Sudoku Strategies: From First Puzzle to X-Wing, Step by Step

The most useful Sudoku strategies, in the order you should reach for them, are: cross-hatching to find hidden singles, naked singles, pencil marks, naked and hidden pairs, pointing and claiming, and finally the X-Wing. Easy puzzles need only the first two. Each technique below comes with a small worked example, so you can see exactly why it works.

How to read the examples

Rows are numbered 1 to 9 from top to bottom and columns 1 to 9 from left to right, so r3c1 means row 3, column 1. A . is an empty cell. Candidates (the digits that could still go in a cell) are written in braces, like {2,5}.

1. Cross-hatching (finding hidden singles in a box)

Cross-hatching is the first technique every solver uses. Pick a digit and a box, then "draw lines" through the rows and columns that already contain that digit. If only one empty cell in the box is left uncrossed, the digit goes there.

Here is the top band of a puzzle (rows 1 to 3):

       c1 c2 c3   c4 c5 c6   c7 c8 c9
  r1    .  .  .    5  .  .    .  .  .
  r2    .  .  .    .  .  .    5  .  .
  r3    .  7  .    .  .  .    .  .  .

Where can 5 go in the top-left box?

1. Row 1 already has a 5 (at r1c4), so none of r1c1, r1c2 or r1c3.
2. Row 2 already has a 5 (at r2c7), so none of r2c1, r2c2 or r2c3.
3. That leaves row 3. r3c2 is taken by the 7, so the 5 is in r3c1 or r3c3.
4. Now suppose column 3 already has a 5 lower down, say at r6c3. That rules out r3c3.

Result: the 5 goes in r3c1.

Run this check for every digit in every box and you will solve most easy puzzles without writing a single note.

2. Naked singles

A naked single is a cell that has only one possible digit left.

Take r5c2. Say:

• its row already contains 1, 3 and 8,
• its column already contains 2, 4 and 9,
• its box already contains 5 and 6.

That's eight different digits ruled out. The only digit left is 7, so r5c2 = 7.

Naked singles are easy to miss without notes, because you have to look in three directions at once. That's where pencil marks help.

3. Pencil marks (candidates)

When scanning stops finding numbers, write the candidates in each empty cell: every digit that doesn't already appear in that cell's row, column or box. On Poozee, switch Notes on (or press N) and tap digits to add them.

Pencil marks turn the puzzle into something you can read instead of something you have to hold in your head. Every technique from here on works by looking at candidates.

Pro tip: Keep notes up to date. When you place a digit, remove it from the candidates of every cell in the same row, column and box. Poozee does this automatically when your number is correct.

4. Hidden singles

A hidden single is a digit that can only go in one cell of a row, column or box, even though that cell has other candidates too.

  Row 4 candidates:
  c1 {2,6}  c2 {2,6,9}  c3 [8]  c4 {1,2}  c5 [5]  c6 {1,6}  c7 [3]  c8 {1,2,4,6}  c9 [7]

([8] means the cell is already filled with 8.) Look for the digit 4: it appears as a candidate only in c8. Every row needs a 4, so r4c8 = 4, even though that cell also lists 1, 2 and 6. Look again and you'll spot a second one: 9 appears only in c2, so r4c2 = 9.

5. Naked pairs

If two cells in the same row, column or box have exactly the same two candidates, those two digits must go in those two cells, in some order. So they can be removed from every other cell in that unit.

  Row 3 candidates:
  c1 {1,2,4}  c2 {1,2}  c3 {4,5}  c4 {3,6}  c5 {1,2}  ...

Cells c2 and c5 are both {1,2}. One is 1 and the other is 2, so no other cell in row 3 can be 1 or 2.

• c1 goes from {1,2,4} to {4}, a naked single: r3c1 = 4.
• That removes 4 from c3, which goes from {4,5} to {5}: r3c3 = 5.

One pair, two solved cells. Naked triples work the same way with three cells whose candidates, together, contain only three digits.

6. Hidden pairs

A hidden pair is the mirror image. If two digits appear as candidates in only the same two cells of a unit, those cells must hold those two digits, so every other candidate in those two cells can be erased.

  Box candidates (only the cells that mention 3 or 7):
  A {1,3,7,8}   B {3,5,7}

In this box, 3 and 7 appear only in cells A and B. So A and B must be 3 and 7. Erase the rest: A becomes {3,7} and B becomes {3,7}. That often creates a naked pair you can use elsewhere.

7. Pointing pairs and claiming

These two techniques use the overlap between a box and a row or column.

Pointing: if a digit's candidates inside a box all sit in one row (or one column), the digit must be in that row within the box. So it can be erased from the rest of that row outside the box.

  Top-left box, candidates for 5 marked with *:
       c1 c2 c3
  r1    .  .  .
  r2    *  *  .
  r3    .  .  .

The 5 for this box must be r2c1 or r2c2, both in row 2. So no other cell in row 2 (c4 to c9) can be 5.

Claiming (also called box-line reduction) is the reverse: if a digit's candidates in a row (or column) all sit inside one box, then the digit must be in that part of the box. So it can be erased from the other cells of that box.

For example, if the only places for 8 in row 5 are r5c4 and r5c6 (both in the middle box), then the middle box's 8 is in row 5, and you can erase 8 from the middle box's cells in rows 4 and 6.

8. X-Wing

X-Wing is the first "advanced" technique most solvers learn, and it is less scary than it sounds.

Look for a digit that has exactly two candidate cells in one row, and exactly two in another row, in the same two columns:

  Candidates for 3 (marked *):
          c4   c7
  r1       *    *     <- the only places for 3 in row 1
  r6       *    *     <- the only places for 3 in row 6

Row 1's 3 is in c4 or c7. Row 6's 3 is in c4 or c7. They can't both be in the same column, so one row takes c4 and the other takes c7. Either way, columns 4 and 7 each get their 3 from rows 1 and 6.

Result: erase 3 from every other cell in column 4 and column 7.

The same pattern works with columns and rows swapped: two columns where a digit has exactly two candidates, in the same two rows, lets you erase it from the rest of those rows.

A solving routine that works

1. Cross-hatch every digit, 1 through 9, box by box.
2. Fill any naked singles you can see.
3. When you get stuck, add pencil marks to every empty cell.
4. Look for hidden singles in each row, column and box.
5. Look for naked pairs and hidden pairs, and erase what they allow.
6. Check for pointing and claiming.
7. Try an X-Wing on the digits with the fewest candidates.
8. After every placement, go back to step 1. New numbers create new singles.

When even X-Wing isn't enough

The very hardest puzzles need bigger patterns: Swordfish (an X-Wing across three rows and three columns), XY-Wing, and various chains that follow "if this cell is X, then that one must be Y" through the grid. On Poozee, every puzzle is graded by the techniques it needs: Easy and Medium can always be solved with singles alone, Hard needs pairs, pointing or claiming, and Expert needs something beyond those, such as an X-Wing, a triple or a chain.

Keep playing

The fastest way to get comfortable with these techniques is to use them. Start a Sudoku on Medium, turn on Notes, and look for one naked pair per puzzle. When that feels easy, move up to Hard and hunt for pointing pairs. For a different kind of logic workout, try Minesweeper and our guide to Minesweeper patterns, or learn how to solve nonograms, picture puzzles you solve one row or column at a time.