Exactly one: at least one, and at most one
∃x (Cube(x) ∧ ∀y (Cube(y) → y = x)) says there is one cube and no other. Nearly every puzzle rule has this shape.
How it works
∃x Cube(x) says there is at least one cube. Two cubes, or ten, still make it true. To say exactly one, add that every cube is that same piece: ∃x (Cube(x) ∧ ∀y (Cube(y) → y = x)).
Read it in two halves. The ∃x part is “at least one”: there is a witness. The ∀y part is “at most one”: any cube you find is the witness again, so a second cube is a counterexample.
Puzzle rules are exactly-one rules. Each sudoku row holds exactly one 7. Each Queens row, column and region holds exactly one queen. “At most one” rules squares out. “At least one” forces the last square in.
Worth memorizing
Exactly one = at least one + at most one.
At most one: ∀x ∀y ((P(x) ∧ P(y)) → x = y).
Any two P pieces are the same piece.
Every sudoku and Queens rule is an exactly-one rule.
At most one rules squares out. At least one forces the last one in.
Practice
0 of 6 right“There is exactly one tetrahedron.”
“There is at most one cube.”
A sudoku row must hold exactly one 6, and one cell in it is already a 6. What does the “at most one” half tell you?