Region shadows

The idea

Take a colour with three open squares left. Wherever its queen ends up, it will be on one of those three. Now pick some other square on the board and ask: would it be attacked if the queen went on the first candidate? On the second? On the third? If the answer is yes all three times, that square is dead — not probably dead, definitively dead, because there is no version of the puzzle in which it survives.

This is worth dwelling on because it is the first technique in Queens that produces certainty out of uncertainty. Everything up to this point required you to pin something down. Here you deliberately refuse to decide where the colour's queen goes, and still walk away with eliminations.

The region-line lock is actually a special case of this. If all of a colour's candidates are in row 6, then every other square in row 6 is attacked by all of them — the shadow is the whole row. Shadows generalise that to candidates that are not neatly lined up.

Where shadows come from

A queen attacks its entire row, its entire column and the eight squares immediately surrounding it. So for a set of candidate squares to cast a shadow, their attack patterns have to overlap somewhere, which happens in three recognisable situations.

First, candidates sharing a row or column: the shared line is entirely shadowed, which is the lock case. Second, candidates clustered in a small block — a two-by-two corner, or three squares in an L — whose adjacency rings overlap heavily. Squares tucked against such a cluster are attacked from every candidate at once. Third, candidates occupying two adjacent columns: any square that sits in both of those columns' reach, or wedged between them, may be covered regardless.

The second case is the one worth hunting for, because it is invisible unless you are looking. A colour pinched into a compact blob in one corner of the board rarely looks constrained — it may still have four or five open squares — but the squares immediately around that blob are often all shadowed, and those eliminations are what make the neighbouring colours collapse.

Doing it without burning a minute

Checking every square against every candidate is too slow to do routinely. Narrow it first. Only colours with two, three or four open squares are worth testing — beyond that the candidates are too spread out to overlap usefully. Then only test squares in the rows and columns those candidates occupy, plus the ring immediately around the candidate cluster. Everything else on the board is out of reach of at least one candidate by definition.

With that narrowing, a shadow check on a three-candidate colour takes a handful of comparisons and frequently produces one to three eliminations. The habit is to run it on the two smallest unresolved colours whenever both sweeps come back empty, before reaching for anything more elaborate.

One caution: shadows are only valid against the colour's current open squares. If your marking is stale and a candidate you have discounted is actually still live, the shadow you compute will be too large and you will cross out a square that should have survived. Shadows amplify marking errors, so make sure the board is current before you use one.

Why it matters on hard boards

Boards that a solver rates as demanding usually have a specific property: no colour is confined to a single line, and the region sizes are fairly even, so there is no obvious smallest region to attack. Those boards give up nothing to the sweeps and nothing to the locks. Shadows are frequently the only technique that produces the first elimination, and once one or two squares come off the rest of the board unravels normally.

That is the argument for learning this before the more famous pairing argument. The pairing argument is more spectacular when it fires, but it needs colours to have already been narrowed down to a few lines. Shadows are what does the narrowing.