Queens: ruling out by touching
Suppose a queen sits in a cell. If it would leave some region with no room, that is a contradiction. The cell is out.
How it works
A queen rules out its eight neighbors. So some cells are dangerous: a queen there would touch every open cell of a small region, and that region would have nowhere left for its queen.
Test a cell like this. Suppose a queen is there. Cross out what it rules out. If some row, column or region is left with no room, the supposition led to a contradiction. So there is no queen in that cell.
That is proof by contradiction. It is not a guess. You never place the queen; you only prove it can’t go there.
Where to test
Test cells next to small regions, or next to a region that has only two or three open cells left. A cell that touches all of them is out.
The same test works for rows and columns. A queen next to the last two open cells of a row would empty the row.
The shape of the argument
Suppose Q is in cell X. Then E has no open cell. But E must have a queen. Both can’t be true, so the supposition is false: no queen in X.
Worth memorizing
A queen touches all eight neighbors.
Corners count.
If a queen would empty a region, row or column, the cell is out.
You never place the test queen.
You only prove where queens can’t go.
Practice
0 of 4 rightA queen on which of these cells would leave region C with no room?
Suppose a queen sat on the ringed cell. What would run out of room?
Region F is two cells in row 6. Which cell is out because a queen there would touch both?
How many cells in row 4 does the touching test rule out on this empty board?