Pencil Marks in Sudoku: What a Note Does—and Does Not—Prove

Keep possibilities separate from answers

Suppose an empty cell shows small 5 and 9 marks. That means neither digit has been excluded by the restrictions you have recorded. It does not mean both extend to a finished grid, and it does not tell you which to enter.

HoDoKu describes much human solving as reducing candidates until placements become available. [2] Think of notes as a record of deductions made so far. If you remove a candidate, be able to say why. If you add one back, check whether you have forgotten an earlier deduction rather than simply rebuilding a list from the visible answers.

Build a list from three houses

Start with digits 1 through 9. Remove those already placed in the target cell's row, column and box. Here, the row excludes 1, 4 and 7; the column excludes 2 and 6; the box excludes 3 and 8. The remaining list is {5,9}. The braces mean possibilities, not the number fifty-nine.

This is a local constraint example, not a complete puzzle. Checking only the row would leave six candidates; checking all three houses leaves two. Neither survivor has yet been proved. You have narrowed the question without answering it prematurely.

Label partial notes honestly

You can note one digit's positions, a single box, or every candidate on the board. Choose whichever view helps the question you are asking. The important distinction is completeness, not a universal rule about when notes should begin.

A cell with one written mark is not automatically a naked single. Perhaps you wrote only the places for 7 and never checked the other digits. Likewise, two cells marked 2 and 8 do not establish a naked pair unless those are their only remaining possibilities. Stuart's naked-candidate definitions rely on that restriction. [3]

Before a pattern-based elimination, fill the missing information needed for its proof. You need not annotate the whole grid; you do need to avoid treating unwritten candidates as impossible.

Follow one cleanup all the way through

The second example shows a row with unresolved c2 {2,5}, c5 {5,9} and c8 {2,9}. The other six cells are settled. Suppose a separate deduction proves r5c5 = 9. That new proof is essential: this row alone did not choose between 5 and 9.

Remove 9 from r5c8 because the row now has its 9. The remaining candidate there is 2, so place 2. That placement removes 2 from r5c2, leaving 5. Each consequence is tied to a particular confirmed answer.

The first step introduces new proved information; the later steps update its consequences. Keeping those roles separate prevents circular reasoning. You cannot justify the first 9 by referring to the later 2 when that 2 depends on your initial choice.

Update every affected peer once

A classic cell has 20 distinct peers: other cells sharing its row, column or box. Norvig defines this relationship explicitly in his solver. [1] Some peers share two houses with the placement; they are still one cell.

After a confirmed placement, inspect those peers for its digit, remove it, and check what changed. Look for a cell with one candidate and a house where a missing digit has one location. If a cell loses every candidate, stop. That is a contradiction to investigate, not a blank list to ignore. Check the placement and your earlier eliminations before proceeding.

Repair the record without erasing the reasoning

Keeping an impossible candidate can hide a deduction. Deleting a possible candidate without proof can invent a false one. A suddenly neat pair deserves the same scrutiny as a messy row.

If notes become suspect, compare them with recent moves. Recompute direct row, column and box exclusions, then retain additional eliminations whose reasons you can still verify. A wholesale reset may restore candidates that a valid deduction removed earlier. If you cannot reconstruct the history, return to a known correct state rather than guessing which notes to trust.

On paper, a small annotation beside an unusual elimination can help you revisit its reason. In an app, distinguish a feature that recalculates basic candidates from one that preserves your manual deductions; do not assume the two behave identically.

Practise the meaning of a mark

Use the quiz to distinguish listing a possibility, proving a value, and removing a possibility after a proof. Then choose one unresolved cell on a Standard board and explain how each note survived.

Once the lists are reliable, the naked-pairs guide shows how two cells can reserve two digits. The reservation is a deduction about complete candidate sets, not a consequence of writing matching marks.