THE CONSTRAINT FRAME
Every Sudoku cell sits at the intersection of three units: a row, a column, and a 3×3 box. That means every empty cell has three independent sources of constraint working on it simultaneously.
Beginners look at a cell and ask "what number goes here?" — then scan for it. Advanced players ask "what numbers can't go here?" — then see what's left. The second question is almost always faster because you're using all three constraints at once.
Elimination is the game. Placement is just what happens at the end.
THINK IN UNITS, NOT CELLS
Instead of looking at one cell and trying to fill it, look at one unit (row, column, or box) and ask: which empty cells can still take this digit?
Work through digits 1 to 9 in each unit. When a digit can only go in one cell of that unit — even if that cell has several other possibilities — you can place it. This is the hidden-single technique, and thinking in units makes it automatic.
Units constrain each other. When you place a digit in a box, it eliminates that digit from a row and a column at the same time, often cascading into a different unit.
PENCIL MARKS AS A THINKING TOOL
Pencil marks are not a sign of weakness. They are how you externalise working memory so your brain can think about relationships instead of remembering possibilities.
When to start pencil-marking: when a row or box is half-full and naked singles are drying up. Mark the remaining candidates for each empty cell. Then look across cells — if two cells in the same unit share exactly the same two candidates, neither of those candidates can appear elsewhere in that unit (naked pair). If a candidate appears in only two cells of a box and both are in the same row, it can be eliminated from the rest of that row.
The marks are not answers. They are a map of what's still possible — and every placement shrinks the map for every unit that cell touches.
WHEN YOU FEEL STUCK
Stuck means you've been staring at the same region. The fix is almost always to change your lens:
- If you've been scanning rows, switch to columns or boxes.
- If you've been thinking about one digit, focus on a different digit.
- If you have pencil marks, look for units where a candidate appears in only two or three cells — those often unlock the next step.
If none of that works, look for naked or hidden pairs. A pair of cells in the same unit that together "own" two candidates can eliminate those candidates from every other cell in that unit, often cracking a section that looked sealed.
The puzzle is always solvable without guessing. If you feel like you need to guess, the right move is to find what you missed — not to branch.
THE ORDER OF ATTACK
A consistent scanning order reduces missed opportunities:
- Scan for naked singles — cells with one remaining candidate.
- Scan for hidden singles — digits that fit only one cell in a given unit.
- Add pencil marks where those two scans leave gaps.
- Look for pointing pairs (candidates in one box all in the same row or column).
- Look for naked or hidden pairs within each unit.
- Repeat from step 1 after every placement — the board has changed.
Most puzzles up to Hard difficulty can be solved by cycling steps 1–3 alone. Steps 4–5 are for when the easy paths have closed.