How it works
To show something is false, suppose it is true and reason forward. If you reach a contradiction, something both true and false, the supposition was wrong. This is proof by contradiction.
The contradiction does not have to be dramatic. “a is both left of b and right of b” is enough. So is “row 3 needs a queen and has no open squares”.
This is how most puzzle guesses should work. Try a queen in a square, follow the consequences, and if some row or region runs out of room, cross that square out. You did not guess. You proved the square is empty.
Worth memorizing
Suppose P. Reach a contradiction. Conclude ¬P.
This is proof by contradiction.
A contradiction is something that can’t be true in any world.
P ∧ ¬P, or a left of b and right of b at once.
A trial that breaks the rules is a proof.
In Queens or sudoku, a placement that leaves a row with no room is ruled out for good.
Practice
0 of 5 rightPremises: Small(a) → Cube(a). Small(a) → ¬Cube(a). Which conclusion follows?
You try a 3 in a sudoku cell and it forces two 8s into the same column. What have you shown?
Why can no islander say “I am a knave”?