough, the next rung is in hard sudoku techniques: X-Wing, forcing chains and the rest.