Proof Writing & Problem Solving · Superchallenge problem

For a function $f$ from $\{1, 2, 3, 4\}$ to $\{1, 2, 3, 4\}$, consider the statement "for every $x$ there is a $y \ne x$ with $f(y) = f(x)$". For how many of the $4^4 = 256$ such functions is this statement true?