When a box decides a whole line

The short version

Pointing happens when, inside a box, every remaining home for a digit lies in one row or one column. The box has to put the digit somewhere on that line, so the line is already supplied — and cannot hold the digit anywhere outside the box.

It is the cheapest elimination in sudoku. One digit, one box, and often no candidate grid at all.

The shape

Pick a box. Pick a digit. Mark where that digit can still go inside the box.

If all of those cells sit in a single row, the digit in this box is on that row — no matter which cell finally takes it. Same for a single column.

Two cells in a row is called a pointing pair. Three cells in a row is a pointing triple. The count does not change the logic, only the number of eliminations inside the box.

What breaks the pattern: one home outside the shared line. If a fourth cell elsewhere in the box can also take the digit, the box is no longer committed to that line and nothing can be concluded.

Why it works

A box must contain exactly one of each digit. That is not a convention, it is the definition of a completed grid.

So if all the homes for a digit in a box lie on one row, the digit is somewhere on that row inside the box. The row now has its copy of that digit. A row can hold only one, so no other cell in the row can take it.

The reasoning is short enough that the risk is not misunderstanding it, but skipping it. Pointing is so cheap that solvers often look straight past it while hunting for something more impressive.

A real board

The claim: in the top-middle box, 1 has exactly two homes, R3C4 and R3C6, and both sit in row 3.

• The box's empty cells that can still take a 1 are those two. Nothing else in the box is available.
• Both are in row 3. The box cannot place its 1 anywhere else.
• Therefore row 3's 1 comes from this box, and R3C3 cannot be 1. It gets deleted.

One digit, one box, one elimination — and note that this required no candidate grid. You only ever asked "where can the 1 go in this box", which is a question you can answer by eye.

How to scan for one

Pointing is found box by box, and it is the fastest sweep on the board if you do it in the right order.

1. Take a box, take a digit. Start with the digit that appears most often on the board, since it has the fewest homes left.
2. Ask one question. Where can this digit still go inside this box? Do not write candidates; just count legal cells.
3. Check the line. If all of the answers share a row, scan that row outside the box for the digit and delete it. If they share a column, scan the column.
4. Only then check the other direction. If the homes do not share a line, the same cells might still be a claiming move from the row's point of view — but that is a different question and needs a separate pass.
5. Move on. Nine boxes times nine digits is eighty-one cheap questions, and most of them are answered in a second.

A practical detail: pointing is most productive in boxes that are nearly full. A box with two empty cells has at most two homes per digit, which means the line condition is satisfied almost by accident. Pointing pairs live in crowded boxes.

Pointing vs claiming

Both are box-line reduction, and they are the same fact read from two ends.

• Pointing: the digit is confined to a line inside a box → eliminate along the line, outside the box.
• Claiming: the digit is confined to a box inside a line → eliminate inside the box, outside the line.

Pointing fires from boxes and shoots outward. Claiming fires from rows and columns and shoots inward. Mixing them up produces eliminations in the wrong unit, which is how a solvable board becomes unsolvable.

Claiming, the inward direction.

The traps

One stray home. Three cells in a row inside the box is a pointing triple only if there is no fourth home elsewhere in that box. A single cell outside the shared line destroys the whole argument, and it is easy to miss because it may be sitting in an unrelated corner.

Wrong direction. Finding the pattern in a box and then deleting from the box. The box is where the digit must go; the box has nothing to give up. All the eliminations are outside it.

Confusing it with a single. If the box has only one home for the digit, you have a hidden single and should place the number, not eliminate. Pointing requires at least two homes.

Where to use it

Pointing is the first technique to check when a puzzle stalls, ahead of anything that needs a candidate grid. It costs a few seconds per box and it resolves a surprising number of medium puzzles on its own.

It also chains well: a pointing elimination frequently creates a hidden single in the affected line, which is why a stalled board can suddenly open up after one cheap deletion. If pointing produces nothing, that is useful information too — the board is genuinely stuck, and it is time to write candidates down.