How to Solve Sudoku Step-by-Step

Most "how to solve sudoku" articles dump a list of advanced techniques on you and call it a day. That isn't how good solvers actually work. Good solvers run through a checklist, in order, from cheapest move to most expensive. They only reach for fancy moves when the simple ones run dry.

Here is the order I use, the same one you'll see if you watch any experienced solver. Read it once. Then keep it in mind for the next puzzle you start, and run it as a loop until the grid is full.

Step 1: place every naked single you can see

A naked single is a cell where eight of the nine possible digits are already eliminated by the row, column or box. Only one digit can legally go there. Place it.

In practice, you find them by glancing at the empty cells in each 3×3 box. For each empty cell, check what its row, column and box already contain. If only one digit hasn't been used yet, that's your placement.

On an easy puzzle, naked singles alone often finish the grid. You won't need anything else. Get comfortable spotting them before you go further. The naked singles strategy page has more examples if you want them.

Step 2: scan for hidden singles, one digit at a time

Naked singles ask "what can go in this cell?" Hidden singles ask "where can this digit go in this row, column or box?" It's the same question turned inside out, and it catches placements naked singles miss.

The fast way: pick a digit, say 5. Look at the whole grid. Find every 5 that's already placed. Now walk through the boxes that don't yet contain a 5. For each one, ask which cells could still hold a 5, given the rows and columns that already have one. If exactly one cell remains, place it.

Then move to 6, then 7, then 8, then 9, then start again at 1. After each pass, placements you've made open up new hidden singles elsewhere. Cycle through the digits two or three times. By the end of it, you've cleared most of an easy or medium grid.

Step 3: stuck? Lay down pencil marks. Properly.

If singles run out and you can't place anything for a minute or two, it's time to write candidates in the empty cells. Every cell gets a list of the digits it could still legally hold, given the constraints. This is what pencil marks (or "notes") are for.

Pencil marks are the bridge to harder techniques. Without them, you can't see naked pairs, hidden pairs, pointing pairs or anything else. With them, those patterns jump off the page.

Step 4: scan the boxes, the most underused move

Even with pencil marks in place, there are still easy placements waiting in the boxes. Take a digit, see which boxes don't yet contain it, and look for boxes where it can only fit in one or two cells.

This is the same move as step 2, but now with pencil marks helping you see candidate cells at a glance. You'll be surprised how often a box-by-box pass reveals placements you missed on the row-and-column scan.

Step 5: pairs and triples

Now the real technique starts. A naked pair is two cells in the same row, column or box that contain exactly the same two candidates and nothing else. Whatever those two cells turn out to be, they will take both of those digits between them. So you can eliminate those two digits from every other cell in the row, column or box.

A hidden pair is the same idea inverted: two digits that, between them, can only appear in two specific cells within a row, column or box. Other candidates in those two cells can be erased.

Both are huge time-savers, because every elimination they make often forces a new single. The cascade restarts. For worked examples, see the strategy index.

Step 6: pointing pairs and box-line reduction

If pairs and triples don't unstick you, the next move is almost always a pointing pair. Inside a 3×3 box, look at the candidates for one digit. If all of them lie on the same row or column, that digit has to appear in that row or column when it appears in the box. You can eliminate the digit from the rest of that row or column outside the box.

Pointing pairs unstick almost every hard puzzle. If you're getting stuck on hard, this is the technique you're missing, more often than not. Practise it on every hard puzzle you play.

Step 7: bigger patterns, sparingly

At expert difficulty and above, the previous steps still do most of the work, but the puzzle will occasionally need a bigger pattern. The main ones are X-wing, swordfish, XY-wing and colouring. Each has a specific shape, a specific elimination it permits, and a specific moment to look for it.

The good news: you don't need them for the vast majority of puzzles. The better news: once you've learned them, you have them forever, and they show up in real puzzles often enough to be worth the time.

"Most of solving sudoku is finding the moves that don't need the fancy techniques. Pencil marks should be the bridge, not the starting point."

The loop, written out

1. 1
   Scan for naked singles

   Place every cell where only one digit fits. Sweep the grid once.
2. 2
   Scan for hidden singles

   Go digit by digit. Walk the boxes first, then rows, then columns.
3. 3
   If singles ran out, mark candidates

   Write every digit that could still legally fit in every empty cell.
4. 4
   Look for pairs and triples

   Eliminate candidates from the rest of the row, column or box.
5. 5
   Look for pointing pairs

   Eliminate candidates from outside the box. Then return to step 1.
6. 6
   If still stuck, try bigger patterns

   X-wing, XY-wing, colouring. Only when the previous steps stop firing.

After every successful placement, jump back to step 1. The cascade is the whole point. One pointing-pair elimination often unlocks three naked singles, which unlock two hidden singles, and now you have momentum again.

A few habits that quietly help

These aren't moves. They're working habits that distinguish a smooth solve from a frustrating one.

• ✓After every placement, scrub the digit from candidates in its row, column and box. Pencil marks rot fast.
• ✓When you can't find anything for a minute, redo the candidates in one box from scratch. You probably missed an elimination.
• ✓If you find yourself wanting to guess, stop. Step away for two minutes. You've missed a deduction, not run out of road.
• ✓Don't switch difficulties mid-week. Pattern recognition compounds when you stay at one level.
• ✓Don't watch the clock while you play. Rushing makes you miss easy moves.

Common moments where the method snaps

Even with the loop, three situations catch experienced solvers out often enough to mention. The first: stale pencil marks. You placed a digit ten minutes ago and forgot to scrub. The candidate list in a cell now lies to you. Fix: clear and redo the marks for one box.

The second: confused boxes versus rows. You're scanning for a digit and conflating the box rule with the row rule. Fix: be deliberate about which constraint you're checking, one at a time.

The third: skipping the candidate step. You're at a hard puzzle and refusing to write pencil marks because you want to do it "in your head". Fix: write the marks. There is no shame.

Finishing well

The last few cells of a sudoku are often easier than the middle. Once the grid is mostly full, every remaining cell has very few candidates left, and naked singles become trivial. Don't rush. A misplaced digit in the closing minute means tracing back through ten earlier moves to find the mistake. Slow the last placements down a half-step and you'll finish cleanly.

Ready to apply this on a real grid? Start with a medium puzzle and run the loop from the top. Or if you've been stuck on hard, see the strategy library for worked examples of every move in this article. If you've never finished a puzzle before, the absolute beginner's guide is the gentler entry.