How to Solve Sudoku Without Guessing

A stalled Sudoku grid is usually a missed check, not an invitation to guess. The workflow below is an order: scan the units, write candidates, place singles, then delete candidates with pairs, locked lines, fish, and a short chain. Each step earns the next one. After every placement or deletion, scan the touched units again.

This page is the order of work under the complete guide to how Sudoku works, which states the row, column, and box rules. Each named pattern has its own page, linked from the section where it belongs. Patterns past this workflow are collected in 9 advanced Sudoku techniques.

Quick answer: Place naked singles and hidden singles first. When nothing is forced, delete candidates with naked pairs, hidden pairs, naked triples, pointing pairs, and box-line reduction. Only then look for an X-Wing, a Swordfish, or an XY-Wing.

Scan before you fill in pencil marks

A scan asks one question of one unit: “Where can 4 go in this row?” Do that before you cover an easy grid with tiny numbers.

r4c7 means row 4, column 7. Boxes are numbered 1–9, left to right and top to bottom. A unit is one row, one column, or one box. Every cell sits in three units, and a digit has to be legal in all three.

1. For each digit from 1 to 9, mark the units that already contain it.
2. In a unit that still lacks it, list empty cells that do not see a placed copy.
3. One cell left means you place the digit.
4. Two or three cells on one line inside a box are the seed of a pointing pair or a claim.

After a placement, rescan the three units you just changed. A new 6 in r2c2 deletes 6 from row 2, column 2, and box 1. Easy and Medium grids are full of these direct placements. Hard hides them between deletions.

Candidates and pencil marks

A candidate is a digit not yet ruled out of a cell. Pencil marks are that list. They are not a guess. Write them for cells a direct scan cannot solve, in a fixed order: 1 2 3, then 4 5 6, then 7 8 9. Notes mode does the same job. Pairs and fish are patterns in those marks.

When you place 4, delete 4 from the rest of its row, column, and box. When a technique deletes 4 without a placement, delete only that mark. Erase a mark when you can name the unit and the pattern. If you cannot name them, the mark stays.

A bivalue cell has two candidates, such as {2, 7}. A bilocal digit has exactly two cells left in one unit. Bivalue cells feed wings. Bilocal digits feed pointing, X-Wings, and strong links.

Singles: the only placements this order needs

A placement is allowed when a cell has one candidate, or when a digit has one home in a unit. Everything else on this page only deletes candidates so that one of those two events can happen.

A naked single is a cell whose list has shrunk to one digit. In the fragment below, column 4 already holds every digit except 4, and row 4’s other cells are filled with everything except 4. r4c4 is {4}. Place it.

A naked single can also sit among other empty cells. You place it because that cell’s list has one digit, not because the row is full.

A hidden single is a digit with one remaining cell in a unit, even if that cell still lists other digits. In this row, 4 appears only in r1c7. The cell also allows 8, but 4 has nowhere else in the row, so r1c7 is 4 and 8 comes off that cell.

Stale marks invent fake singles, which is why the scan comes first. Use singles until they stop before you keep a pair “for later.”

Pairs and triples

When singles dry up, look for a small set of cells that has reserved a small set of digits.

A naked pair is two cells in one unit whose lists are exactly the same two digits. In row 3, r3c2 and r3c7 are both {2, 9}. Those cells will be 2 and 9 in some order, so every other 2 or 9 in the row is impossible. r3c4 is left with {5}, a new naked single. r3c8 is left with {3, 5}.

If those two cells also share a box, delete the pair from both units. Leave the order unknown.

A hidden pair is the inward cut: two digits live in only two cells, and you delete the extra candidates from those cells. A naked triple is three cells whose candidates use only three digits. {1, 4}, {1, 8}, and {4, 8} in one row qualify, even though no cell holds all three. Delete those digits from the other cells in the unit. Take a pair before a triple when both are present. The pair is smaller.

Pointing and claiming

Locked candidates delete marks. They do not place a digit by themselves. A pointing pair or triple starts in one box: if a digit’s remaining cells in the box share a row or column, delete it from the rest of that line, outside the box. Box 1 below holds 6 only in r1c1 and r1c2, so r1c6 and r1c8 lose 6.

Box-line reduction, or claiming, runs the other way. If every remaining cell for a digit on a line sits inside one box, delete the digit from the rest of that box. Row 8’s 5s all sit in columns 7 and 9, which are in box 9. r8c8 is in that box and does not contain 5; a claim does not need every cell of the band to hold the digit. Box 9 cannot also place 5 in r7c8 or r9c7.

Pointing clears the line outside the box. Claiming clears the box off the line. Do both before any fish.

Fish

A fish locks one digit across more than one line. An X-Wing is two rows that each hold the digit in the same two columns, or the column version of that rectangle. Those columns lose the digit outside the two rows. Rows 2 and 6 below each contain 7 only in columns 3 and 8, so r4c3 and r9c8 lose 7. r4c5 keeps 7: column 5 is not in the fish.

A Swordfish uses three rows and three columns. A base row may use two or three of those columns, so empty corners are allowed. A fourth column in a base row breaks it. Delete the digit from the three columns outside the three rows. Jellyfish and finned fish are in the advanced techniques guide. Check one digit at a time. A rectangle of mixed candidates is not an X-Wing.

Chains

A chain is a sequence of links you can point at. A strong link means at least one end is true: a digit with exactly two cells in a unit, or the two candidates in a bivalue cell. A weak link means both ends cannot be true, which is any two candidates in one cell, or two cells in one unit that both might hold the same digit.

An XY-Wing is the short chain to learn first. The pivot is {X, Y}. One wing the pivot sees is {X, Z}. The other wing the pivot sees is {Y, Z}. One of those wings must be Z, so any other cell that sees both wings loses Z.

Here X is 2, Y is 7, and Z is 9. r4c9 contains 9 but does not see r7c4, so that 9 stays. The wings do not have to see each other. Coloring, XY-chains, and alternating inference chains are the same links on a longer path, in the advanced techniques guide. Make the deletion, then go back to singles.

When each technique applies

Skip a step only when you are sure that pattern is absent.

Return to the top after every deletion. The pointing pair above is the whole method in miniature: deleting 6 from r1c8 leaves {7}, a naked single. Fish and chains filter candidates. Singles make the placements.

Where to practice

Easy and Medium puzzles make singles and pencil marks automatic. Hard is where pointing, claiming, and pairs keep a puzzle moving. Expert, Master, and Extreme assume those tools and add fish and chains. The daily challenge is one shared puzzle per UTC day when you want a single grid instead of a stream of new ones.

The Sudoku solver can confirm a placement you have already justified. Use it after you can name the technique. If the solver is the first thing that finds the digit, you skipped a scan. None of these techniques adds a fourth rule to how Sudoku works. Each one is a consequence of rows, columns, and boxes containing 1–9 once.

Frequently asked questions

Can every Sudoku be solved without guessing?

A puzzle with one solution can be solved by deductions, sometimes deeper than singles and pairs. A grid with two solutions cannot, once the shared steps run out, because both endings are legal. On a published puzzle, walk the table above before you test a cell.

Should every pencil mark go in at the start?

Mark the cells a direct scan cannot solve, and keep the marks exact. A missing candidate hides a single. An extra candidate invents a pair.

What comes after XY-Wing?

Re-scan for singles, learn Swordfish once X-Wing is easy to see, then use the advanced guide for coloring and longer chains. Unique rectangles assume one solution, so they are a separate family.

Is erasing a trial digit a chain?

A trial with no link is a guess. A chain follows a strong link, such as the only two cells for 7 in a row, and deletes only what both branches forbid. If you cannot point at every step, it is not ready.

Open a medium puzzle or today’s daily challenge and run one pass in this order. When the grid stops, name the last pattern you finished. That name is the next page to open.