Sudoku Solving Techniques, From Easy to Hard

Rows are numbered 1 to 9 from the top and columns 1 to 9 from the left, so r1c6 means row 1, column 6. Large black digits are the givens. Small gray digits are pencil marks, also called candidates: the digits still possible in that cell.

Scanning is where every solve starts. Pick a digit, then pick a box that does not have it yet. Look along the rows and columns that pass through the box. Every copy of your digit in those lines rules out a whole row or column of the box. If only one cell survives, the digit goes there.

Work through the digits 1 to 9 in order and sweep every box for each one. It sounds slow, but your eyes learn the motion quickly. Digits that already appear six or seven times on the board are the fastest wins.

Scanning asks where a digit can go. A naked single asks the opposite: which digit can go in this cell? Collect every digit in the cell's row, column and box. If eight different digits appear, the ninth is the answer.

Naked singles hide in crowded areas, where a row and a column with lots of givens cross. Check the cells there when scanning stops finding anything.

A hidden single is scanning applied to a row or a column instead of a box. A cell can have several candidates and still be solved, because it is the only place in its row or column where one digit fits.

Scanning, naked singles and hidden singles together solve most easy and many medium puzzles. Our easy puzzles start with 43 givens, so you can get a long way with these three alone.

Pencil marks are for when singles run dry. If you have swept every digit through every box and found nothing, it is time to write down the candidates for the empty cells. Every technique below works on pencil marks, so they are worth the effort.

• Fill them in one box at a time, checking the row and column of each cell. Doing it carefully once beats fixing mistakes later.
• Keep them current. Every time you place a digit, erase it from the pencil marks of its row, column and box. Most errors come from stale marks.
• Look for cells with two candidates. They feed pairs, triples and X-Wings, and they are the first cells to collapse into singles.
• Do not start too early. On an easy puzzle, marking everything takes longer than scanning. Save it for medium and hard.

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. You do not know which way round yet, but you know no other cell in that unit can use them. Erase both digits from the rest of the unit.

A naked triple is the same idea with three cells sharing three digits. Each cell does not need all three. In the next example the cells hold 1 and 9, 1 and 5, and 5 and 9. Together they use exactly 1, 5 and 9, so those three digits are locked inside them.

Notice how one elimination often creates a single somewhere else. That chain reaction is how hard puzzles open up.

Now the techniques work across two units at once. Look at one digit inside one box. If all its candidates in that box sit in the same row or column, the digit has to be in that line, inside this box. So it cannot be anywhere else in that line. The pair "points" along the line and clears it.

The same works with three cells in a line, sometimes called a pointing triple. Showing marks for a single digit, as here, is the easiest way to spot it.

Box-line reduction is the pointing pair turned around. Look at one digit in a row or column. If all its candidates in that line fall inside one box, the line's copy of that digit must be in that box. That fills the box's need for the digit, so erase it from the rest of the box.

A simple way to remember the pair: pointing goes from box to line, box-line reduction goes from line to box. Check both directions for every digit before reaching for anything harder.

The X-Wing is the first advanced technique most solvers learn. Find a digit that fits in exactly two cells in one row, then another row where it also fits in exactly two cells, in the same two columns. The four cells form a rectangle.

Each row needs the digit once, so it goes in two opposite corners of the rectangle. Either way, each of the two columns gets its copy from those corners. That means the digit can be erased from every other cell in both columns.

It works with columns and rows swapped too: two columns where a digit fits only in the same two rows clears those rows. Hunt for X-Wings one digit at a time, starting with rows that have exactly two marks for it.

1. Scan every digit through every box until nothing new appears.
2. Check crowded rows and columns for naked and hidden singles.
3. Fill in pencil marks for the whole grid.
4. Look for naked pairs and triples, then pointing pairs and box-line reductions.
5. Try an X-Wing on each digit.
6. After every placement, go back to step 1. Simple moves are faster to spot and usually reappear.

Every puzzle in our game has exactly one solution, so it can always be solved by logic. If you are stuck, a pencil mark is probably wrong or missing. Practice the routine on the daily Sudoku, which is the same medium puzzle for everyone, then step up to Hard in Sudoku.