Aligned Pair Exclusion

Aligned Pair Exclusion works backwards from most techniques: instead of hunting a shape, you take two cells, list every pair of digits they could hold between them, and cross out the pairs that would break something else. Whatever survives tells you which candidates are still possible.

What the pattern is

Two cells are aligned when they sit in the same unit, so they can never hold the same digit. The pattern is at its strongest when the two cells share a box and a line at the same time, because that gives them a large set of common peers.

Now look at the bi-value cells — cells with exactly two candidates — that both of your chosen cells can see. Each one is a small veto. A bi-value cell holding {a, b} must end up as a or b. If your first cell took a and your second took b, that bi-value cell would have nothing left. So the combination a-b is impossible, and so is b-a.

Run that veto over every pair, and any candidate that survives in no combination at all can be removed.

When to look for it

Two conditions make Aligned Pair Exclusion worth the effort:

• Two cells in the same box with three or four candidates each, so the list of pairs stays short.
• A cluster of bi-value cells around them. Three or four vetoes are usually enough to bite.

An empty box corner facing a busy box is the classic setting.

How to apply it, step by step

1. Pick two unsolved cells in the same unit. Keep them small: two or three candidates each is ideal.
2. List every ordered pair of digits they could take. Drop the pairs where both cells take the same digit — the shared unit forbids it.
3. List the bi-value cells that both chosen cells can see.
4. Delete every pair matching a bi-value cell's two digits, in either order.
5. Read the surviving pairs. A digit that appears in no surviving pair for its cell is eliminated.
6. If only one digit survives for a cell, place it.

A worked example

Both cells sit in row 1 and in box 1, so they can never hold the same digit.

r1c1 {2,4,7}   r1c2 {2,4,9}

bi-value cells seen by both (all in box 1):
  r2c3 {2,4}     r3c1 {4,9}     r3c3 {2,9}

pairs (r1c1, r1c2):
  2-4  x   r2c3 would be empty
  2-9  x   r3c3 would be empty
  4-2  x   r2c3 would be empty
  4-9  x   r3c1 would be empty
  7-2  ok      7-4  ok      7-9  ok

2 and 4 survive nowhere for r1c1  =>  r1c1 is 7

Every combination starting with 2 or 4 kills one of the three bi-value cells, so r1c1 can be neither. Only 7 is left, which is a placement rather than a plain elimination — a good day for this technique. More often you will remove one candidate from one cell, which is still enough to restart a stalled grid.

Common mistakes

• Choosing two cells that do not share a unit. Then they may hold the same digit and half your vetoes disappear.
• Using a bi-value cell that only one of the two cells can see. It must see both.
• Forgetting the reverse order. {a, b} vetoes a-b and b-a.
• Eliminating a digit that still appears in one surviving pair. One survivor is enough to keep a candidate alive.
• Testing cells with five candidates each. Twenty-five pairs is a bookkeeping exercise, not a technique.

The same reasoning extends to three aligned cells and to almost locked sets used in place of bi-value cells, but the two-cell form is where the value-per-minute is highest.

Where you meet it

Aligned Pair Exclusion is a diabolical technique. It first becomes necessary in Expert puzzles, typically once the wings and fish have run dry and the grid still has a dense pocket of bi-value cells. Easier levels never need it.