What Are Pencil Marks
Pencil marks are small candidate numbers written in empty cells. They show which digits are still possible for that cell. To create pencil marks, look at the row, column, and box for each empty cell. Any digit from 1 to 9 that does not already appear in those groups is a candidate. Write them small.
Why You Need Them
Easy puzzles can be solved by scanning alone. Medium and hard puzzles cannot. You need to see the candidates to spot patterns like naked pairs, hidden pairs, and pointing pairs. Without pencil marks, these patterns are invisible. Every experienced solver uses them.
How to Write Them
Most solvers write candidates in a 3x3 pattern inside the cell, matching the digit position to a mental keypad layout. 1 goes in the top left. 5 goes in the center. 9 goes in the bottom right. This makes it easy to scan for specific numbers. In apps, pencil marks are usually toggled with a tap.
Naked Pairs Explained
A naked pair is two cells in the same row, column, or box that share exactly the same two candidates. For example, if cell A has candidates 3 and 7, and cell B in the same row also has candidates 3 and 7, those two numbers must go in those two cells. Remove 3 and 7 from all other cells in that row.
Why Naked Pairs Matter
Eliminating candidates with naked pairs often reveals new naked singles elsewhere in the grid. One good pair can trigger a chain of easy fills. This is why pencil marks are so important. Without them, you would never see the pair. With them, it jumps out.
Practice
Solve medium puzzles with full pencil marks. Every time you get stuck, look for naked pairs first. Then check for hidden pairs and pointing pairs. Most medium puzzles need only these three techniques beyond basic singles. Our medium puzzle book (Volume 2) is perfect for practicing this skill.