Sudoku with no solution
A Sudoku position can have no repeated digits and still have no solution. Duplicate checks only test the entries already present. A full solution must also fill every blank consistently; a cell with no possible digit, or a required digit with nowhere to go in a unit, proves that completion is impossible.
The worked example
Coordinates start at the top left: r = row, c = column. Small digits are candidates; outlined cells are discussed below.
Check this exampleView puzzle text
Read left to right across each row, then top to bottom. A 0 means a blank.
013056789056789123789023456234567891567891234891234567345678912678912345902345678A missing 2 with nowhere to go
This position starts from the hidden-single example, then inserts 1 at r1c2. That entry creates no repeated placed digit: there was no other 1 in its row, column or box. The board nevertheless cannot be completed.
You can also find a contradiction at r9c2: row 9 needs 1 there, but column 2 now contains the inserted 1 at r1c2. An impossible board can expose the same underlying error in several places. The location where you notice the contradiction need not be the location of the error.
Step 1. Row 1 reads blank, 1, 3, blank, 5, 6, 7, 8, 9. It is missing 2 and 4.
Step 2. r1c1 cannot take 2 because r4c1 already contains 2 in column 1.
Step 3. r1c4 cannot take 2 because r6c4 already contains 2 in column 4.
Step 4. These are the only blanks in row 1. There is no position for its required 2, so the board has no solution.
Understand the diagnosis
Check the diagnosis to decide which entries to review.
| Diagnosis | What it establishes | Useful next step |
|---|---|---|
| Malformed input | The text does not describe 81 valid cells | Check length, order and blank symbols |
| Conflicting entries | A placed digit repeats in a unit | Compare the highlighted entries with the source |
| No solution | The conflict-free entries have no complete extension | Check copied givens and undo unsupported moves |
Check givens and recent moves
If you have the original puzzle, enter only its givens into the solver first. When those have a unique solution but your current position has none, at least one of your added digits is inconsistent. Undo your own moves back to a position you can justify.
If the copied givens already have no solution, inspect the transcription row by row. Pay particular attention to skipped blanks or a digit entered one column to the side. Do not silently change a given to make a solver produce an answer; that would solve a different puzzle.
In this constructed example we know the inserted r1c2 = 1 caused the failure. Clearing it restores the original position, where row 1 forces r1c2 = 2. On a board with unknown history, the solver’s status alone does not identify a single wrong cell.
Being stuck is not proof of impossibility
Failing to find a naked or hidden single means those techniques did not produce a move in your current scan. It does not prove that no solution exists. A harder deduction may be available, or the puzzle may have multiple solutions.
A complete search can establish no solution by exhausting every allowed branch. An incomplete scan cannot. If a tool stops because of a time limit or another technical limit, that is an unknown result and must not be reported as ‘no solution’.
Check your understanding
Both blanks in row 1 allow 4. Could placing one 4 reveal where the 2 goes?
Show the reasoning
No. Neither blank can ever take 2 while the existing column entries remain fixed. Placing a 4 in either cell only leaves the same impossible requirement for the other. The contradiction is already complete.
About this example
This teaching board keeps many givens to make the deduction clear and has no difficulty rating. Diagnosis examples deliberately include the stated conflicts or ambiguity.
Rules and terminology
- Nikoli: Sudoku rules
The standard row, column and 3 × 3 box rules used in these examples. The board and worked explanation on this page were created separately.