Intermediate

Killer pairs

A killer pair is any two cells, in the same row, column, or box, that are down to the same two digits. Those two digits cannot appear anywhere else in the house. The cage rules are simply the most common way those pairs are born.

Unique two-cell cages are pairs from the start. 1 and 2 in a sum-3 cage, written in both cells, are a naked pair the moment both cells sit in the same house — which they must, if the cage is inside one row, column, or box. Erase 1 and 2 from the other seven cells. This is the first elimination that is not just “fill in the last cell,” and it is worth doing even when you cannot yet see which cell is which.

Hidden pairs are quieter. Suppose 4 and 7 can each go in only two cells of a row, and those two cells are the same. Then those cells are 4 and 7, even if they also still look like they could hold other digits. Erase the other candidates from just those two cells. Cages create hidden pairs when every remaining combination of a cage uses the same two digits and both digits are confined to two cells.

A cage does not have to be a pair itself to produce one. If every legal combination of a three-cell cage contains both 6 and 9, then 6 and 9 are inside the cage. Should the cage have only two cells in a particular box, those two cells are a hidden pair in the box. The rest of the box loses 6 and 9. Players who only look at cage sums miss this, because the sum never became a unique total. The overlap of the combinations did the locking.

Triples are the same story with three digits and three cells. They show up once the pairs have been cleared and a house is still crowded. Do not hunt triples first. Clear every naked and hidden pair you can see, then look again. Most triples are pairs that were hiding under an extra candidate.

The hint on a hard puzzle will often say “pair” or “locked digits” rather than hand you the number. Follow the cells it highlights, erase the digits it names from the rest of the house, and only then look for the single that falls out.

Use this technique on a puzzle

Keep the combination table open while you read, and the rules if the cages themselves are still new.