Tier 3
The rule of clumps
Queens in touching rows need two columns of clearance. When one option would swallow the whole row next door, it is not an option.
This is the technique that uses the adjacency rule as a counting constraint rather than a crossing-off rule, and it is where hard boards give way.
Two neighbouring rows, both still needing queens. Pick a surviving square in the upper row and ask: if the queen went here, what survives in the row below? She kills the square directly beneath and the two diagonals. If the row below has nothing left after that, the square you picked is impossible. Cross it off.
You have eliminated a square without ever placing a queen, using nothing but the geometry of two adjacent rows.
Why touching lines are special
Anywhere else on the board, two queens only need different rows and different columns. In neighbouring rows they need columns at least two apart. That extra unit of clearance is what makes the squeeze possible — pack the surviving squares of two adjacent rows into a narrow band of columns and there is physically no room.
The same applies to neighbouring columns, with rows playing the squeezed role.
When to reach for it
Late, on a board that has stopped yielding to counting. Look for two adjacent lines whose surviving squares sit in a column band barely wider than the lines themselves. If the band is narrower than 2k − 1 columns for k touching lines, something has to give — and often several somethings.
If clumps finds nothing either, the board wants exclusion logic.