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Expert Sudoku: forcing chains and how to think about them
Expert puzzles are designed to resist every pattern-based technique. To finish them you need forcing chains: logical arguments that follow the consequences of a candidate until they prove something must be true or false.
What makes a puzzle expert level
The difficulty of a Sudoku is set by its single hardest step. Expert puzzles contain at least one point where no wing, fish or uniqueness pattern helps, and the only way forward is a chain of reasoning across many cells. In the Sudoku Explainer scale, forcing chains start around 7.0, and dynamic forcing chains, used by the very hardest puzzles, sit above 8.8.
Forcing chains are not guessing
Guessing means writing a digit and hoping. A forcing chain is a proof. You take a cell with two candidates and follow each one: "if this cell is 4, then this one is 7, then that one is 2…" If both branches lead to the same conclusion, that conclusion is true. If one branch ends in a contradiction, such as a unit with no place for a digit, that branch is false and its starting candidate can be removed.
Techniques you will need
- Alternating inference chains: chains that alternate strong links (one of two must be true) and weak links (both cannot be true).
- Cell and region forcing chains (8.2–8.7): try every candidate of one cell, or every position of one digit in a unit; whatever all of them imply is certain.
- Nishio (7.6–8.1): assume a single digit in a cell and follow that one digit until it creates a contradiction.
- Dynamic forcing chains (8.8 and above): chains in which each new conclusion is used for the next step. Very few puzzles need them.
How to approach an expert puzzle
- Apply every simpler technique first; expert puzzles usually need only one or two chain steps between long runs of ordinary solving.
- Start chains from cells with exactly two candidates and from digits with only two positions in a unit. Short chains are easier to check.
- Write the chain down or trace it with colors. Long chains held only in memory lead to mistakes.
- After every elimination, return to singles and pairs before looking for the next chain.
Expert puzzles can take a long time even for experienced solvers. Taking a break and coming back with fresh eyes often reveals a link that was missed.
Solving Techniques
View All Techniques →X-Wing and Swordfish
When a digit in two (or three) rows is confined to the same columns, it is removed from the rest of those columns. Key to hard puzzles.
Learn →Naked and hidden singles
When a cell has one candidate left, or a digit has only one place in a unit. Almost every easy puzzle is solved with these.
Learn →Locked candidates
When a box’s candidate is confined to one row or column (pointing), or the reverse (claiming), the same candidates outside are removed.
Learn →Pairs and triples
If two cells share the same two candidates (naked pair) or two digits fit only two cells (hidden pair), other candidates in the unit go.
Learn →Tools & Features
Sudoku SE Rating Calculator
Calculate the difficulty level of your Sudoku puzzles with the SE Rating system. Learn the objective difficulty score of your puzzles using the Sudoku Explainer algorithm.
Calculate →Printable Sudoku
Download printable Sudoku puzzles to solve offline. PDF puzzles in different difficulty levels with solutions included.
Download →Daily Sudoku
A new puzzle every day, the same for everyone. Use the calendar to solve past days too.
Today's puzzle →Frequently Asked Questions
How do you play Sudoku?
Fill the empty cells of the 9×9 grid with the digits 1–9 so that every row, column and 3×3 box contains each digit exactly once. No guessing or arithmetic is needed, only logic.
Read the rules and a worked example →How are the difficulty levels decided?
Not by the number of clues but by the hardest technique the solution needs. Every puzzle is rated with the Sudoku Explainer (SE) system: easy puzzles need only singles, hard ones need patterns such as X-Wing.
SE Rating guide →Do I need to be good at math to play Sudoku?
No. The digits are only nine different symbols; there is no adding or multiplying. What you need is careful scanning and logical elimination.
Is Sudoku good for your brain?
Sudoku is an enjoyable mental workout that uses attention and working memory, and observational studies link regular puzzle solving with better scores on cognitive tests. It has not been proven to prevent dementia on its own.
What does the research say? →