Queens solving techniques
Every puzzle on this site is checked by a solver that only uses the 5 techniques below, easiest first. A puzzle’s difficulty is simply the hardest technique it needs. Hints use the same names: each hint tells you which technique it used, why it works, and links back here.
In the examples: the outlined area is what the reasoning is about. A green queen or red ✕ is what it proves. A gray ✕ was already ruled out.
1. Last Cell
A row, column or region has only one open cell left, so the queen goes there.
If a row, column or region has only one cell that isn’t ruled out, the queen goes there. This includes one-cell regions at the start of a puzzle.
After placing any queen, its row, its column, its region and the eight cells around it are ruled out (auto-mark shows these as faded ✕s). That often leaves another area with one cell, and the solve snowballs.
Worked example (5×5 puzzle #2): The green region has only one open cell left, so its queen must go there.
2. Region Squeeze
A region fits inside one row or column (or a line inside one region), so the rest of it is out.
Sometimes all the open cells of a region sit in one row (or one column). The region needs a queen, and it can only go in that row, so that row’s queen belongs to the region. Every other cell in the row, outside the region, is ruled out.
It works the other way round too. If all the open cells of a row sit inside one region, that row’s queen is also that region’s queen, so the rest of the region can be ruled out.
Worked example (5×5 puzzle #1): Every open cell of the gray region sits in row 5, so row 5’s queen has to be the gray region’s, and no other cell in row 5 can hold a queen.
3. No Room
A queen on a cell would wipe out every open cell of some area, so that cell is out.
Pick an open cell and imagine a queen on it. It would rule out its row, its column, its region and its neighbors. If that would leave some other row, column or region with no open cells, the queen can’t go there, so rule the cell out.
The common case is a small region. If a region’s last two or three open cells are close together, any cell touching all of them is out. So is any cell that shares a row or column with all of them.
Worked example (5×5 puzzle #9): A queen on the marked cell would rule out every open cell of column 1, leaving it no place for its queen.
4. Pair Squeeze
Two regions fit inside two rows or columns (or the reverse), so the rest of those lines is out.
This is a Region Squeeze for two regions at once. If two regions only have open cells in the same two rows, those two rows’ queens must be those two regions’ queens. Every other cell in those two rows is ruled out.
The same works with columns, and the other way round (two rows whose open cells all sit inside two regions). It’s the key move on Hard puzzles. Look for regions that are long and thin, or that sit side by side across the same band of rows.
Worked example (5×5 puzzle #11): Rows 1 and 2 only have open cells in the purple and orange regions, so those two regions’ queens must be in rows 1 and 2, and their other cells can’t hold a queen.
5. Triple Squeeze
Three regions fit inside three rows or columns (or the reverse), so the rest of those lines is out.
The same idea with three: three regions whose open cells fit inside three rows (or columns) use up all three of those queens, so every other cell in those lines is ruled out. It also works the other way round.
Triples are what make a puzzle Expert. When nothing else works, count: pick a band of three rows and check whether exactly three regions live entirely inside it.
Worked example (10×10 puzzle #29): Rows 8, 9 and 10 only have open cells in the red, pink and tan regions, so those three regions’ queens must be in rows 8, 9 and 10, and their other cells can’t hold a queen.
Difficulty levels
| Technique | First needed at |
|---|---|
| Last Cell | Easy |
| Region Squeeze | Easy |
| No Room | Medium |
| Pair Squeeze | Hard |
| Triple Squeeze | Expert |
General tips
- Start with the smallest regions. They have the fewest options.
- Mark as you go. Auto-mark covers the cells each queen rules out. Add your own ✕s for what you deduce, and drag to mark many cells at once.
- Look at the edges. Regions squeezed against the border are often confined to one row or column.
- Count. If three regions fit inside three rows, those rows are full up.
- Use hints to learn. Each hint names its technique (Region Squeeze, No Room, …), outlines the area that causes it, and explains why the cells are out.
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