Can Sudoku have no solution without any repeated digits?
Yes. 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
Read coordinates from the top left: r = row, c = column. Small digits are candidates. Outlines identify the cells discussed in the caption.
Open this example in the solverView the exact 81-cell input
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.
Three different reasons a board cannot be accepted
The solver separates input errors from the number of possible completions. That distinction tells you what to check next.
| 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 |
Find the bad assumption, not an arbitrary replacement
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 example was constructed for Sudoku Slate from a completed grid. Many givens are retained to make the specific deduction easy to inspect. Diagnostic examples are deliberately conflicting or ambiguous where stated; these are teaching positions, not difficulty-rated game puzzles.
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.