BrainPuzzles

Box/line reduction

The mirror of the pointing pair: a digit confined to one box within a line, which pushes it out of the rest of that box.

How to spot one

  1. Pick a row (or column) and a digit, and mark every cell in the line where the digit can still go.
  2. If they all sit inside one box, the digit is somewhere in that box, on that line.
  3. So it cannot be anywhere else in the box. Cross it out of the rest of the box.

A real example

This grid is not a drawing. It came out of our generator, and the position below is one a solver actually reaches: every digit the reasoning uses is on the board, so you can check it yourself without taking anything on trust. The reasoning sits in row 8, shaded blue. The cells it looks at are green; the cells it changes are amber.

6423323694812375527493181374923598377632
In row 8, 4 can only sit inside box 7, so 4 is gone from the rest of that box. So r7c2 loses that candidate.

The same move on a different grid

One example can be a fluke of the grid it sits in, so here is another, dug at hard from the seed second:claiming:148 with 28 starting digits. The technique fires in a different place, for a different digit, and the point is what it buys you afterwards.

6283591413529687975618234891212699131843737124931
In row 4, 7 can only sit inside box 5, so 7 is gone from the rest of that box. So r5c5, r6c4, r6c5 lose those candidates.

What the solver does next, from exactly that position:

  1. hidden single — In column 4, 8 has only one cell left it can go in. r6c4 takes 8, the only digit it had left.
  2. hidden single — In row 5, 8 has only one cell left it can go in. r5c9 takes 8, the only digit it had left.
  3. hidden single — In row 9, 8 has only one cell left it can go in. r9c5 takes 8, the only digit it had left.

That is the pay-off a box/line reduction is worth: not the one cell it names, but the run of cheap moves it unlocks behind it.

Where you meet it

Our grids are graded by the hardest technique they force, so a box/line reduction is what makes a hard grid what it is. Needs a pair or a locked digit. When you ask for a hint on one of them, the first thing the game says is the name of the technique and which row, column or box to use it in, and that step costs nothing — you still get to find the cell yourself.

The other techniques

Ready to use it? Play hard sudoku, take on the daily grid, or start from how to play.