Solving techniques · An interactive read
How to Solve Sudoku Without Guessing: Make Every Move Explainable
Distinguish a legal candidate from a proved answer, use a concrete recovery routine, and understand how a contradiction can justify an elimination.
The idea to take away
A candidate that fits locally is not automatically the answer. Make a placement only when a valid argument rules out its alternatives.
Two plausible numbers are still two possibilities
A row is missing 4 and 9. Its two empty cells, c2 and c5, both allow {4,9}. The first diagram shows the two orders: 4 then 9, or 9 then 4. Both satisfy that row. Nothing in the row alone chooses between them.
You might find another restriction in a crossing column or box. Until then, entering the number that looks more convenient is a guess. This does not mean it is necessarily wrong; it means your current explanation has not proved it.
The diagram shows two compatible rows, not two full solutions of a particular Sudoku. A full board can contain information that distinguishes them. Keeping the scope of an example explicit prevents a small ambiguity from being mistaken for a claim about the whole puzzle.
Ask whether the conclusion survives every allowed case
A sound placement is true in every completion compatible with the puzzle and your valid deductions. A sound elimination removes a candidate that cannot occur in any such completion. In practice, a small local argument can often establish this without listing complete grids.
Norvig's solver explanation separates candidate reduction from the search needed when its elementary deductions stall. [1] That distinction is useful for human solving too: a surviving candidate is not automatically a confirmed value.
For the two-cell row, both cases reserve digits 4 and 9 for those cells, but neither fixes their order. State the conclusion at exactly that strength. A correct reservation becomes an incorrect placement if you quietly add a preference that the proof never supplied.
Name the premise, target and reason
Before deleting a note, identify the facts doing the work. For a naked pair, are the two complete lists restricted to the same two digits? Do both cells share the house containing the proposed target? For a hidden single, have you checked every other possible location for that digit in the house?
Then say what changes: a value is placed, or a particular candidate is removed. Those actions are not interchangeable. A proof that a cell cannot be 4 may still leave 2, 7 and 9.
Finally, identify what would break your argument. An extra candidate can break a naked pair; an extra location can break a hidden single. This countercheck makes a half-seen pattern easier to catch before it affects later moves.
Make a bounded pass instead of filling a cell for motion
First verify recent answers and note cleanup. Next inspect the most constrained cells and houses for singles. If none appear, look for one candidate relationship you understand, such as a pair or a box-line interaction. After a justified elimination, return to the changed area.
This is a suggested search routine, not a theorem that those techniques solve every puzzle. Stuart's strategy-family index includes many different forms of deduction. [3] A stalled pass may mean that you missed something, that your notes need repair, or that the puzzle needs a technique you have not learned.
A hint that explains one move can help distinguish those situations. Try to reconstruct its premise and target before continuing. Seeing the finished grid alone does not tell you which part of your search was missing.
Reasoning with an assumption is not the same as keeping a guess
Suppose two cells in one row are A {4,9} and B {4}. Assume A were 4. Then B could not be 4 because the row would repeat it, leaving B with no candidate. That contradiction rules out A = 4; A must be 9.
The second diagram makes each step visible. Here, placing the naked single B = 4 is simpler. The assumption example is deliberately elementary so the difference between a temporary premise and a concluded answer is clear.
HoDoKu's forcing-chain discussion describes deductions obtained from contradictions or consequences shared by alternative cases. [2] Such an argument needs valid intermediate implications. Merely trying a number and continuing as though it were established is a different process; a later mistake does not automatically prove the original assumption false.
Separate puzzle guarantees from your current toolkit
A unique solution does not promise an easy route using your favourite techniques. Conversely, difficulty in finding a move does not establish that the puzzle has multiple solutions. Those are different questions: how many completed grids fit, and which deductions you can currently recognise.
This guide makes no blanket claim that every ZUDOKU puzzle has a particular human solving path. If an imported or transcribed puzzle seems inconsistent, check its givens before treating the problem as a lack of skill.
Try the quiz, then return to the strategy roadmap for a practical search order. The aim is to keep the distinction between possibility and proof visible, including when the honest next step is to seek an explanation.
Go to the evidence
Sources & further reading
Research checked on 14 October 2026. Follow the original sources to explore the evidence and its limits.
- Peter Norvig — Solving Every Sudoku Puzzle
Original explanation and code for units, peers, constraint propagation and search.
- Bernhard Hobiger — HoDoKu: Last Resort
Explanation of assumption-based forcing chains and nets, including contradiction and common-result logic.
- Andrew Stuart — Strategy Families
A solver-specific ordering of techniques, useful as an example rather than a universal difficulty scale.
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