Sudoku Candidates: How to Use Pencil Marks
Step 4 of the step-by-step guide says to start writing pencil marks when scanning stops paying. This is what to write, how to keep it true, and how to avoid the version of the habit that costs more time than it saves.
What a candidate is
A candidate is a digit that could still go in an empty cell without breaking the rule. A pencil mark is that candidate written small in the cell, so that you do not have to work it out a second time.
Working one out is the same three-way check you already do. Take the cell in the eighth row, first column.
Its row already holds 4, 1, 9 and 5. Its column holds 5, 6, 8, 4 and 7. Its box adds a 6. Between them that is seven different digits — 1, 4, 5, 6, 7, 8 and 9 — and the two nobody has claimed are 2 and 3.
So the cell gets marked 23. It is not solved, but it is no longer open. Seven
of the nine possibilities are gone and you will never have to rule them out
again.
Mark a unit, not a cell
One marked cell is not worth the ink. The unit around it is, because marks in the same unit talk to each other.
Here is that whole eighth row.
Read it across: 23, 278, 237, then the givens 4, 1 and 9, then 36, then
3, then the given 5.
That second-to-last cell has one candidate. It is a 3, and you can write it in now. No cleverness was involved — it fell out of the bookkeeping. Marking the row did the work that staring at the row would not have done.
Keeping them true
Pencil marks are worth exactly what their upkeep costs. The moment you place a digit it is dead everywhere it can see, which is its row, its column and its box. Cross it out of every mark in those three units before you do anything else.
Writing that 3 in means the other 3s in the row go.
Three strokes, and the marked cell to the left of the new 3 is down to a single 6. Place that, and the first cell of the row is down to a single 2. Place that, and the row unzips the rest of the way on its own.
That cascade is the return on the whole exercise, and it only happens if the marks are current. A stale mark is worse than no mark at all: it claims a digit is still possible when the board has already ruled it out, and sooner or later you will believe it.
When to start writing
Not at the beginning. A puzzle that three scans would have cracked does not need eighty-one cells of notes, and filling them in is twenty minutes of arithmetic you did instead of solving.
Wait until scanning stops producing placements. Then mark one unit — the one you are stuck on, or the emptiest one with the most givens around it — and see what falls out. Mark another when you need another. Most puzzles below the hardest tiers never need the whole grid marked.
The exception is a puzzle you already know is hard. If it is going to demand a technique that reads several cells at once, mark the grid up front and accept the cost, because those techniques are unreadable without it.
What the marks let you see
Two things straight away, and they are the subject of the next guide: a cell with one candidate left, and a digit with only one home left in a unit. Almost every puzzle below expert level is those two, repeated.
Past that the patterns get more interesting. Two cells in a unit holding the same
pair and nothing else — 36 and 36 — own those two digits between them, so
the 3 and the 6 can be crossed out of every other cell in that unit even though
you cannot yet say which cell takes which. A digit that appears in only two
places in a row is the first half of a longer argument. None of that is visible
on an unmarked grid. Marks are what turn a board into something you can reason
about a unit at a time.
Four ways this goes wrong
Marking all eighty-one cells on move one. It feels thorough and it is mostly wasted, because the first dozen placements would have deleted half of what you just wrote.
Letting them go stale. Marks you do not maintain will lie to you, and the lie surfaces as a contradiction ten placements later, by which point you cannot tell which placement was wrong.
Treating a mark as an answer. 23 means two digits survive, not that you get
to pick one. If you are choosing, you are guessing, and
the guide before this one explains why that is a
signal rather than a move.
Marking the cells and never re-reading the unit. The notation is only useful once you look across the row you just wrote. The point is the comparison, not the record.
Where to go next
Naked singles and hidden singles names the two patterns above and covers how to hunt each one. The rules themselves are in What is Sudoku.