Sudoku Techniques From Beginner to Expert, in the Order You Need Them

The vocabulary and the ladder

A house is any row, column or 3x3 box; each must hold 1 to 9 once. A cell's candidates are the digits it could still take, written small in the corner as pencil marks. Cells are named by row and column, so r4c7 is row 4, column 7. Two cells see each other when they share a house.

Every technique below does one of two things: places a digit, or removes a candidate. Elimination techniques never place anything directly, but each removal brings a single closer.

The table shows the usual order of difficulty. Always try the easier techniques first; they are faster to spot and solve most of the grid.

Scanning

Scanning, also called cross-hatching, needs no pencil marks. Choose a digit, say 7, and a box that does not yet contain it. Look along every row and column that passes through the box. Each 7 already in the grid rules out its whole row and column inside your box.

If only one empty cell in the box survives, the 7 goes there. Sweep each digit from 1 to 9 across all nine boxes, starting with the digits that already appear most often; they rule out the most cells.

Use scanning at the start of every puzzle and every time you place a new digit. Easy puzzles are often finished with nothing else.

Naked singles and hidden singles

A naked single is a cell with only one candidate left. Look at everything its row, column and box already contain; if eight digits are taken, the ninth is the answer. The quickest case is the last empty cell of a house.

A hidden single is a digit with only one possible cell in a house. The cell itself may still have several candidates, but if 4 can go nowhere else in row 6, that cell is the 4. Scanning is simply a fast way of finding hidden singles in boxes.

Check rows and columns for hidden singles, not just boxes. They are easy to miss there. Once you start writing candidates, scan the pencil marks of each house for a digit that appears only once.

Pointing pairs and triples

Look inside one box at where a digit can go. If all its candidates in that box lie in the same row, the box's copy of that digit must be in that row. So it cannot appear anywhere else in that row, outside the box.

Example: in box 1, the digit 5 is possible only in r2c1 and r2c3. Whichever is right, row 2's 5 is inside box 1. Remove 5 from r2c4 to r2c9. The same works for columns, and with three cells instead of two.

Reach for pointing as soon as scanning dries up. It is the most common step in medium puzzles, and its eliminations often create a hidden single in a neighbouring box.

Box-line reduction

This is pointing in reverse. Look along a row or column at where a digit can go. If all its candidates in that line fall inside one box, the line's copy of the digit is in that box, so the digit cannot sit in the box's other cells.

Example: in column 8, the digit 2 is possible only in r4c8 and r6c8, both in box 6. Remove 2 from every other cell of box 6.

Use it alongside pointing. Some solvers call the pair of techniques locked candidates; checking both directions catches eliminations that one alone would miss.

Naked pairs, triples and quads

If two cells in a house both have exactly the candidates {3,8}, one is 3 and the other is 8. No other cell in that house can be 3 or 8, so remove those candidates from the rest of the house.

The idea extends: three cells whose candidates together use only three digits form a naked triple, and four cells using only four digits form a naked quad. The cells do not each need every digit. The triple {1,4}, {4,9}, {1,9} is valid because together they use just 1, 4 and 9.

Pairs show up often in intermediate puzzles, triples in hard ones, and quads are rare. Look for them when a house has many pencil marks and singles have run out.

Hidden pairs, triples and quads

Now flip the view from cells to digits. If two digits can each go in only the same two cells of a house, those cells must hold those two digits, and every other candidate in those two cells can be erased.

Example: in row 3, the digits 6 and 7 appear only in r3c2 and r3c5. Those cells may show long candidate lists, but they are now {6,7}. Hidden triples and quads work the same way with three or four digits confined to as many cells.

Hidden subsets are harder to see than naked ones because the extra candidates disguise them. A useful fact: in any house, a naked subset and a hidden subset come in matching pairs. If a house has seven empty cells containing a naked triple, the other four cells form a hidden quad. When one is hard to see, look for its partner.

X-Wing and Swordfish

X-Wing is the first technique that links houses far apart. Find a digit that has exactly two candidate cells in each of two rows, with those cells in the same two columns. Say digit 4 is possible only in c3 and c7 of row 2, and only in c3 and c7 of row 6.

Row 2's 4 is in c3 or c7, and row 6's 4 must take the other column, because two 4s cannot share a column. Either way, columns 3 and 7 each get their 4 from rows 2 and 6. Remove 4 from every other cell of columns 3 and 7. The pattern works just as well with columns and rows swapped.

Swordfish is the same idea across three rows and three columns. Find three rows where a digit's candidates all fall within the same three columns, with two or three candidates per row. The three rows supply the digit to all three columns, so remove it from those columns everywhere else.

Look for these fish when one digit stubbornly has many candidates left and nothing simpler works. Filter your view to that single digit; the rectangle or the three-by-three pattern stands out much more clearly.

XY-Wing

XY-Wing uses three cells that each have exactly two candidates. The pivot holds {X,Y}. One pincer sees the pivot and holds {X,Z}. Another pincer also sees the pivot and holds {Y,Z}. The pincers do not need to see each other.

If the pivot is X, the first pincer becomes Z. If the pivot is Y, the second pincer becomes Z. One of the two pincers is Z either way, so any cell that sees both pincers cannot be Z.

Example: the pivot r5c5 is {2,7}. In the same row, r5c1 is {2,9}. In the same column, r2c5 is {7,9}. The cell r2c1 shares a row with r2c5 and a column with r5c1, so it sees both pincers. Remove 9 from r2c1.

Try XY-Wing when the grid is full of two-candidate cells. It is usually easier to find than a Swordfish, which is why many solvers learn it first.

A routine that keeps you moving

Speed in sudoku comes from always trying the cheapest step first. After every placement, go back to scanning; a new digit almost always creates another single nearby.

If you want to drill one technique at a time, practice puzzles built around it help more than random grids. Nonoku, our free offline sudoku and nonogram app for Android, has hints that name and explain the technique behind each next step, from naked singles to quads and beyond.

• Scan every digit across every box.
• Fill in pencil marks and look for naked and hidden singles in rows, columns and boxes.
• Check pointing pairs and box-line reduction for each digit.
• Look for naked pairs, then hidden pairs, then triples and quads.
• Filter by digit and search for X-Wing and Swordfish.
• Scan two-candidate cells for an XY-Wing.
• Beyond this lie XYZ-Wing, Jellyfish and chains. Most published puzzles never need them.