How to solve a queens puzzle without guessing
A queens puzzle gives you a square grid split into coloured regions, and asks you to place one queen in every row, every column and every colour — with no two queens touching, not even diagonally. An eight-by-eight board has eight regions and needs eight queens.
It looks like a game of trial and error. It isn't. A well-made queens grid has exactly one solution, and you can reach it without ever placing a queen you're unsure of. The trick is that you spend most of the puzzle placing crosses, not queens.
Start by marking, not placing
Every queen you place rules out a lot of cells at once:
- the rest of its row,
- the rest of its column,
- the rest of its colour region,
- and the eight cells around it, because queens can't touch.
Mark every ruled-out cell with a cross the moment you place a queen. Solvers who skip this step end up re-checking the same cells in their head, and that's where the guessing starts. The crosses are the solve; the queens are just what's left when the crosses run out.
Technique 1: the region with nowhere else to go
Look for the smallest colour region first. A region of one cell is a free queen. A region of two or three cells often has only one cell that survives the crosses already on the board.
More generally: when a region has exactly one cell without a cross, that cell is its queen. The same goes for rows and columns — a row with one open cell left gets its queen there.
This sounds too obvious to count as a technique, but it's the engine of the whole puzzle. Every other technique exists only to produce more crosses so that this one fires again.
Technique 2: a region that lives in one line
Sometimes a colour region is spread over several cells, but all of them sit in the same row. You don't know which cell gets the queen yet — but you know the queen for that row is going to be inside that region.
So every other cell in that row, belonging to any other colour, gets a cross.
The same works for columns, and it's the single most productive move on most boards. A long thin region lying along a row quietly empties that row for everyone else, and those crosses often leave another region with only one cell.
Technique 3: count regions against rows
Technique 2 scales up. Suppose two colour regions sit entirely inside the same two rows. Those two regions need two queens, those two rows hold two queens, so the two rows are fully spoken for — every cell in those rows from any other region gets a cross.
It works for any number: if three regions fit inside three columns, those columns belong to those regions. It also works in reverse — if two rows only contain cells from two regions, those regions' queens must be in those rows, so the regions' other cells get crossed.
This is the technique that cracks the hard boards. When you're stuck, scan for a group of regions squeezed into the same number of rows or columns.
Technique 4: the cell that would kill a region
The no-touching rule creates a quieter kind of elimination. Take any open cell and imagine putting a queen there. It would cross out its eight neighbours. If those neighbours include every remaining open cell of some region, that region would be left with nowhere to go — so the cell you imagined can't hold a queen. Cross it.
The common case is a small region hugging a corner or an edge. The cells next to it are poisonous: a queen on any of them would wipe the little region out.
A close relative: if a region's open cells are two neighbours side by side in a row, its queen is on one of them — and either way, the cells directly above and below both of them are touched. Cross them all.
The loop that solves the board
- Place the obvious queens — single-cell regions and rows or columns with one open cell.
- Cross everything each queen rules out: row, column, region and all eight neighbours.
- Find regions that sit in one row or column and clear the rest of that line.
- Count regions against lines — two regions in two rows, three in three columns.
- Cross any cell whose neighbours would swallow a whole region.
- Go back to step 1. Something new is always obvious now.
If you run the loop and genuinely nothing changes, recheck your crosses before you consider guessing. On a puzzle with a unique solution, a stuck board almost always means a cross that should be there isn't.
Why it's better with two
Queens splits well between two people, which is rare for a logic puzzle. The techniques are local — one person can work the regions along the top while the other counts columns at the bottom — and every queen one of you places drops a fresh set of crosses into the other's half. You end up feeding each other deductions.
It also turns the slowest part of the solve, the stuck moment, into a conversation. "I can't see anything" gets answered by someone who has been looking at a different corner. That's the whole appeal of solving it together: nobody wins, and the board falls about twice as fast.