It's impossible for a bird to be fixated on more than one bird. Show why.
This seems obviously true on its face: when a bird is fixated on another, it returns only that bird. Smullyan proves it by assuming that a bird A is fixated on both x and y, then showing that x and y are equal.
Take a third bird z. A(z) === x, because A is fixated on x. But A(z) === y, because A is fixated on y. Because both x and y are equal to A(z), they are equal to each other.