Every Pencil Grids Queens puzzle can be solved by logic alone. These are the six ideas our solver uses, from easiest to hardest. The hint button uses exactly the same ideas, in the same words.
1. Last cell standing
If a row, column or region has only one cell that can still take a queen, the queen goes there. It sounds obvious, but scanning for it after every move is what keeps a solve flowing.
2. Crossing out around a queen
Every queen rules out its row, its column, its region and its eight neighbours. Do this every time, straight away. Most mistakes in Queens come from a missed diagonal neighbour.
3. A region trapped in one line
When all the open cells of a region lie in one row (or one column), the region's queen must be that row's queen. Cross out everything else in the row. Look for this especially with long thin regions and with regions whose other cells you've already crossed out.
It works backwards too: when all of a row's open cells belong to one region, that region's queen is in this row, so cross out the region's cells in other rows.
4. The cell that would break something
Pick an open cell and imagine a queen there. Would it wipe out every remaining cell of some other row, column or region? Then it can't be a queen. This catches cells next to small regions: a cell that touches both cells of a two-cell region can never hold a queen, because it would leave that region empty.
5. Groups of regions squeezed into the same rows
This is the technique that separates medium from hard. If two regions only have open cells in the same two rows, those two rows' queens are used up by those two regions. Every other cell in those two rows can be crossed out. The same goes for three regions in three rows, and for columns.
The reverse works too: if two rows can only take queens from the same two regions, cross out those regions' cells everywhere else.
To find these, scan for regions that are short and wide (they hug a couple of rows) or tall and thin (they hug a couple of columns).
6. Short what-if chains
Expert grids sometimes need a little look-ahead. Choose a cell, imagine a queen there, and follow techniques 1 to 5 for a few steps. If some row, column or region ends up with nowhere to go, the cell must be empty.
Good candidates are cells in rows or regions with only two options left. One of the two must be right, so testing one tells you about both. Keep these chains short. If you need more than four or five steps, there's usually a simpler squeeze you've missed.
Practise the ideas
- Techniques 1–4 solve every easy and medium puzzle.
- Hard puzzles need technique 5.
- Expert puzzles may need technique 6.