Last Possible Number Technique
The Last Possible Number technique, also known as "Naked Singles," involves systematically eliminating possible numbers for a cell until only one candidate remains. This is a process of elimination that focuses on individual cells rather than groups.
How It Works
For any empty cell, start with all possible numbers (1-9) and eliminate those that already appear in the same row, column, or 3×3 box. When only one number remains possible, that's the answer for that cell.
Example 1: Basic Elimination Process
Let's find what number goes in the highlighted cell:
Before: Initial Candidates
After: Final Answer
Step-by-Step Elimination Process
For the cell at row 9, column 5 (highlighted in yellow):
1Check Row 9: Contains 9, 4, 5, 6, 8, 7, 4, 3
- 3, 4, 5, 6, 7, 8, 9 - eliminated by row
- 1, 2 - still possible
2Check Column 5: Contains 1, 7, 2, 8, 4, 3, 9, 5
- 1, 2, 3, 4, 5, 7, 8, 9 - eliminated by column
- 6 - still possible from our previous list
3Check Bottom-Center Box: Contains 4, 5, 1, 2, 9, 6, 7, 8, 3
- Wait, column 5 already has 1, 2, 3, 4, 5, 7, 8, 9
- From row elimination: only 1, 2 were possible
- From column elimination: 2 is eliminated (present in column)
- 1 - the only remaining possibility!
4Final Answer: Only 1 can go in this cell
Example 2: Multiple Eliminations
Here's a cell with more complex elimination:
The elimination cells (highlighted in red) show all the numbers that block our target cell:
- Row 9: Contains 9, 4, 5, 6, 8, 7, 4, 3 - eliminates these numbers
- Column 5: Contains 1, 7, 2, 8, 4, 3, 9, 5 - eliminates these numbers
- Bottom-center box: Contains 2, 9, 6, 3, 8, 7, 4, 5, 1 - eliminates these numbers
- Result: After all eliminations, only 1 remains possible
Step-by-Step Process
- 1Select an empty cell to analyze
- 2List all possibilities (1, 2, 3, 4, 5, 6, 7, 8, 9)
- 3Check the row - eliminate numbers already present
- 4Check the column - eliminate numbers already present
- 5Check the 3×3 box - eliminate numbers already present
- 6Count remaining candidates - if only one remains, that's your answer!
When to Use This Technique
- Early in solving: Good for cells with many constraints
- Systematic approach: Go through each empty cell methodically
- When other techniques fail: Sometimes elimination reveals hidden singles
- Candidate tracking: Essential for maintaining candidate lists in complex puzzles
⚠️ Common Mistakes to Avoid
- Missing constraints: Forgetting to check all three constraints (row, column, box)
- Calculation errors: Miscounting which numbers are already present
- Premature conclusions: Placing a number when multiple candidates still exist
- Incomplete elimination: Not checking all related cells thoroughly
- Overcomplication: Using this technique when simpler ones would work
Advanced Applications
This technique becomes the foundation for more advanced strategies:
- Candidate notation: Write small numbers in cells to track possibilities
- Pencil marks: Use systematic notation to track eliminations
- Chain eliminations: One elimination might trigger others
- Pattern recognition: Spot when eliminations create obvious next moves
Practice Tips
- Start simple: Pick cells with the most filled neighbors
- Be systematic: Always check row, then column, then box
- Double-check: Verify your eliminations before concluding
- Use pencil marks: Write small candidate numbers to track possibilities
- Practice pattern recognition: Learn to quickly spot which numbers are blocked