sudoku · EN
Sudoku Candidate Elimination: A Practical Solving Guide
Sudoku candidate elimination removes digits that cannot legally occupy a cell, then uses relationships inside rows, columns, and 3×3 boxes to narrow the remaining options. Start with complete candidate lists, look for singles, and apply pairs or box-line restrictions only when their conditions are fully satisfied.
Study the method step by step: Sudoku Locked Candidates: Worked Example · Sudoku Naked Pairs: Worked Example · Sudoku Hidden Singles: Worked Example · Sudoku X-Wing: Verified Worked Example.
Build candidates from all three units
In standard 9×9 Sudoku, every row, column, and 3×3 box contains each digit from 1 to 9 exactly once. These regions are called units. For an empty cell, remove digits already present in its three units.
Suppose a cell's row already contains {1,3,5,7}, its column contains {2,5,8}, and its box contains {1,4,6}. Together they exclude {1,2,3,4,5,6,7,8}; only 9 remains. This is a naked single: the cell has one legal candidate after all direct restrictions are applied.
Use a consistent notation. Pencil marks describe remaining possibilities, while a large entry means the value has been proved. A missing pencil mark is meaningful only if your lists are complete and current. Partial notes can make a digit appear unique when you simply forgot to write it elsewhere.
Distinguish naked and hidden singles
A naked single is one cell with one candidate. A hidden single is one digit with only one available position in a unit, even if that cell has several candidates. Check both perspectives: what can this cell contain, and where can this digit go?
For example, imagine the empty cells in a row have candidates A = {2,5}, B = {2,5,8}, C = {2,7}, and D = {5,7}. The row is missing {2,5,7,8}. Only B can contain 8, so B = 8 despite its three pencil marks.
After entering a single, remove its digit from every unsolved cell sharing its row, column, or box. Check those units again: one placement can create a naked single in its column and a hidden single in its box.
Work through a naked pair
Consider one row with five blank cells. Its missing digits are {1,2,4,6,9}, and their candidate lists are A = {2,6}, B = {1,2,6}, C = {2,6}, D = {1,4,9}, and E = {4,9}. A and C form a naked pair: each can take only 2 or 6.
Because A and C must contain 2 and 6 between them, no other cell in that row can use either digit. Remove 2 and 6 from B, leaving B = 1. You still do not know whether A is 2 or 6, but the pair has produced a confirmed value elsewhere.
After B becomes 1, remove 1 from D, leaving D = {4,9}. D and E now reserve 4 and 9, with their order unresolved. Further progress must come from their columns or boxes.
The pair works because two cells reserve exactly two digits in one unit. Cells containing extra candidates are not a naked pair. If both paired cells share a box, the same reservation applies there.
Use box-line restrictions
Focus on a single digit in one 3×3 box. If every remaining position for that digit lies in the same row, the digit must appear somewhere in that row inside the box. You can therefore remove it from cells in the same row outside the box.
For example, in the top-left box, suppose 7 can appear only at r2c1 or r2c3. The box needs a 7, so row 2's 7 must occupy one of those two cells. Remove candidate 7 from r2c4 through r2c9 wherever it appears. Neither of the two original positions is solved yet.
The reverse direction also works. If all remaining positions for 7 in row 2 lie inside one box, remove 7 from the other rows of that box. In either direction, first inspect every relevant candidate position. One overlooked position outside the claimed intersection invalidates the elimination.
Recognise a hidden pair carefully
A hidden pair occurs when two digits can appear only in the same two cells of a unit. Those cells may contain other candidates, but they must reserve the two restricted digits between them. Remove their extra candidates; do not remove the pair's digits from the paired cells.
Suppose the digits 3 and 8 appear as candidates only in cells A and B of a column. A has {1,3,6,8} and B has {3,5,8}, while no other cell in the column allows 3 or 8. Reduce both A and B to {3,8}. Their order remains unresolved.
Check both digits separately. If 8 also appears in a third cell, the hidden pair does not exist. Check the entire unit before acting on a pattern, especially when notes may be incomplete.
Keep elimination auditable
Each removed candidate should have a short reason: an existing value in a shared unit, a reserved pair, or a digit confined to a box-line intersection. If you cannot identify the exact cells and rule behind an elimination, leave the candidate until you can.
An empty candidate list indicates an inconsistency in the current entries or notes. Likewise, a unit missing a digit with no possible position signals trouble. Review your last deduction and rebuild the affected candidates from confirmed entries before attempting a more advanced technique.
Check a completed board by verifying all 27 rows, columns, and boxes independently. Each must satisfy the 1–9 rule, and every original given must remain unchanged.
- Create complete candidate lists before searching for hidden singles or pairs.
- After every entry, update its row, column, and box immediately.
- Name the shared unit when applying a pair elimination.
- Check every location for a digit before using a box-line restriction.
Frequently asked questions
Should I write every Sudoku candidate?
Complete candidates are useful when learning elimination or solving a difficult grid because hidden singles and pairs depend on knowing every possible position. You can use lighter notes on easy puzzles, but do not treat an unwritten digit as impossible unless you have actually checked the cell's row, column, and box.
What is the difference between a naked pair and a hidden pair?
A naked pair is two cells in one unit containing only the same two candidates. A hidden pair is two digits whose only possible positions in a unit are the same two cells, even if those cells contain extra notes. Both reserve two digits; the immediate eliminations occur in different places.
Does eliminating a candidate always produce a number?
No. Many useful eliminations only narrow a cell or remove an entire alternative pattern. Keep the remaining candidates and update the affected units. A later deduction may turn that earlier reduction into a single. Never choose between two surviving candidates merely because the current technique has stopped producing entries.
