Let $c$ be a complex number.  Suppose there exist distinct complex numbers $r$, $s$, and $t$ such that for every complex number $z$, we have
\[
  (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]	
Compute the number of distinct possible values of $c$.
1. We start with the given equation:
   \[
   (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct)
   \]
   for all complex numbers \( z \).

2. Since the polynomials are equal for all \( z \), their roots must be the same. This means that the sets \(\{r, s, t\}\) and \(\{cr, cs, ct\}\) must be permutations of each other.

3. First, consider the case where \( c = 1 \). Clearly, if \( c = 1 \), then \(\{cr, cs, ct\} = \{r, s, t\}\), so \( c = 1 \) is a solution.

4. Now, assume \( c \neq 1 \). We need to find other possible values of \( c \).

5. Consider the case where one of \( r, s, t \) is zero. Without loss of generality, let \( r = 0 \). Then the equation simplifies to:
   \[
   (z - 0)(z - s)(z - t) = (z - 0)(z - cs)(z - ct)
   \]
   This implies that \(\{s, t\}\) must be a permutation of \(\{cs, ct\}\). Since \( s \) and \( t \) are distinct, we must have \( cs = t \) and \( ct = s \). Solving these equations, we get:
   \[
   c^2 = 1 \implies c = \pm 1
   \]
   Since we already considered \( c = 1 \), we now have \( c = -1 \). This works for the general case where \( (r, s, t) = (0, -t, t) \).

6. Next, consider the case where none of \( r, s, t \) is zero. In this case, \( cr \neq r \), \( cs \neq s \), and \( ct \neq t \). Without loss of generality, assume \( cr = s \), \( cs = t \), and \( ct = r \). This implies:
   \[
   c^3 = 1
   \]
   The solutions to \( c^3 = 1 \) are the cube roots of unity, which are \( 1, \omega, \overline{\omega} \), where \( \omega = e^{2\pi i / 3} \) and \( \overline{\omega} = e^{-2\pi i / 3} \).

7. We have already considered \( c = 1 \). Therefore, the additional solutions are \( c = \omega \) and \( c = \overline{\omega} \).

8. Combining all the distinct possible values of \( c \), we have \( c = 1, -1, \omega, \overline{\omega} \).

Conclusion:
The number of distinct possible values of \( c \) is \( \boxed{4} \).