How to solve Sudoku without guessing: the three techniques that cover almost every grid

Every Sudoku has exactly one solution, and every cell in it can be reached by logic. That is not a slogan; it is what makes the puzzle a puzzle. If you have ever pencilled a 4 into a cell “to see what happens”, you didn't solve that grid. You searched it. Three techniques cover almost every grid you will meet, from the easy ones on the back page to the ones an app labels hard. Here they are, in the order you should reach for them.

First, the one habit that makes the techniques work

Before any technique: write in the candidates. A candidate is a small digit in the corner of an empty cell meaning “this could still be a 3”. Most people skip this because it feels slow. It is the opposite. Every technique below is a rule about candidates, and if the candidates are not on the board you are running the rule in your head, badly, and calling the result a guess.

You do not need to fill every cell with all nine. Start with the rows, columns and boxes that are already busy — six or seven digits placed — because their empty cells have the fewest candidates and are where the first moves live.

Technique one: the only candidate

Look at a single empty cell. Cross off every digit that already appears in its row, its column, and its 3×3 box. If exactly one digit survives, that is the answer. Write it in.

This is the technique everyone discovers on their own, and it solves the whole of an easy grid on its own. The trick to using it well is not to scan the grid at random. Go to the most crowded box, take its emptiest-looking cell, and count. A cell whose row has five digits, column has four and box has six will often be down to one candidate before you have finished counting.

Each digit you place removes a candidate from up to twenty other cells — the rest of its row, column and box. So after every placement, look again at the cells nearest to it. Solving is not nine separate searches; it is a chain, and the next link is usually next door.

Technique two: the only place

Now flip the question. Instead of asking “what can go in this cell?”, pick a digit and a region and ask “where in this box can the 7 go?”

Take one box. Find the 7s in the three rows and three columns that cross it; each one rules out a full line of cells inside the box. Whatever cells are left are the only places a 7 can live in that box. If one cell is left, the 7 goes there — even if that cell still has four other candidates written in it. The other candidates were never real. They just had not been ruled out yet.

This is the technique that unlocks medium grids, and it is the one most people are missing when they say a puzzle “got stuck”. A cell can have three candidates and still be forced, because the digit has nowhere else to go. Run it digit by digit: pick a number that already appears five or six times on the board, and check each box that still lacks it. The more copies of a digit are placed, the more constrained the remaining ones are, so the well-populated digits fall first.

The same rule works on rows and columns, not just boxes. “Where in this row can the 2 go?” is just as legal, and on some grids it is the row version that breaks the deadlock.

Technique three: the pair that locks a line

The first two techniques place digits. The third one places nothing. It removes candidates, and the removal is what lets the first two techniques start again. This is the technique that separates people who finish hard grids from people who reach for a guess.

There are two forms, and they are the same idea from two directions.

The pointing pair. Inside one box, find a digit whose only remaining candidates sit in a single row (or a single column). Say the 5s in the top-left box can only be in the top row of that box. Then the 5 for that box is in that row, somewhere in those cells — which means it cannot be anywhere else in the row. Cross off every 5 candidate in the rest of that row, in the other two boxes. You have not placed a 5. You have just made the other boxes simpler, and very often one of them now has an “only place” move waiting.

The naked pair. Find two cells in the same row, column or box that have exactly the same two candidates and nothing else — both say “3 or 8”. Between them those two cells must hold the 3 and the 8, in some order. So no other cell in that region can be a 3 or an 8. Cross them off. Again, nothing is placed, but a cell elsewhere in the region that was “3, 8 or 9” is now simply 9.

Both forms are about the same thing: a digit that is confined to a small set of cells is excluded from everywhere else those cells can see. Once that sentence sits comfortably in your head, you will start spotting the triple versions and the box-line versions without needing their names.

The order that avoids guessing

When you feel the urge to guess, it is nearly always because you skipped a step. Work the loop in this order, every time:

1. Only candidate. Sweep the cells nearest your last placement. Place anything that is down to one.
2. Only place. Take the digits that already appear most often and check each region that lacks them.
3. Lock a line. Look for a digit confined to one row or column within a box, or two cells sharing the same two candidates. Cross off, then go back to step one.

If all three genuinely produce nothing, you have either miscounted a candidate — this is the answer far more often than people like — or you have met one of the rare grids that needs a heavier technique. Newspaper grids and the easy-to-hard range of most apps do not. Recount before you reach for anything else.

Why guessing feels like it works

A guess on a Sudoku often does lead to a finished grid, which is why the habit survives. But there are only two outcomes and neither is solving. If the guess was right, you got there by luck and learned nothing about the position. If it was wrong, you discover it ten cells later, and now you are erasing back to a point you can no longer identify, because you did not mark where the guess began.

The three techniques above are slower per cell and faster per grid, because they never send you backwards. That is the whole argument for them.

The same techniques, with a second person

Sudoku is one of the few puzzles that splits cleanly between two people — the board has regions, and every digit one of you places is a constraint the other can use. Played on a shared grid, the three techniques turn into a division of labour: one of you sweeps for only-candidates around the last few placements while the other hunts a digit through the boxes. The pointing pair is the move that feels best to call out loud, because it is a removal the other person could not see from their side of the grid. If you want the longer case for why that works and for which puzzles it does not, it is in the co-op Sudoku piece and the three shapes a two-player puzzle can take.

Either way, alone or shared: candidates on the board, then only candidate, only place, lock a line. Nothing in a standard grid needs more than that, and nothing in it ever needs a guess.