Sudoku Solver Using Backtracking in DSA C# - Execution Trace

Concept Flow - Sudoku Solver Using Backtracking

Start with empty cell

↓

Try number 1 to 9

↓

Check if number is safe

↓

Place number

↓

Move to next cell

↓

Repeat until board complete

↓

If stuck, backtrack

↓

Remove number, try next

↓

Done when all cells filled

The solver tries numbers 1-9 in empty cells, checks if safe, places if yes, moves forward, backtracks if stuck, repeats until complete.

Execution Sample

DSA C#

bool SolveSudoku(int[,] board) {
  for (int row = 0; row < 9; row++) {
    for (int col = 0; col < 9; col++) {
      if (board[row, col] == 0) {
        for (int num = 1; num <= 9; num++) {
          if (IsSafe(board, row, col, num)) {
            board[row, col] = num;
            if (SolveSudoku(board)) return true;
            board[row, col] = 0;
          }
        }
        return false;
      }
    }
  }
  return true;
}

This code tries to fill empty cells with numbers 1-9 safely, recurses forward, backtracks if no number fits.

Execution Table

Step	Operation	Cell (row,col)	Number Tried	Is Safe?	Action	Board State Snapshot
1	Find empty cell	(0,2)	-	-	Try numbers 1 to 9	[5 3 0 | 0 7 0 | 0 0 0] ...
2	Try number	(0,2)	1	No	Try next number	[5 3 0 | 0 7 0 | 0 0 0] ...
3	Try number	(0,2)	2	No	Try next number	[5 3 0 | 0 7 0 | 0 0 0] ...
4	Try number	(0,2)	4	Yes	Place number 4	[5 3 4 | 0 7 0 | 0 0 0] ...
5	Move to next empty	(0,3)	-	-	Try numbers 1 to 9	[5 3 4 | 0 7 0 | 0 0 0] ...
6	Try number	(0,3)	1	Yes	Place number 1	[5 3 4 | 1 7 0 | 0 0 0] ...
7	Move to next empty	(0,5)	-	-	Try numbers 1 to 9	[5 3 4 | 1 7 0 | 0 0 0] ...
8	Try number	(0,5)	2	No	Try next number	[5 3 4 | 1 7 0 | 0 0 0] ...
9	Try number	(0,5)	6	Yes	Place number 6	[5 3 4 | 1 7 6 | 0 0 0] ...
10	Move to next empty	(0,6)	-	-	Try numbers 1 to 9	[5 3 4 | 1 7 6 | 0 0 0] ...
11	Try number	(0,6)	7	No	Try next number	[5 3 4 | 1 7 6 | 0 0 0] ...
12	Try number	(0,6)	8	Yes	Place number 8	[5 3 4 | 1 7 6 | 8 0 0] ...
13	Move to next empty	(0,7)	-	-	Try numbers 1 to 9	[5 3 4 | 1 7 6 | 8 0 0] ...
14	Try number	(0,7)	9	Yes	Place number 9	[5 3 4 | 1 7 6 | 8 9 0] ...
15	Move to next empty	(0,8)	-	-	Try numbers 1 to 9	[5 3 4 | 1 7 6 | 8 9 0] ...
16	Try number	(0,8)	2	No	Try next number	[5 3 4 | 1 7 6 | 8 9 0] ...
17	Try number	(0,8)	3	No	Try next number	[5 3 4 | 1 7 6 | 8 9 0] ...
18	Try number	(0,8)	5	No	Try next number	[5 3 4 | 1 7 6 | 8 9 0] ...
19	Try number	(0,8)	7	No	Try next number	[5 3 4 | 1 7 6 | 8 9 0] ...
20	Try number	(0,8)	1	No	Try next number	[5 3 4 | 1 7 6 | 8 9 0] ...
21	Try number	(0,8)	4	No	Try next number	[5 3 4 | 1 7 6 | 8 9 0] ...
22	Try number	(0,8)	6	No	Try next number	[5 3 4 | 1 7 6 | 8 9 0] ...
23	Try number	(0,8)	8	No	Try next number	[5 3 4 | 1 7 6 | 8 9 0] ...
24	Try number	(0,8)	9	No	Backtrack	[5 3 4 | 1 7 6 | 8 9 0] ...
25	Backtrack	(0,7)	Remove 9	N/A	Try next number	[5 3 4 | 1 7 6 | 8 0 0] ...
26	Try number	(0,7)	1	No	Try next number	[5 3 4 | 1 7 6 | 8 0 0] ...
27	Try number	(0,7)	2	Yes	Place number 2	[5 3 4 | 1 7 6 | 8 2 0] ...
28	Move forward	(0,8)	-	-	Try numbers 1 to 9	[5 3 4 | 1 7 6 | 8 2 0] ...
29	Try number	(0,8)	1	Yes	Place number 1	[5 3 4 | 1 7 6 | 8 2 1] ...
30	Move forward	(1,0)	-	-	Continue solving	[5 3 4 | 1 7 6 | 8 2 1] ...
...	...	...	...	...	...	...
N	All cells filled	-	-	-	Return true, solved	[Solved board state]

💡 All cells filled with valid numbers, Sudoku solved successfully

Variable Tracker

Variable	Start	After Step 4	After Step 9	After Step 14	After Step 24	After Step 29	Final
board[0,2]	0	4	4	4	4	4	4
board[0,3]	0	0	1	1	1	1	1
board[0,5]	0	6	6	6	6	6	6
board[0,6]	0	0	8	8	8	8	8
board[0,7]	0	0	0	9	0	2	2
board[0,8]	0	0	0	0	0	1	1
current_cell	(0,2)	(0,3)	(0,5)	(0,7)	(0,7)	(0,8)	(1,0)

Key Moments - 3 Insights

Why do we backtrack when no number fits in a cell?

When no number 1-9 is safe in a cell (see step 24), we remove the last placed number and try a different number in the previous cell to find a valid path.

How do we know if a number is safe to place?

We check the row, column, and 3x3 box for the number's presence before placing it (see 'Is Safe?' column in steps 2-4).

Why do we move forward only after placing a safe number?

Moving forward only after placing a safe number ensures partial solutions are valid, preventing wasted work and enabling correct backtracking (see steps 4, 6, 9).

Visual Quiz - 3 Questions

Look at the execution table at step 4. What number was placed in cell (0,2)?

A 4

B 1

C 2

D 3

💡 Hint

Check the 'Number Tried' and 'Action' columns at step 4.

At which step does the solver backtrack from cell (0,8)?

A Step 14

B Step 24

C Step 29

D Step 9

💡 Hint

Look for the 'Backtrack' action and cell (0,8) in the execution table.

If the number 9 was safe at cell (0,8), how would the execution table change after step 15?

A The solver would place 9 and move forward without backtracking.

B The solver would backtrack immediately.

C The solver would try numbers 1 to 8 again.

D The solver would skip cell (0,8).

💡 Hint

Refer to the 'Is Safe?' and 'Action' columns around step 15.

Concept Snapshot

Sudoku Solver Using Backtracking:
- Find empty cell
- Try numbers 1-9
- Check if safe (row, col, box)
- Place number if safe
- Move to next cell
- If stuck, backtrack
- Repeat until board complete

Full Transcript

The Sudoku solver uses backtracking to fill empty cells. It scans the board for empty cells, tries numbers 1 to 9, and checks if placing a number is safe by verifying the row, column, and 3x3 box. If safe, it places the number and moves to the next empty cell. If no number fits, it backtracks by removing the last placed number and tries the next possible number. This process repeats until all cells are filled with valid numbers, solving the Sudoku puzzle.

Practice

1. What is the main purpose of using backtracking in a Sudoku solver?

easy

A. To try possible numbers and undo choices if they lead to conflicts

B. To randomly fill the Sudoku grid without checking rules

C. To sort the numbers in each row before solving

D. To count how many empty cells are in the Sudoku grid

Solution

1. Step 1: Understand backtracking role

   Backtracking tries numbers in empty cells and removes them if they cause conflicts.

2. Step 2: Compare other options

   Options A, B, and C do not describe backtracking's trial and error approach.

3. Final Answer:

   To try possible numbers and undo choices if they lead to conflicts -> Option A

4. Quick Check:

   Backtracking = Try and undo choices [OK]

Hint: Backtracking means try and undo wrong choices [OK]

Common Mistakes:

• Thinking backtracking fills randomly without checks
• Confusing backtracking with sorting or counting
• Ignoring the undo step in backtracking

2. Which C# syntax correctly defines a method header for a Sudoku solver using backtracking that returns a boolean?

easy

A. public bool SolveSudoku(int[,] board)

B. bool SolveSudoku(int[][] board)

C. public void SolveSudoku(int[] board)

D. bool SolveSudoku(int[] board);

Solution

1. Step 1: Check method signature correctness

   public bool SolveSudoku(int[,] board) uses correct C# syntax: public access, bool return, 2D array parameter.

2. Step 2: Identify errors in other options

   bool SolveSudoku(int[][] board) uses wrong array syntax for C#, C uses void return instead of bool, D has semicolon ending which is invalid for method definition.

3. Final Answer:

   public bool SolveSudoku(int[,] board) -> Option A

4. Quick Check:

   Correct C# method header = public bool SolveSudoku(int[,] board) [OK]

Hint: Use public bool with int[,] for 2D arrays in C# [OK]

Common Mistakes:

• Using jagged array syntax int[][] instead of int[,]
• Missing public keyword or wrong return type
• Ending method header with semicolon

3. What will be the output of the following code snippet when trying to solve a partially filled Sudoku board?

int[,] board = new int[9,9] {
 {5,3,0,0,7,0,0,0,0},
 {6,0,0,1,9,5,0,0,0},
 {0,9,8,0,0,0,0,6,0},
 {8,0,0,0,6,0,0,0,3},
 {4,0,0,8,0,3,0,0,1},
 {7,0,0,0,2,0,0,0,6},
 {0,6,0,0,0,0,2,8,0},
 {0,0,0,4,1,9,0,0,5},
 {0,0,0,0,8,0,0,7,9}
};
SudokuSolver solver = new SudokuSolver();
bool solved = solver.SolveSudoku(board);
Console.WriteLine(solved);

medium

A. false

B. true

C. Compilation error

D. Runtime exception

Solution

1. Step 1: Understand the input board

   The board is a valid partially filled Sudoku puzzle with zeros as empty cells.

2. Step 2: Analyze SolveSudoku method behavior

   Assuming correct backtracking implementation, the solver will find a solution and return true.

3. Final Answer:

   true -> Option B

4. Quick Check:

   Valid puzzle solved = true [OK]

Hint: Valid puzzles return true after solving [OK]

Common Mistakes:

• Assuming zeros cause errors instead of empty cells
• Thinking solver returns false without trying all options
• Confusing compilation/runtime errors with logic

4. Identify the bug in this backtracking Sudoku solver snippet:

bool IsSafe(int[,] board, int row, int col, int num) {
  for (int x = 0; x < 9; x++) {
    if (board[row, x] == num || board[x, col] == num) return false;
  }
  int startRow = row - row % 3;
  int startCol = col - col % 3;
  for (int i = 0; i < 3; i++) {
    for (int j = 0; j < 3; j++) {
      if (board[startRow + i, startCol + j] == num) return false;
    }
  }
  return true;
}

medium

A. The method does not check if the cell is already filled

B. The row and column checks are incomplete

C. No bug; the method correctly checks safety

D. The 3x3 box check uses wrong start indices calculation

Solution

1. Step 1: Review row and column checks

   The method checks all cells in the row and column for the number, which is correct.

2. Step 2: Review 3x3 box check

   Calculates startRow and startCol correctly using modulo, then checks all 3x3 cells.

3. Step 3: Confirm no missing checks

   Method returns true only if no conflicts found, so it is correct.

4. Final Answer:

   No bug; the method correctly checks safety -> Option C

5. Quick Check:

   Safety check correct = No bug; the method correctly checks safety [OK]

Hint: Check row, column, and 3x3 box carefully [OK]

Common Mistakes:

• Miscomputing start indices for 3x3 box
• Forgetting to check both row and column
• Assuming cell fill check is needed here

5. Given a Sudoku board with multiple empty cells, which strategy improves the backtracking solver's efficiency the most?

hard

A. Fill cells randomly to reduce search space

B. Always try numbers 1 to 9 in order without checks

C. Skip checking 3x3 boxes to save time

D. Choose the empty cell with the fewest possible valid numbers first

Solution

1. Step 1: Understand backtracking efficiency

   Choosing the cell with fewest valid options reduces branching and speeds solving.

2. Step 2: Evaluate other options

   Random filling (A) wastes time, trying numbers blindly (B) is inefficient, skipping box checks (D) breaks rules.

3. Final Answer:

   Choose the empty cell with the fewest possible valid numbers first -> Option D

4. Quick Check:

   Minimum options first = faster solving [OK]

Hint: Pick cell with fewest options first to speed solving [OK]

Common Mistakes:

• Trying numbers in fixed order without heuristics
• Ignoring 3x3 box constraints
• Randomly filling cells hoping for quick solve