The no-touching rule

What the rule actually says

No two queens may occupy squares that touch, in any direction. Side by side horizontally, one above the other vertically, or corner to corner diagonally — all forbidden. A placed queen therefore removes its entire row, its entire column, and the ring of up to eight squares surrounding it.

Note that this is weaker than the adjacency rule in classical chess-queens problems, where queens attack along entire diagonals to the edge of the board. In Queens the diagonal restriction reaches exactly one square. A queen on row 4, column 4 forbids row 5, column 5, but says nothing about row 6, column 6. Players arriving from the classical eight-queens puzzle routinely over-eliminate on this point and make the board unsolvable.

Of the eight squares in the ring, four — the ones directly above, below, left and right — were already eliminated by the row and column rules. The four that matter are the diagonal corners. Those four squares are the entire practical content of the no-touching rule, and they are the ones that get skipped.

Why they get skipped

The row and column sweeps are visually easy: you run your eye along a straight line. The diagonal neighbours require you to stop and look at four isolated squares in four different directions, which breaks the rhythm of the sweep. Under any kind of time pressure, that is the step that gets dropped.

The consequence is delayed and therefore hard to trace. A missing diagonal mark does not cause an immediate error. It causes a colour to appear to have two open squares when it really has one, so a forced queen stays hidden. Ten moves later the board looks stuck, and nothing about the stuck position points back at the omission.

This is why the standard advice for a stalled board is to audit the marking before hunting for a clever technique. In the overwhelming majority of cases the board is not hard; it is out of date.

The marking routine

Place the queen. Then, in a fixed order every time: clear the row, clear the column, clear the remaining squares of the colour, then explicitly touch each of the four diagonal neighbours. Saying the four out loud, or counting them on your fingers, is not excessive on a 12x12 board — the point is to make the step something you complete rather than something you intend.

Fixed order matters more than which order. The value is that a routine you always perform the same way is one you notice yourself skipping. An ad-hoc sweep that varies by position is one you cannot audit.

When a board does stall, re-run the audit on every queen already placed, not just the recent ones. The missing mark is as likely to be from the second queen as the eighth, and an audit that only covers what you remember doing is not an audit.

Using adjacency positively

The rule is not only a bookkeeping chore. It generates deductions of its own. Two adjacent rows cannot hold queens in the same column or in neighbouring columns, which means a colour spread across two adjacent rows is more constrained than its square count suggests. A colour drawn as a two-by-two block has four squares but, in effect, far fewer useful positions relative to its neighbours.

This is also the mechanism behind region shadows. Candidate squares clustered together have heavily overlapping adjacency rings, so the squares tucked against the cluster are attacked whichever candidate wins. Without the no-touching rule that technique would not exist — the overlap would only come from shared rows and columns.