Pointing pair

What is a pointing pair?

A pointing pair happens when, inside one box, every possible place for a digit lies on the same row or the same column. The digit has to go in one of those cells, so the box’s copy of it ends up on that line — and the line cannot have another copy anywhere outside the box. You remove the digit from those cells.

It is a pair when there are two candidate cells and a triple when there are three; the logic is the same. Unlike a single, it places nothing by itself. It clears candidates, and the payoff is usually a single a move or two later.

How to spot a pointing pair

You need pencil marks for this one, at least for the digit you are checking. Go box by box and, for each digit a box is missing, look at where its candidates sit. If they all fall in one row or one column of the box, follow that line out of the box in both directions and cross out the digit wherever it appears.

The tell-tale is a box where the rest of the cells are already blocked for that digit by copies of it in neighbouring rows and columns, leaving two or three candidates lined up.

A worked example

Cells are named by row and column: r4c1 is row 4, column 1. Rows count from the top, columns from the left, and boxes are numbered 1 to 9 reading like a book, so box 1 is top-left and box 9 is bottom-right.

This position is from a Hard puzzle in our game, at the point where both kinds of single have run out. Look at box 4 and the digit 7. The box has four empty cells, but only two can take a 7: r4c1 and r6c1, both in column 1. Whichever of them is the 7, column 1’s 7 is inside box 4.

So no other cell in column 1 can be a 7. Below the box, r7c1 and r9c1 still list 7 as a candidate, and both lose it. The effect is immediate: r7c1 had only 1 and 7 left, so it becomes a naked single, and r7c1 is 1.

Why it works

Box 4 needs a 7, and the only cells that can take it are in column 1. Column 1 can hold just one 7. Whichever of r4c1 and r6c1 gets the box’s 7, that is also column 1’s 7, so a 7 anywhere else in the column would be a second one. You never need to know which of the two cells is right — only that both are on the same line.

Which difficulty levels need it

Hard and Expert. Our Hard puzzles are graded so that singles run out before the grid is full, and a pointing pair, a box-line reduction or a naked pair has to break the deadlock.

In 199 Hard puzzles we sampled, the solver used a pointing pair in 20, and could not have finished 5 of them without it. Those numbers are low because pointing and box-line reduction overlap: the solver checks rows and columns before boxes, so it usually reaches the same elimination the other way round. For a person, the box view is often the quicker one to see. Expert puzzles need this and more.