Box/Line Reduction

Every box in the grid overlaps three rows and three columns. Box and line share exactly three cells, and those three cells belong to both at once. That overlap is where the technique lives.

Box/line reduction, sometimes called claiming, works along one line — a single row or a single column. Take a digit that still needs a home somewhere in that line. Look at the cells in the line where the digit could still go. If all of those candidate cells happen to sit inside one box, you have learned something concrete: the digit belongs to that box, and specifically to the stretch of the line that passes through it. It cannot be anywhere else in the line, so it cannot be anywhere else in the box either. You are entitled to remove it from the box's other two rows or columns.

The line is where the constraint comes from. The box is where the eliminations land. Two candidate cells will do the job, and three work just as well.

If you have met the pointing pair, you already know half of this. The two techniques are the same shape read in opposite directions, and holding them side by side is the quickest way to keep them straight.

Pointing reasons box to line. You notice that inside a box, a digit's only candidates all lie on one line, so the digit is claimed by that line within the box, and you clear it from the rest of the line outside the box.

Box/line reduction reasons line to box. You notice that along a line, a digit's only candidates all lie inside one box, so the digit is claimed by that box within the line, and you clear it from the rest of the box outside the line.

The overlap between box and line does the reasoning both times. In pointing you look out from the box along the line. Here you look in from the line into the box. Once the symmetry clicks, you stop memorising two separate rules and start seeing one situation from whichever end presents itself first.

Look at the grid above. The line in question is column 4, and the digit is 7. Run down the column and mark every cell where 7 is still a live candidate — those are the cells outlined in blue. In this puzzle they all fall in the top stretch of the column, at r1c4 and r3c4. Both of those cells sit inside box 2, the top-middle box.

That is the whole trigger. Column 4 needs a 7 somewhere, and the only places left are those two blue cells. Wherever the 7 finally lands, it lands inside box 2. So box 2 has now used up its 7 on the column-4 line, even though you do not yet know which of the two cells takes it.

Now turn your attention to the rest of box 2 — the six cells in columns 5 and 6. None of them can hold a 7 any longer, because the box's 7 is spoken for by column 4. That is where the eliminations come from. The cell shaded red, r1c5, had 7 among its candidates; the digit is struck through because you can now cross it off. Any other 7 candidates sitting in columns 5 or 6 of box 2 go the same way.

Notice what you did not need. You never placed the 7. You never worked out which blue cell wins. The mere fact that both candidates are trapped in one box was enough to tidy up the neighbouring cells, and those smaller candidate lists often unlock the next move.

This one hides more easily than pointing, because you are scanning lines rather than the tidy nine-cell blocks of a box. A steady routine helps.

Work one line at a time, row or column, and one digit at a time. For the line you are on, ask where that digit can still go. If the answer is two or three cells and they share a box, you have a box/line reduction. Glance into the other two lines of that box and remove the digit wherever it appears.

The telling pattern is a digit's candidates in a line huddled at one end — all in the left third, the middle third, or the right third of a row, or the top, middle, or bottom third of a column. That huddle is the box boundary showing through. If the candidates straddle two boxes, there is nothing to claim and you move on. It is often quickest to hunt right after you have filled a digit in, when a fresh placement has just squeezed that digit's options in the neighbouring lines down to a single box.

Box/line reduction rarely finishes a puzzle by itself. It is a clearing tool: it trims candidate lists so that a hidden single, a naked pair, or another intersection becomes visible a move or two later. On its own it feels modest, which is why beginners skip past it. That is a mistake — much of the middle game is small reductions like this one compounding into a real breakthrough.

Because it shares its geometry with pointing, practise the two together. Whenever you settle on a box-line overlap, check it from both ends: is a digit trapped in one line of the box, or trapped in one box of the line? Reading the same intersection both ways doubles what a single glance earns you.

Keep your notes honest. The technique is only as reliable as your candidate marks, so if a pencilmark is stale the elimination will be wrong. Confirm that the line's candidates for the digit genuinely all sit in one box before you cross anything out, and the move is airtight every time.