Do You Need Trial and Error to Solve Sudoku?

What is trial and error?

With trial and error, you choose one number in a cell with two or more candidates, assume it is correct and continue solving. If you then repeat a number in the same Row or Column, or reach a cell with no candidates (a contradiction), your chosen number was wrong and you can eliminate that candidate. If the cell had only two candidates, the remaining one is the answer. If it had three or more, you have ruled out just one candidate and still do not know the answer. You may also hear this called “guessing” or “trial and error.”

Trial and error is not inherently wrong. A contradiction gives you evidence too. The problem is the cost. To test one assumption, you may need to fill several cells and follow the consequences. The later a contradiction appears, the more cells you have to undo.

What uniqueness means for a puzzle

When DailySudoku generates a puzzle, it removes givens one at a time, keeping each removal only if exactly one solution remains. Every puzzle therefore has uniqueness: the original givens determine exactly one number for every cell. Uniqueness does not guarantee a solving path without trial and error, however. Some puzzles are difficult to finish using only common human solving techniques.

The level of technique you need also varies between puzzles. This app sets difficulty by the number of givens at the start: from 40 for Easy to 24 for Expert, with a target of 22 for Master. Puzzles at the same difficulty can therefore require different levels of technique.

Why testing assumptions is costly in this app

• A wrong number counts toward Mistakes as soon as you enter it. If you enter an assumed number directly into a cell and it is wrong, your mistake count immediately increases by one.
• Undo does not reduce your mistake count. The cell returns to its previous state, but a mistake already counted stays counted.
• If “Mistake limit” is enabled (on by default, with a limit of 3), the game ends when you reach that limit.

Entering a number to test an assumption in this app therefore risks a mistake immediately. If you still want to test one, follow it using Notes only, without entering answers into cells. Notes do not count as mistakes.

Instead of guessing — a cell that looks like a 50–50 choice

In the example below, r2c6 has only two candidates: 4 and 5. Row 2 contains 9, 6, 2 and 1, and column 6 contains 1, 6, 3, 9, 8 and 7, leaving just 4 and 5. It is tempting to take a 50–50 guess.

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

6

8

2

1

9

34

9

34

6

578

45

34578

2

1

1

7

5

6

7

8

4

6

2

2

6

7

1

3

4

7

6

5

3

1

4

6

9

2

8

7

9

7

1

8

6

3

8

6

3

7

Before guessing, write all the candidates in the other empty cells in row 2. You will find that r2c1 and r2c3 both have exactly {3, 4}. This is the “Naked Pair” from How to Use Sudoku Candidate Notes. Row 2's 3 and 4 must occupy those two cells, so no other cell in row 2 can contain 4.

Remove 4 from r2c6's candidates of 4 and 5, and only 5 remains. Guessing would have given you a one-in-two chance of adding a mistake. Writing the candidates for just one row lets you determine the answer with certainty.

When you still need trial and error

Some puzzles will not let you move forward with basic techniques and Naked Pairs alone. If you decide to test an assumption, you can reduce the cost by following these guidelines.

• Choose a cell with only two candidates. With three or more, you have too many possibilities to check.
• Follow the assumption in your notes, not by entering answers. In this app, a wrong answer counts as a mistake the moment you enter it.
• If you reach a contradiction, confirm the other candidate. When the starting cell has only two candidates, a contradiction from one means the other is the answer.
• If you would rather avoid testing an assumption, you can use Hint. It fills the answer in one selected empty cell, with a limited number of uses per puzzle (3 by default).

Common mistakes

• You guess before writing all the candidates. As in the example, writing the candidates for one row or column often removes the need to guess.
• You enter your assumed number directly as an answer. If it is wrong, your mistake count increases, and undoing it does not restore the count.
• You lose track of the branch you followed. If you do not know which assumption led to the cells you filled, you will not know how far to undo when a contradiction appears.

Practice puzzle

c1c2c3c4c5c6c7c8c9

r1r2r3r4r5r6r7r8r9

1

9

3

8

2

7

3

2

6

1

8

7

5

9

4

8

2

1

3

6

1

8

47

4679

2469

3

247

47

5

3

8

6

1

6

2

1

9

8

3

2

1

3

5

8

3

8

7

6

1

2

6

8

2

5

1

3

9

What number goes in r4c7? Find the two cells in row 4 with exactly the same candidates.

Show answer

2. Both r4c3 and r4c8 have candidates {4, 7}, so row 4's 4 and 7 must occupy those two cells. Removing 4 and 7 from r4c7's candidates of 2, 4 and 7 leaves only 2. For the same reason, you can remove 4 and 7 from r4c4, and 4 from r4c5.