Aligned Pair Exclusion

How an aligned pair works

Whatever two digits a pair of cells ends up with form one combination. So list every combination their candidates allow and test each against the grid.

The rules come down to two checks. Two cells that see each other can't hold the same digit. And no other cell may be left without a candidate, which happens when the pair takes both digits of a bivalue cell that sees both cells, or two digits of an almost locked set that does.

Each candidate of the pair needs a combination that passes. A candidate in none goes.

The bivalue-cell case

A bivalue cell has two candidates. If it sees both cells of the pair and the pair takes exactly those two digits, it is left empty.

In the figure below, row four, column two holds 1 and 9, and row five, column two holds 2 and 6. Row eight, column two holds 1 and 2 and sees both. So those two cells can't be the 1 and the 2 together.

When a bivalue cell is not enough

An almost locked set works the same way, one size up: N cells in one house with N + 1 candidates. Take two of its digits away and N cells are left with N − 1 digits. So if every cell of the set sees both cells of the pair, a combination that uses two of its digits fails.

A bivalue cell is the smallest such set. This grid needs bigger ones. On it, every strike that works through bivalue cells alone removes a candidate that a naked pair or naked triple also removes.

Testing a combination against a set is also the first step toward ALS chains, later in this chapter.

How to spot one

Look for two cells that see each other with two to four candidates each. Test against the cells that see both. Keep the pencil marks complete, since a missing candidate can hide a combination that survives. The definition also allows two cells that only see the same group of cells, but the game checks only pairs that see each other.

The six combinations

Row four, column two first, then row six, column two:

• 1 and 1: struck, the same digit in two cells that see each other.
• 1 and 4: survives.
• 1 and 6: struck, the set in rows five and eight keeps only the 2 for two cells.
• 9 and 1: survives.
• 9 and 4: struck, the set in rows two and seven keeps only the 7 for two cells.
• 9 and 6: struck, the box-four set keeps only the 2 and the 5 for three cells.

Only the 6 appears in no surviving line.

Why it works

In the solution the two cells hold some combination, and it breaks no rule. Every combination that puts the 6 in row six, column two leaves a set with fewer digits than cells. So none of them is the real one, and the 6 can't be right.

How it relates to other moves

When bivalue cells do all the striking, the move can be an XYZ-wing seen from another side. On that article's grid, the game's check for this move pairs the cell that loses the 2 with the pivot, the wings strike its combinations, and the same 2 goes.

With almost locked sets the move often overlaps the restricted common candidate, but not here. On the starting grid no pair of sets removes this 6, and none of the game's wing, coloring, chain or cycle checks from earlier chapters does.

What the in-game hint shows

The in-game hint may show a different or simpler step first. On this grid it opens with a naked pair: row six, columns six and nine hold only 5 and 8, so the rest of row six loses its 5s and 8s. No cell of the pair or its three sets changes.

A naked triple and another naked pair follow, then a 3D Medusa step that strikes this 6 along with many other candidates.

On the starting grid, no step the hint checks before Nishio strikes that 6 on its own. Nishio does. If row six, column two is the 6, row six, column three must be the 4 and row four, column two the 1. Then rows five and eight of column two both need the 2. An ALS chain also removes the 6, and the game checks for one before this move. So the hint seldom shows an aligned pair. A different route can still be right.

Where this sits on the path

This is the third technique of the last chapter, after the restricted common candidate. Aligned triple exclusion comes next, the same test with three cells. ALS chains and death blossoms follow later in the chapter.