Sudoku solving techniques: from first scan to Conjecture

This is Professor Hoot’s complete guide to sudoku solving techniques, and to the logic behind every other puzzle on the site. It starts with the first scan and builds up to naked pairs, pointing pairs, cage arithmetic, line logic and Latin-square reasoning.

If you have ever wondered how to solve hard sudoku without guessing, the answer is a small toolkit used patiently. Each technique below comes with a worked example or a link to the puzzle guide where it matters most.

Candidates and pencil marks

A candidate is a digit that could still go in a cell. Logical solving is the art of removing candidates until only one remains. The game supports two styles of pencil mark: corner marks, conventionally used for “this digit goes in one of these few cells of the house”, and centre marks, used for “this cell is one of these few digits”. You don’t need to fill every cell with marks; mark where it helps you think.

Every puzzle on Hoot’s Puzzles has exactly one solution and can be solved by logic alone. That matters: if you are stuck, there is always a step you haven’t found yet, and guessing is never required.

Singles: the first sudoku solving techniques

Hidden single

Pick a house (a row, column or box, or any extra region a variant adds) and a digit. If only one cell in the house can take that digit, it goes there. Hidden singles are found by scanning: each copy of the digit elsewhere rules out its row, column and box.

Naked single

Pick a cell. If every digit but one is ruled out by the cell’s houses, the last one goes there. Naked singles are easiest to see once you have centre marks.

Between them, singles solve most easy puzzles, and every Lemma on the site can be finished with singles plus the variant’s own rule.

Pointing pairs and claiming

When a box’s candidates for a digit all lie in one row or column, the digit must be in that line within that box, so it can be removed from the rest of the line. This is a pointing pair (or pointing triple, with three cells).

Claiming, also called box/line reduction, is the mirror image: when a row’s or column’s candidates for a digit all lie inside one box, the digit can be removed from the rest of that box. Proposition-rank puzzles may call for these moves.

Naked pairs, hidden pairs and triples

Naked pairs

If two cells in a house have exactly the same two candidates, those two digits must go in those two cells, in some order. Remove both digits from every other cell in the house.

Hidden pairs

If two digits can only go in the same two cells of a house, those cells must hold those digits, and every other candidate can be removed from them. Hidden pairs are naked pairs seen from the other side, and they are often easier to spot with corner marks.

Triples

The idea extends to three cells and three digits. A naked triple does not need every cell to show all three digits: cells with {1, 2}, {2, 3} and {1, 3} form a triple, because together they use exactly three digits. Theorem and Conjecture puzzles may need pairs; triples are a useful extra eye.

Cage arithmetic and the rule of 45

Every row, column and box of a 9×9 sudoku contains 1 to 9, which add up to 45. In killer sudoku, subtract the cages lying wholly inside a house from 45 to get the total of the cells left over. A single leftover cell inside the house is an innie; a cell of a cage that pokes out is an outie. The rule scales: two houses total 90, three total 135.

Partition arithmetic means thinking about which sets of digits can make a total. A two-cell cage of 10 can be 1 + 9, 2 + 8, 3 + 7 or 4 + 6, but never contains a 5. Small and large totals are often forced; the full list is in the combinations chart below. The same thinking powers Killer, Little Killer, Sandwich, Arrow, Calcudoku and Kakuro.

Parity

The digits 1 to 9 contain five odds and four evens. That simple count goes a long way. A two-cell cage with an odd total must hold one odd and one even digit. In XV sudoku, an X (sum 10) joins two odds or two evens, while a V (sum 5) always joins one of each. In Chimera puzzles, parity shading makes it explicit: a grey square is even, a grey circle odd. And since a full house holds five odd digits, if four cells of a row are known to be even, the other five are all odd.

Line logic: thermometers, arrows, whispers and more

Thermometers. On a thermometer of L cells, the k-th cell from the bulb lies between k and 9 − L + k. Any digit landing on or next to a thermometer tightens every cell on it.

Arrows. The circle is at least the smallest possible sum of its arrow and at most 9. Arrow cells that share a house must differ, so three of them sum to at least 6, and four can’t all differ.

German whispers. Neighbours differ by at least 5, so 5 never appears, the line alternates low (1–4) and high (6–9), a 4 can only touch 9 and a 6 can only touch 1.

Renban. A line of L cells is a run of L consecutive digits. Five-cell lines must contain 5; six-cell lines must contain 4, 5 and 6.

Between lines. Interior digits are never 1 or 9, and the circles must be far enough apart to fit every interior digit that shares a house.

Palindromes. Mirrored cells hold the same digit, so they can’t share a house, and any candidate removed from one twin is removed from the other.

Dots and grid-wide rules have their own patterns: black Kropki dots allow only 1–2, 2–4, 3–6 and 4–8; non-consecutive cells push neighbours away; anti-knight adds eight more cells to every scan; Sudoku X and Windoku add extra houses, and Windoku’s four windows quietly create five hidden regions more.

Latin-square reasoning

Calcudoku, Skyscrapers and Futoshiki are Latin squares: an N×N grid where each row and column contains 1 to N once, with no boxes. Singles, pairs and triples work exactly as in sudoku, using rows and columns only. Some extra tools:

• Row totals. Each row and column adds up to N(N + 1)/2 and multiplies to N!: 15 and 120 on a 5×5, 21 and 720 on a 6×6, 28 and 5,040 on a 7×7, 36 and 40,320 on an 8×8. In Calcudoku that finds the odd cell out, like the rule of 45.
• Prime factors. In a × cage, a target divisible by 5 needs a 5 and one divisible by 7 needs a 7 (on grids up to 8×8).
• Visibility bounds. In Skyscrapers, the cell k steps from a clue c is at most N − c + k, and opposite clues add to at most N + 1.
• Inequality chains. In Futoshiki, a rising chain of k cells starts at most N − k + 1 and ends at least k.
• Every digit in every line. Because each row needs all of 1 to N, ask “where does the N go in this row?” as often as “what goes in this cell?”.

Kakuro and killer combinations chart

These totals can be made in exactly one way from distinct digits 1 to 9. They apply to Kakuro runs, Killer sudoku cages, and any other region whose digits can’t repeat. For two cells: 3 is always 1 + 2, 4 is 1 + 3, 16 is 7 + 9 and 17 is 8 + 9. For three: 6 is 1 + 2 + 3, 7 is 1 + 2 + 4, 23 is 6 + 8 + 9 and 24 is 7 + 8 + 9. For four: 10 is 1 + 2 + 3 + 4, 11 is 1 + 2 + 3 + 5, 29 is 5 + 7 + 8 + 9 and 30 is 6 + 7 + 8 + 9.

Notice the symmetry: a run of n cells totalling t is unique exactly when a run of 9 − n cells totalling 45 − t is, because the two sets are complements. Every entry was generated by checking all sets of distinct digits by computer.

How to solve hard sudoku when you’re stuck

1. Re-read the rules beside the board, especially in a Chimera. A forgotten rule is the most common cause of being stuck.
2. Do a clean scan for hidden singles in every house, including diagonals, windows and hidden regions where the variant has them.
3. Check your pencil marks are up to date. A stale mark can hide a single or fake a pair.
4. Look for pointing pairs and claiming in each box.
5. Look for naked and hidden pairs, then triples.
6. Revisit every clue mark: cages near 45, thermometers near a newly placed digit, dots and missing dots.
7. Ask Professor Hoot. The hint shows the next logical step and explains why it works.

Guides to every puzzle type

• Classic SudokuRows, columns and boxes, nothing else. Pure logic at its cleanest.
• Killer SudokuCages with sums instead of given digits. Partition arithmetic and the rule of 45.
• Thermo SudokuDigits strictly increase from bulb to tip. Count the cells, bound the ends.
• Arrow SudokuThe circled digit equals the sum along its arrow. Bound circles and arrows.
• Kropki SudokuWhite dots for consecutive pairs, black for a 1:2 ratio. Missing dots matter.
• XV SudokuX marks a pair summing to 10, V a pair summing to 5. Every one is shown.
• Sandwich SudokuOuter clues sum the digits trapped between the 1 and the 9.
• German WhispersNeighbours on a green line differ by at least 5. Fives are impossible there.
• Sudoku XBoth main diagonals are extra houses. Also called diagonal sudoku.
• Anti-Knight SudokuNo digit repeats a chess knight’s move away.
• Renban SudokuPurple lines hold runs of consecutive digits, in any order.
• Palindrome SudokuGrey lines read the same from either end. Twin cells share a digit.
• Between LinesEvery digit on a line sits strictly between its two circled ends.
• Little Killer SudokuOutside clues add up the diagonals their arrows point along.
• WindokuFour shaded windows are extra 3×3 houses. Also known as hyper sudoku.
• Non-Consecutive SudokuSide-by-side cells never hold consecutive digits.
• Chimera SudokuTwo or three rule sets layered into one puzzle. The daily puzzle.
• CalcudokuA Latin square with arithmetic cages: + − × ÷. Sizes 5×5 to 8×8.
• KakuroA numerical crossword: runs of distinct digits 1–9 that hit their sums.
• SkyscrapersEdge clues count the buildings you can see. Taller ones hide shorter.
• FutoshikiA Latin square with < and > signs between neighbouring cells.