The pairing argument

Two colours, two rows

Suppose the amber region has open squares only in rows 3 and 6, and the blue region also has open squares only in rows 3 and 6. Neither colour can leave those two rows. Each needs exactly one queen. Two colours, two rows — between them they must fill both, one each, in one order or the other.

Which order does not matter and you will probably never find out. What matters is the consequence: rows 3 and 6 are now fully accounted for. Every other colour on the board is excluded from both of them, and every square in rows 3 and 6 that belongs to a third colour can be crossed out at once.

On a large grid that is a huge move. Two rows of a 12x12 board contain 24 squares; if amber and blue between them hold six of the open ones, the other however-many all die in a single stroke. It is not unusual for a pairing to remove more squares than the previous five moves combined.

Scaling up

The argument does not stop at two. Three colours whose open squares all fall within the same three rows must take one row each, excluding everyone else from all three. Four colours across four rows do the same. The condition is always the same shape: N colours, N lines, no open squares outside them.

In other logic puzzles this pattern is called a naked set — naked pair, naked triple, naked quad. The name is worth knowing because the underlying counting argument is identical wherever it appears, and players who have met it in another puzzle usually find it easier to spot here.

Triples are harder to see than pairs, and the reason is arithmetic rather than logic. A pair requires you to notice two colours that happen to share two lines; a triple requires you to notice three colours whose combined lines number exactly three, which may mean one colour uses rows 2 and 5, another rows 5 and 9, and a third rows 2 and 9. No single colour looks constrained. The set does.

Columns, and the inverse

Everything above applies to columns exactly as written. Two colours confined to columns 1 and 2 take one each and lock the rest of the board out of both. In practice column pairings are found less often than row pairings purely because most people scan horizontally by habit, which makes them a reliable source of missed moves. Deliberately scanning columns is worth a lot on boards where nothing else is moving.

There is also an inverse form. Instead of starting from colours, start from lines: if some set of N rows can only be served by N colours between them — that is, the open squares in those rows belong to only N distinct colours — then those colours are committed to those rows and cannot appear anywhere else on the board. The eliminations land outside the rows rather than inside them.

The forward and inverse forms find different things, for the same reason the region sweep and the line sweep do. Checking both is what makes the technique reliable rather than occasional.

Where to look

Pairings need narrow colours, so the search starts with the colours that have the fewest distinct rows or columns left — not the fewest squares. A colour with six open squares all in two rows is a much better pairing candidate than a colour with three open squares spread over three rows.

So: after each round of eliminations, note for each unresolved colour which rows it can still reach. Look for two colours with the same pair, then for three colours whose combined rows number three. Repeat for columns. This sounds laborious and takes about twenty seconds once you have done it a few times, because most colours are eliminated as candidates instantly on the count.

The payoff is concentrated on hard boards. On an easy grid the sweeps and locks finish the puzzle before a pairing ever becomes relevant. On a demanding one, the pairing is often the single move that separates a ten-minute solve from a twenty-minute one — or from an abandoned board.