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, since 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

The pair is D2 and D3. Both sit in row D and in box 4, so they can never hold the same digit. Watch the four bi-value cells that see both of them.

123456789

ABCDEFGHJ

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Step 0 of 5

1. Easier techniques are used up. Look at D2 and D3, which share row D and box 4.

2. 1

   D2 and D3 share row D and box 4, so they can never hold the same digit.

   D2 has 1, 5, 7, 9 and D3 has 2, 5, 9. Test which pairs of values they can take together.

3. 2

   Four bivalue cells see both D2 and D3: D7 {1,2} and D4 {2,9} through row D, E1 {2,7} and F3 {2,5} through box 4.

   If D2 and D3 took both digits of one of these cells, that cell would be left empty.

4. 3

   Try D3 = 2 with D2 = 1. D7 sees both cells through row D, so it loses 2 and 1 and has nothing left.

5. 4

   The other partners fail the same way: D2 = 5 empties F3, D2 = 7 empties E1, D2 = 9 empties D4.

   So 2 in D3 has no partner in D2 at all. The values 5 and 9 in D3 each keep a partner that leaves every one of these cells a digit.

6. 5

   2 removed from D3 because every value of D2 paired with it would empty one of D7, F3, E1 or D4.

Easier techniques are used up. Look at D2 and D3, which share row D and box 4.

D2 holds 1, 5, 7 and 9, and D3 holds 2, 5 and 9. Four bi-value cells see both: D7 and D4 through row D, E1 and F3 through box 4. Test D3 = 2 first. Paired with D2 = 1 it empties D7, with D2 = 5 it empties F3, with D2 = 7 it empties E1 and with D2 = 9 it empties D4. So 2 has no partner in D2 at all and comes out of D3, which becomes . The 5 and 9 in D3 each keep a partner that leaves every excluder a digit. This time the result is one elimination rather than a placement, 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.

Aligned Triplet Exclusion

Take the same idea to three cells at once and list every combination of three digits they could hold together, skipping any combination that repeats a digit in two cells sharing a unit. The excluders are now cells with two or three candidates that see all three chosen cells: a combination is impossible if the three chosen cells would take every candidate such an excluder has.

Strike out any combination that would leave one of those excluder cells with no candidates left, the same as before. A digit surviving in no combination for its cell is eliminated.

Aligned Triplet Exclusion is harder to find and rarer to pay off than the two-cell form: the combination list grows fast, and every excluder has to see all three chosen cells. It sits above Aligned Pair Exclusion on the difficulty ladder, closer to the extreme end, so worth a look only after the pair form has failed.

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.