Aligned Pair Exclusion
Aligned Pair Exclusion — APE for short — is a precise late-game technique. It looks at two cells that see each other and asks which pairs of digits they could jointly take. Some of those pairs turn out to be forbidden by other cells the two can both see, and ruling them out often erases a candidate that no simpler method could reach.
The idea
Two aligned cells share a unit, so they can never hold the same digit. List every ordered pair of candidates they might take. Now bring in the cells that both aligned cells can see — especially any near-locked group whose digits are confined to those shared peers. Any combination the aligned pair proposes that would leave such a group with nowhere to go is impossible, so cross it off the list. When every surviving combination avoids a particular digit in one of the cells, that digit can be eliminated there.
A worked example
Let r4c4 hold {2,5,8} and r4c7 hold {5,8,9}; they share row 4, so they can never be equal. Both of them also see the bivalue cell r4c1 = {2,9}. Now test one combination: r4c4 = 2 and r4c7 = 9. That would take both of r4c1's only digits — 2 by r4c4 and 9 by r4c7, each of which r4c1 can see — leaving r4c1 with nowhere to go. So the pair (r4c4 = 2, r4c7 = 9) is impossible, even though 2 is fine in r4c4 and 9 is fine in r4c7 on their own. That is the heart of APE: a combination struck down not by either cell alone but by a peer they share. Chase enough forbidden pairs and a candidate that no surviving pair can partner drops out of one of the cells.
The aligned pair r4c4 and r4c7 (blue) both see the bivalue r4c1 = {2,9} (green). The combination r4c4 = 2 with r4c7 = 9 (red link) would starve r4c1 of both its digits — so that pair is forbidden, though each value is legal alone.
Step by step
- Choose two cells that see each other and still hold several candidates.
- List their possible pairs, remembering they cannot share a digit.
- Bring in shared peers — locked sets both cells can see.
- Strike impossible pairs, then remove any digit no surviving pair allows.
Common mistakes
- Cells that don't align. The two cells must share a unit; otherwise they could take equal digits and the reasoning breaks.
- Ignoring shared peers. The exclusions come entirely from cells both aligned cells can see — miss those and nothing is forbidden.
- Stopping at one dead pair. You eliminate a digit only when no surviving combination uses it in that cell.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.