Sudoku Puzzles: Valid or Invalid

A Sudoku puzzle is generally a 9x9 grid with some pre-filled digits from 1 to 9. But the pre-filled digits cannot be arbitrary; otherwise, it may be an invalid Sudoku puzzle.

Invalid Sudoku Puzzles

Puzzles in the following three cases are invalid Sudoku puzzles.

Case 1: The pre-filled digits violate the Sudoku rules. For example, the same digit appears more than once in a single row, in a single column, or in a single 3x3 block.

R1
R2
R3
R4
R5
R6
R7
R8
R9
1
5
7
3
8
6
9
4
3
5
6
8
1
5
8
3
6
2
4
7
2
5
1
4
2
9
Figure 1: Invalid Puzzle - Row Violation

The puzzle in Figure 1 contains the digit 2 in two cells of row R8. The clues in this puzzle violate the Sudoku rule for rows. Therefore, it cannot be a valid Sudoku puzzle.

C1
C2
C3
C4
C5
C6
C7
C8
C9
6
5
3
8
1
7
2
8
1
4
7
8
3
6
9
2
4
7
5
9
4
7
3
1
Figure 2: Invalid Puzzle - Column Violation

The puzzle in Figure 2 contains the digit 7 in two cells of column C5. The clues in this puzzle violate the Sudoku rule for columns. Therefore, it cannot be a valid Sudoku puzzle.

8
1
2
6
4
5
8
2
7
9
5
9
3
4
1
2
6
7
9
1
7
4
3
5
Figure 3: Invalid Puzzle - Block Violation

The puzzle in Figure 3 contains the digit 9 in two cells of block B4. The clues in this puzzle violate the Sudoku rule for blocks. Therefore, it cannot be a valid Sudoku puzzle.

The invalid puzzles in this case are relatively easy to identify. All we need to do is see if there are any violations of the Sudoku rules for the pre-filled digits. In the examples in Figures 1-3 above, check the clues (pre-filled digits), and there is a violation of the Sudoku rules in each puzzle. Therefore, each of them is an invalid Sudoku puzzle.

Case 2: The pre-filled digits do not violate the Sudoku rules, but there are no solutions for the puzzle. That means, based on the pre-filled digit positions, there are no digit placement combinations that can satisfy the Sudoku rules.

Unlike the puzzles in Case 1, the pre-filled digits for a puzzle in this case do not violate the Sudoku rules. So, it is hard to tell it is an invalid puzzle by just looking at the positions of the pre-filled digits.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
9
8
1
6
2
4
9
9
3
6
7
3
2
6
8
5
2
7
8
5
3
6
4
4
3
9
Figure 4: A Puzzle With No Solutions

At first glance, all the pre-filled digits for the puzzle in Figure 4 above do not violate any Sudoku rules. So, it is hard to tell if it is an invalid puzzle. However, if we try to solve this puzzle, at some point, we will reach a situation where there are no possible positions for a certain digit in a 9-cell unit (a row, a column, or a 3x3 block), or there are no candidate digits for an empty cell.

For the puzzle in Figure 4, three of the nine cells in Block B6 have pre-filled digits 5, 6, and 8. We try to find a place for the digit 9 in one of the remaining six empty cells. Figure 5 below shows that we cannot find a possible place for the digit 9 in block B6.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
9
8
1
6
2
4
9
9
3
6
7
3
2
9
6
9
9
9
9
8
5
9
2
7
8
5
3
6
4
4
3
9
Figure 5: No Cells Can Accommodate Digit 9 in Block B6

Three of the nine cells in block B6 of this puzzle have been pre-filled with digits. The other six cells are empty. Let's try to find a cell for the digit 9, as it does not appear in this block yet.

  1. The cell (R4,C7) is empty. But the cell (R3,C7) in the same column is 9. So the cell (R4,C7) cannot be 9.
  2. The cell (R4,C8) has been pre-filled with the digit 6. So, of course, it cannot be 9.
  3. The cell (R4,C9) is empty. But the cell (R9,C9) in the same column is 9. So the cell (R4,C9) cannot be 9.
  4. The cell (R5,C7) is empty. But the cell (R5,C2) in the same row is 9, and the cell (R3,C7) in the same column is 9. So the cell (R5,C7) cannot be 9.
  5. The cell (R5,C8) is empty. But the cell (R5,C2) in the same row is 9. So the cell (R5,C8) cannot be 9.
  6. The cell (R5,C9) is empty. But the cell (R5,C2) in the same row is 9, and the cell (R9,C9) in the same column is 9. So the cell (R5,C9) cannot be 9.
  7. The cell (R6,C7) has been pre-filled with the digit 8. So, of course, it cannot be 9.
  8. The cell (R6,C8) has been pre-filled with the digit 5. So, of course, it cannot be 9.
  9. The cell (R6,C9) is empty. But the cell (R9,C9) in the same column is 9. So the cell (R6,C9) cannot be 9.

As you can see now, no cells in block B6 can be filled with the digit 9. So, there is no solution to this puzzle. Therefore, it is not a valid puzzle.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
7
5
8
4
8
1
8
6
3
1
4
6
3
1
9
4
7
5
6
2
2
7
1
4
3
1
5
Figure 6: Another Puzzle With No Solutions

The puzzle in Figure 6 is another example of how the pre-filled digits do not violate any Sudoku rules, but you cannot find a digit placement combination that can fulfill the Sudoku rules with these pre-filled digits. That means there is no solution to this puzzle.

Just like the previous example, it is hard to tell this is an invalid puzzle by just looking at the puzzle itself since all pre-filled digits do not violate the Sudoku rules. However, when we try to check what digits can possibly be filled in the empty cell (R6,C6), we see that no digits can be candidates for that cell.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
7
5
8
4
8
1
8
6
3
1
4
6
3
1
9
?
4
7
5
6
2
2
7
1
4
3
1
5
Figure 7: No Candidates for Cell (R6,C6)

The cell (R6,C6) is located in row R6, column C6, and block B5, where they are all highlighted in yellow as shown in Figure 7. Let's see why no digits can be candidates for the cell (R6,C6).

  1. The digit 1 cannot be filled in the cell (R6,C6) since it has appeared in the same block B5 (and in the same column C6 as well).
  2. The digit 2 cannot be filled in the cell (R6,C6) since it has appeared in the same column C6.
  3. The digit 3 cannot be filled in the cell (R6,C6) since it has appeared in the same row R6.
  4. The digit 4 cannot be filled in the cell (R6,C6) since it has appeared in the same column C6.
  5. The digit 5 cannot be filled in the cell (R6,C6) since it has appeared in the same row R6.
  6. The digit 6 cannot be filled in the cell (R6,C6) since it has appeared in the same row R6.
  7. The digit 7 cannot be filled in the cell (R6,C6) since it has appeared in the same column C6.
  8. The digit 8 cannot be filled in the cell (R6,C6) since it has appeared in the same column C6.
  9. The digit 9 cannot be filled in the cell (R6,C6) since it has appeared in the same block B5.

Therefore, no digits can be candidates for the cell (R6,C6), and as a result, there is no solution to the puzzle. Therfore, it is an invalid puzzle.

Case 3: The pre-filled digits do not violate the Sudoku rules, but there is more than one solution for the pre-filled grid. That means, based on the pre-filled digit positions, there are at least two combinations of digit placements that can satisfy the Sudoku rules.

The puzzle in Figure 8 below is an example.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
1
7
9
4
8
2
4
1
9
3
9
6
1
7
5
5
3
8
7
2
6
4
8
3
1
4
6
5
Figure 8: A Puzzle With More Than One Solution

The pre-filled digits in this puzzle do not violate the Sudoku rules. So, it is hard to tell if it is an invalid puzzle. However, when we try to solve this puzzle, we reach the following situation, as shown in Figure 9.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
2
1
7
3
9
6
4
5
8
9

5?
8?

3
4

5?
8?

2
7
6
1

5?
8?

4
6

5?
8?

1
7
9
3
2
5
3
9
8
6
1
7
2
4
6
7
8
2
9
4
3
1
5
1
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5
3
6
9
8
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6
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5
9
8
3
5
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8
3
9
1
2
7
3
8
9
2
7
1
4
6
5
Figure 9: Cannot Use Logic to Proceed for Last Four Cells

We use logic to solve this puzzle until the last four cells: (R1,C5), (R2,C5), (R1,C7), and (R2,C7). We need to fill in two more 5's and two more 8's to complete the puzzle. Based on the Sudoku rules, there are two possible combinations:

  1. The two cells (R1,C5) and (R2,C7) are filled with the digit 5, and the other two cells (R2,C5) and (R1,C7) are filled with the digit 8, or
  2. The two cells (R1,C5) and (R2,C7) are filled with the digit 8, and the other two cells (R2,C5) and (R1,C7) are filled with the digit 5.

At this point, logic cannot determine a unique answer anymore. We find that both two combinations of digit placements can satisfy the Sudoku rules, as shown below in Figure 10.

2
1
7
3
9
6
4
5
8
9
5
3
4
8
2
7
6
1
8
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6
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9
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1
5
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8
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9
1
2
7
3
8
9
2
7
1
4
6
5
Figure 10: Both Two Combinations Satisfy the Sudoku Rules

To verify that each of the two combinations of digit placements in Figure 10 above can be a solution to the puzzle, we check each row, each column, and each 3x3 block of these two grids and find that all of them satisfy the Sudoku rules. So, either one can be a solution to the puzzle in Figure 8. This puzzle has more than one solution. Therefore, it is an invalid puzzle. Logic alone cannot give a unique solution.

The three cases discussed above are considered invalid puzzles. The puzzles in the first two cases have no solutions. Therefore, it is meaningless to play with them. Although the puzzles in the third case have solutions, they have more than one, and therefore, they cannot be solved by logic alone. As Sudoku is a logic-based game, all invalid puzzles should not be included in the Sudoku database. We only offer valid Sudoku puzzles. Every Sudoku puzzle you play on our website is guaranteed to have only one solution.

Valid Sudoku Puzzles

As we know now, invalid Sudoku puzzles are those that have no solutions or have more than one solution. Conversely, valid Sudoku puzzles refer to those puzzles that have only one solution. The following is an example of a valid Sudoku puzzle.

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
3
2
2
5
4
8
5
6
1
3
9
9
6
5
8
3
7
9
5
2
4
6
Figure 11: A Valid Sudoku Puzzle

The Sudoku puzzle above has 22 pre-filled digits at the positions shown in Figure 11. After using logic to fill up the empty cells according to the Sudoku rules, we reach a unique solution as follows:

C1
C2
C3
C4
C5
C6
C7
C8
C9
R1
R2
R3
R4
R5
R6
R7
R8
R9
5
3
4
1
8
6
2
9
7
1
2
7
9
3
5
4
8
6
6
9
8
4
7
2
3
1
5
9
4
2
8
5
1
6
7
3
3
7
8
6
9
4
5
1
2
5
6
1
7
2
3
9
8
4
4
2
9
3
6
8
7
1
5
8
6
3
7
5
1
2
4
9
1
5
7
2
4
9
8
3
6
Figure 12: The Unique Solution to the Puzzle in Figure 11

The digit placements above satisfy the Sudoku rules. Each row, each column, and each 3x3 block has each of the 9 digits from 1 to 9, and there are no repeated digits in any rows, any columns, or any 3x3 blocks. For each empty cell of the puzzle in Figure 11, we can use logic to determine the unique correct digit. Therefore, this is the unique solution to the Sudoku puzzle. No other digit placement combination can satisfy the Sudoku rules with the pre-filled digits of the puzzle other than the one shown in Figure 12.

How to distinguish between valid Sudoku puzzles and invalid Sudoku puzzles?

You can tell that it is an invalid Sudoku puzzle if the pre-filled digits violate the Sudoku rules. If you see the same digit appear more than once in a row, in a column, or in a 3x3 block, then this is an invalid puzzle. However, in most cases, the pre-filled digits do not violate any Sudoku rules. It is almost impossible to tell if it is a valid Sudoku puzzle or an invalid Sudoku puzzle by just looking at the pre-filled digits of the puzzle. Only after trying to solve this puzzle with logic will we reach a situation where there are no possible positions for a certain digit in a 9-cell unit (a row, a column, or a 3x3 block) or there are no candidate digits for an empty cell. Then we realize that it is an invalid puzzle. In fact, no one is willing to spend so much time trying to solve a Sudoku puzzle, only to find out that it is unsolvable or cannot be solved only with logic.

Is there a tool for us to distinguish between valid Sudoku puzzles and invalid Sudoku puzzles? Yes, there is. Our online Sudoku solver can do this job. Enter your puzzle on the blank grid there and click "Solution Only", and you will get an alert if the puzzle is invalid (either no solutions or more than one solution). Actually, what you really hope for is that there is no need to worry about playing with invalid Sudoku puzzles. The good thing is that every Sudoku puzzle you play on our website has only one solution. So, you can use your logic and solving skills here to play every Sudoku puzzle, and you don't need to worry that you may waste your time on invalid Sudoku puzzles.

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