Beginner

Last Free Cell Technique

One of the most fundamental and straightforward Sudoku solving strategies that involves identifying when only one empty cell remains in a row, column, or 3×3 box.

What is the Last Free Cell Technique?

The Last Free Cell technique is the most fundamental Sudoku solving strategy. It involves identifying when only one empty cell remains in a row, column, or 3×3 box, making it easy to determine what number must go there.

When there's only one empty cell left in any row, column, or 3×3 box, you can immediately determine what number belongs there by finding which number from 1-9 is missing from that group.

Key insight: Since each row, column, and 3×3 box must contain all numbers from 1 to 9 exactly once, when 8 out of 9 numbers are already placed, only one number can possibly fit in the remaining cell.

How Last Free Cell Works

Requirements

  • 1 Find a row, column, or 3×3 box with only one empty cell
  • 2 Count which numbers from 1-9 are already present
  • 3 The missing number must go in the empty cell

Result

You can immediately fill the empty cell with the missing number. This technique is 100% reliable - there's no guessing involved.

Each new number you place might complete another row, column, or box, creating more last free cell opportunities.

Step-by-Step Examples

Example 1: Last Free Cell in a Row

Look at the middle row in this grid - it has only one empty cell:

5
3
7
8
1
9
2
6
4
8
1
9
3
7
5
6
2
4
2
6
4
9
2
3
1
7
8

1 The middle row (highlighted in blue) has numbers: 8, 1, 9, 3, 7, 5, 6, 2, and one empty cell.

2 Looking at which numbers are missing from 1-9: We have 1, 2, 3, 5, 6, 7, 8, 9. The only missing number is 4.

3 Therefore, the empty cell (highlighted in yellow) must contain 4.

Example 2: Last Free Cell in a Column

Look at the rightmost column in this grid:

5
3
7
8
1
9
2
6
4
8
1
9
3
7
5
6
2
1
2
6
4
9
2
3
1
7
8
4
8
1
7
5
9
2
3
6
7
9
5
6
4
8
1
3
2
3
2
6
1
8
4
9
7
5
1
4
3
2
6
7
5
8
9
6
7
8
4
3
1
2
9
7
9
5
2
7
4
6
8
1
3

1 The rightmost column (highlighted in blue) has numbers: 4, 1, 8, 6, 2, 5, 7, 9, and one empty cell.

2 Looking at which numbers are missing from 1-9: We have 1, 2, 4, 5, 6, 7, 8, 9. The only missing number is 3.

3 Therefore, the empty cell (highlighted in yellow) must contain 3.

Example 3: Last Free Cell in a 3×3 Box

Look at the top-left 3×3 box in this grid:

5
3
7
8
1
9
2
6
4
8
1
9
3
7
5
6
2
1
2
6
4
9
2
3
1
7
8
4
8
1
7
5
9
2
3
6
7
9
5
6
4
8
1
3
2
3
2
6
1
8
4
9
7
5
1
4
3
2
6
7
5
8
9
6
7
8
4
3
1
2
9
5
9
5
2
7
4
6
8
1
3

1 The top-left 3×3 box (highlighted in blue) has numbers: 5, 3, 7, 8, 1, 9, 2, 6, and one empty cell.

2 Looking at which numbers are missing from 1-9: We have 1, 2, 3, 5, 6, 7, 8, 9. The only missing number is 4.

3 Therefore, the empty cell (highlighted in yellow) must contain 4.

When to Use This Technique

Timing

  • At the beginning: Look for rows, columns, or boxes that are almost complete
  • After filling numbers: Each new number you place might complete a row, column, or box
  • Regular scanning: Periodically scan for last free cells as you work through the puzzle

Strategy

  • Before trying harder techniques: Always exhaust last free cells before moving to more complex strategies
  • Systematic approach: Check all rows, then columns, then 3×3 boxes
  • Chain reactions: One completion often creates new last free cells

⚠️ Common Mistakes to Avoid

  • Miscounting: Double-check which numbers are present in the row/column/box
  • Missing the obvious: Sometimes the last free cell is so obvious you might overlook it
  • Not checking all groups: Remember to check rows, columns, AND 3×3 boxes
  • Calculation errors: Carefully identify which number from 1-9 is missing

💡 Practice Tips

Systematic Approach

  • Systematic scanning: Develop a habit of checking all rows, then all columns, then all 3×3 boxes
  • Start simple: Begin each puzzle by finding all the last free cells first

Verification

  • Double-check: Always verify your answer makes sense before moving on
  • Use elimination: Cross off numbers mentally as you count what's already present

Why This Technique Works

The Last Free Cell technique works because of Sudoku's fundamental constraint: each row, column, and 3×3 box must contain all numbers from 1 to 9 exactly once. When 8 out of 9 numbers are already placed, only one number can possibly fit in the remaining cell.

This technique is also 100% reliable - there's no guessing involved. When you identify a last free cell, you can be absolutely certain of the answer.

Related Techniques

Build From Here

Prerequisites

  • Basic Rules - Understanding Sudoku constraints
  • Number Recognition - Counting 1-9
  • Pattern Recognition - Spotting empty cells
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