How to Solve Sudoku Step by Step

To solve a sudoku, repeat one loop: scan for a digit that has only one place to go, write it in, and scan again. When the scan finds nothing, write out the candidates in every empty cell and look for pairs, then pointing pairs. This guide is for beginners who know the rules of Sudoku and want a method that always tells them what to do next. Below you get the method in six steps, then a real Easy puzzle solved move by move.

The method in six steps

1. Scan boxes for hidden singles. Pick the fullest box, list the digits it is missing, and for each one find the cells where it can still go. One cell left means the digit goes there.
2. Scan cells for naked singles. A cell whose row, column and box already hold eight different digits takes the ninth.
3. Rescan after every placement. A new digit blocks that digit in 20 other cells, so the next single is usually right beside it.
4. Stuck? Write pencil marks. Note every legal digit in every empty cell, as small candidates.
5. Look for pairs. Two cells in one unit that share the same two candidates, and nothing else, let you strike those digits from the rest of the unit.
6. Look for pointing pairs. A digit confined to one line inside a box can be struck from the rest of that line. Any elimination can create a new single, so go back to step 1.

Guessing is not on the list. Every cell in a well-made puzzle can be proved.

Worked example: an Easy puzzle from our catalogue

This is a classic 9x9 Easy puzzle from our own puzzle set. Sudoku Explainer, the solver we use to rate puzzles, gives it 1.2 (the Easy band runs from 1.0 to 1.9). It has 36 givens, and the engine solves it in 45 steps. Every one of them is a hidden single. Dots are empty cells.

      c1 c2 c3   c4 c5 c6   c7 c8 c9
r1     3  6  5    .  .  2    4  9  .
r2     1  .  .    .  3  4    2  7  .
r3     .  2  .    9  5  .    .  .  1

r4     .  .  3    .  .  .    8  2  .
r5     9  .  2    .  .  6    .  .  .
r6     .  7  .    3  2  .    .  .  4

r7     .  .  1    2  .  3    5  6  .
r8     .  5  .    .  .  7    .  .  .
r9     2  .  9    .  6  .    .  .  7

Cells are written r2c2: row 2, column 2. The first steps, in the order the engine takes them:

1. r2c2 = 9. Box 1 (top left) is missing 4, 7, 8 and 9. The 9 cannot go in r3c1 or r3c3 because row 3 already has a 9 at r3c4. It cannot go in r2c3 because column 3 already has a 9 at r9c3. Only r2c2 is left.
2. r2c4 = 6. In box 2 (top middle) the 6 is blocked in row 1 by the 6 at r1c2, and in r3c6 by the 6 at r5c6 in column 6. That leaves r2c4.
3. r2c9 = 5. In box 3 (top right) the 5 is blocked in row 1 by the 5 at r1c3 and in row 3 by the 5 at r3c5. Only r2c9 can take it.
4. r2c3 = 8. Row 2 now reads 1, 9, _, 6, 3, 4, 2, 7, 5. Only the 8 is missing, so it goes in the last empty cell. This is a hidden single across a whole row, and the engine names it that way.
5. r3c7 = 6. Box 3 still needs 3, 6 and 8. Row 1 already has a 6 and column 8 has a 6 at r7c8, so the only home for the 6 is r3c7.
6. r3c8 = 3. Box 3 now needs 3 and 8. Row 1 has a 3 at r1c1, so the 3 cannot go in the top row of the box, which leaves r3c8.

Steps 1 to 4 solved the whole of row 2 without a single pencil mark. That is the pattern on Easy puzzles: the givens are dense enough that scanning boxes keeps producing placements. The engine's next placements are r1c9 = 8, r3c6 = 8 and r4c9 = 6, and it goes on like this to the end. The finished grid begins 365712498, 198634275, 427958631.

Notice what the method never needed here: pairs, pointing pairs or any guess. Solving by scanning worked because the puzzle is Easy. Harder puzzles reach a point where the scan runs dry, and that is the next section.

What to do when you are stuck

Being stuck means a full sweep of every box, row and column finds no single. Do not guess. Run this loop instead:

1. Write pencil marks. Fill every empty cell with its legal candidates. This is slow once and saves time for the rest of the puzzle.2. Check for naked singles. With candidates written down, any cell showing one digit is a placement.
2. Look for pairs. Two cells in a unit that hold the same two candidates: those digits belong to those two cells, so erase them from the other cells of the unit. Details, plus the hidden version and triples, are in naked and hidden pairs.
3. Look for pointing pairs. If a digit's candidates inside a box all sit in one row or column, erase that digit from the rest of the row or column. The reverse, a line whose candidates all sit in one box, is Box Line Reduction.
4. Return to singles. Each elimination may leave a cell with one candidate or a digit with one home. Sweep again from step 1 of the method above.

If you finish the loop with a full grid of candidates and no pair or pointing pair, the puzzle is above your current toolkit, not broken. Use the tips for beginners to check your pencil marks for mistakes first, since a missed erase hides many real singles.

When to move up a difficulty level

Difficulty in our game is the hardest technique a puzzle needs, measured by Sudoku Explainer: Easy is 1.0 to 1.9, Medium 2.0 to 2.9, Hard 3.0 to 4.9, Expert 5.0 to 6.9. Move up when:

• You finish Easy puzzles by singles alone, with no pencil marks, and without errors.
• Medium puzzles now feel routine, meaning you meet a stuck point, add pencil marks, and get through it with pairs and pointing pairs.
• You are curious about the next technique, not just faster at the current one.

Stay put if you are still guessing or if the same techniques keep failing you. Speed at one level is worth more than a jump to a level whose tools you have not learned. Try a fresh Easy puzzle to check yourself, and when the loop above starts to feel automatic, the Medium level is the next step.