Sudoku tips for when you are stuck

When a grid stops moving, the answer is almost never a flash of insight. It is a list, worked in order, from the cheapest technique to the most expensive. Here is that list, with each technique explained on a concrete position, and none of them involves guessing.

The order to work in

Do these in sequence and stop as soon as something falls out, then start again from the top. Restarting matters: one new digit often unlocks a technique three steps cheaper than the one you were about to reach for.

1. Re-scan all nine digits across all nine boxes.
2. Look for a unit with one empty cell.
3. Look for hidden singles.
4. Pencil in candidates, but only where you need them.
5. Naked pairs and triples.
6. Hidden pairs.
7. Locked candidates, also called pointing pairs.
8. X-wing.

Notation below is the usual one: r4c7 means row 4, column 7. Rows count from the top, columns from the left.

1. Re-scan, properly this time

Most stuck grids are not stuck. They have a digit sitting in plain sight that the solver stopped looking for twenty moves ago, because attention drifted to the interesting corner of the grid.

Do it systematically. Take the digit 1. Find every 1 already on the grid and rule out its row, its column and its box. Look at the boxes that still need a 1 and ask whether only one cell survives. Then take 2. Then 3. All nine, every box, without skipping. It is dull, it takes two minutes, and it breaks the majority of Easy and Medium grids on its own.

The trick people miss: scan the digit, not the cell. Asking “what goes in this cell” is a much weaker question than asking “where can this digit go in this box”.

2. The unit with one cell left

Any row, column or box with eight digits placed has exactly one answer left, and it is whichever digit is missing. Trivial, and worth a deliberate pass whenever the grid is more than half full, because these appear and then sit unnoticed.

3. Hidden singles, the workhorse

A hidden single is a digit that has only one legal cell left in a row, column or box, even though that cell could still take several digits as far as its own row and column are concerned.

Say box 5, the middle box, still needs a 7. Rows 4 and 6 already carry a 7 elsewhere, so within box 5 the 7 can only sit in row 5. Columns 4 and 6 also already carry a 7. That leaves r5c5 as the only cell in the box where a 7 can go, so the 7 goes there. It does not matter that r5c5 might also accept a 2 and a 9. The box has run out of alternatives for the 7, and that is enough.

Hidden singles are the single most commonly missed move in Sudoku, because the eye is drawn to cells with few candidates rather than to digits with few homes.

4. Pencil marks, but not everywhere

Filling every candidate into every empty cell on a half-finished grid is a lot of writing and most of it will be wrong within three moves. Pencil in one region at a time: pick the box, row or column with the fewest empty cells and mark only that. Techniques 5 to 8 all operate inside a single unit, so a single marked unit is enough to run them.

A cell that ends up with exactly one candidate is a naked single. Write it in. Then re-scan, because you are back at step 1 with new information.

5. Naked pairs and triples

Two cells in the same unit, each with the same two candidates and nothing else. Suppose r1c1 and r1c4 both hold exactly {3,8}. You do not know which is which, and you do not need to: between them they use up the 3 and the 8 for the whole of row 1. So every other cell in row 1 loses 3 and 8 as candidates.

That elimination is the point. It rarely places a digit by itself, and it very often turns some other cell into a naked single.

A naked triple is the same idea one size up: three cells in a unit whose candidates, pooled together, are exactly three digits. The cells do not each need all three. {2,5}, {5,9} and {2,9} is a valid triple on 2, 5 and 9, and all three digits come out of every other cell in that unit.

6. Hidden pairs

The mirror image, and harder to see. Look for two digits that can only go in the same two cells of a unit, even though those cells carry other candidates as well.

If in one box only r7c1 and r9c1 can take a 4, and only those same two cells can take a 6, then those cells hold the 4 and the 6 between them. Every other candidate in those two cells can be rubbed out, even though nothing was placed. That cleanup often exposes a naked pair elsewhere.

7. Locked candidates, or pointing pairs

The technique that unsticks most Hard grids, and the one most people have never been shown.

Inside one box, find a digit whose remaining cells all lie in the same row. Suppose in box 1 the digit 5 can only go in r3c1 or r3c3. Wherever it lands, the 5 for box 1 is somewhere in row 3. Therefore row 3 cannot take a 5 anywhere else, so the 5 comes out of every cell of row 3 in boxes 2 and 3. You have eliminated across the grid on the strength of an uncertainty.

It works in both directions. If a digit in a row is confined to cells inside a single box, that digit comes out of the rest of that box. Same idea with columns.

8. The X-wing

The first technique that looks at two units at once, and the point at which most people stop. It is worth learning because it is the only advanced pattern that shows up regularly at Hard.

Take a digit, say 9. Find a row where 9 has exactly two possible cells. Find a second row where 9 also has exactly two possible cells, in the same two columns. Those four cells form a rectangle. The 9s must sit on one diagonal of it: either both top-left/bottom-right, or both top-right/bottom-left. Either way, both of those columns get their 9 from inside the rectangle, so 9 can be removed from every other cell in both columns.

Swap rows and columns and the pattern works exactly the same way.

When none of it works

Three possibilities, in order of likelihood:

• You put a digit in the wrong place earlier. This is by far the most common. If the grid has become genuinely contradictory, undo back to the last placement you can prove.
• Your pencil marks are stale. A candidate you should have rubbed out three moves ago is hiding the answer. Re-derive one unit from the grid itself rather than from your own marks.
• The puzzle genuinely needs more. Above X-wing there are swordfish, XY-wings, colouring and the rest, but they appear almost only at Expert. At Easy through Hard, it is nearly always one of the first two.

And the thing to hold on to: a properly made Sudoku has exactly one solution and never requires a guess. If it feels like it does, the information you need is on the grid and you have not found it yet.

Scan the digit, not the cell. Ask where a number can go, not what a square can take.

Practising this without wrecking your daily grid

Techniques 5 to 8 need repetition on grids that actually contain them, and one puzzle a day is a slow way to get it. In Puzzle Lantern, the Daily Sudoku is one grid, the same for everyone that day, and More puzzles is unlimited on a difficulty you choose: Easy, Medium, Hard or Expert. Pointing pairs turn up constantly at Hard, which is where to go if that is the one you want to drill.

Pencil marks, unlimited undo and hints are all there. Mistake highlighting can be switched off if you would rather find your own errors, which is the honest way to practise technique 1 of the three above.

For how a grid gets made and checked before it reaches anyone, see how one Sudoku a day is generated and checked. For what the difficulty names actually measure, see Sudoku difficulty levels, explained.

Questions people ask

What should I do when I am stuck on a Sudoku?

Re-scan all nine digits across all nine boxes, check every unit for a digit with one cell left, pencil in candidates for the emptiest unit, then look for naked pairs, hidden pairs and locked candidates in that order.

Do you ever have to guess in Sudoku?

No. A properly made grid has one solution reachable by logic alone. If you are guessing, either a technique is missing or an earlier placement is wrong.

What is a naked pair?

Two cells in one unit holding the same two candidates and nothing else. Those two digits are used up by those two cells, so they can be removed from every other cell in the unit.

What is the difference between a naked single and a hidden single?

A naked single is a cell with one candidate left. A hidden single is a digit with one cell left in a unit. Hidden singles are far more common and far more often missed.

What is an X-wing?

A digit confined to the same two columns in two different rows. It must sit on one diagonal of the rectangle they form, so it can be removed from the rest of both columns.