Sudoku Techniques from Easy to Hard

How to use this page

A unit is a row, a column or a box. Rows are counted from the top and columns from the left. R4C7 means the cell in row 4 and column 7. Boxes are counted from the top left: box 1 covers rows 1 to 3 and columns 1 to 3, box 2 lies to its right, and box 9 is at the bottom right.

A candidate is a digit that may still go into a cell. From the second step on, the techniques do not place digits. They remove candidates, and that makes a single appear somewhere.

Step 1: naked single

Collect the digits in the row, column and box of an empty cell. If eight different digits are there, the ninth goes into the cell.

Example: the row of a cell contains 1, 4 and 7, its column contains 2, 5 and 8, and its box adds 3 and 6. Only 9 is missing, so the cell is 9.

Naked singles are easy to find where a row, column or box is nearly full.

Step 1: hidden single

Choose a unit and a digit that is still missing in it. Rule out every cell of the unit where the digit cannot go. If one cell is left, the digit goes there, even if the cell has other candidates.

Example: box 5 has no 3 yet and four empty cells. Row 4 and row 6 both contain a 3 outside the box, which rules out the empty cells in these rows. Only one empty cell of the box lies in row 5, so the 3 goes there.

This is the technique of scanning: follow one digit across the grid and watch where its rows and columns leave room.

Step 2: locked candidates

Pointing: if all places for a digit inside a box lie in one row, that row’s copy of the digit must lie inside the box. Remove the digit from the candidates of the other cells of that row. The same holds for a column.

Example: in box 1, the digit 7 fits only in R1C2 and R1C3. Row 1 gets its 7 inside box 1, so 7 can be removed from the candidates of R1C4 to R1C9.

Claiming: if all places for a digit inside a row or column lie in one box, remove the digit from the other cells of that box.

Example: in row 5, the digit 2 fits only in R5C7 and R5C9, which are both in box 6. Box 6 gets its 2 from row 5, so 2 can be removed from the other empty cells of box 6.

Step 2: naked pair

If two cells of a unit have the same two candidates and no others, these two digits will fill these two cells. Remove both digits from every other cell of the unit.

Example: in column 4, the cells R2C4 and R8C4 both have the candidates 3 and 6. A third cell, R5C4, has 3, 6 and 9. Since 3 and 6 are used up by the pair, R5C4 is 9.

Step 3: hidden pair

If two digits can go into the same two cells of a unit and nowhere else in it, these cells hold these digits. Remove all other candidates from the two cells.

Example: in row 7, the digits 1 and 8 fit only in R7C2 and R7C5. R7C2 has the candidates 1, 4 and 8, and R7C5 has 1, 8 and 9. Remove 4 from the first cell and 9 from the second. The cells are now a naked pair, and 4 or 9 may have become a single elsewhere in the row.

Hidden pairs are hard to see because the two cells are crowded with other candidates. Look for digits that appear only twice in the notes of a unit.

Step 3: naked triple

If three cells of a unit contain, between them, only three different candidates, these three digits fill these three cells. Remove them from every other cell of the unit. Not every one of the three cells needs all three digits.

Example: in box 9, three cells have the candidates 2 5, 5 7 and 2 7. Together they use only 2, 5 and 7. A fourth cell of the box with the candidates 2, 5 and 6 loses 2 and 5 and becomes 6.

Step 3: X-wing

Find a digit that has exactly two possible cells in each of two rows, and these cells lie in the same two columns. The four cells are the corners of a rectangle. The digit will sit on two opposite corners, so both columns get the digit from these rows. Remove the digit from all other cells of the two columns.

Example: in row 2, the digit 4 fits only in columns 3 and 8. In row 6, it also fits only in columns 3 and 8. Whichever corner row 2 uses, row 6 uses the other column. Remove 4 from the candidates of every other cell in columns 3 and 8. It also works the other way round: two columns with the digit in the same two rows clear it from the rest of those rows.

• X-wing explained, with two practice positions

A routine that works

1. Scan for hidden singles digit by digit, from 1 to 9.
2. Look at nearly full rows, columns and boxes for naked singles.
3. When nothing is left, write notes into all empty cells. On this site, placing a digit then removes it from the notes of its peers.
4. Look for locked candidates in every box, then for pairs in every unit.
5. After every removal, look for singles again (points 1 and 2). One removed candidate often opens a chain of singles.

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Common mistakes

• Incomplete notes. The techniques beyond singles assume that every possible candidate is written down. A missing note creates pairs that do not exist.
• Removing the pair itself. A naked pair removes its digits from the other cells of the unit, not from the two cells.
• Using a pair across units. Two cells form a pair only for a unit that contains both of them.
• An X-wing with three places. If one of the two rows has a third possible cell for the digit, there is no X-wing.