In the diagram, $\triangle ABE$, $\triangle BCE$ and $\triangle CDE$ are right-angled, with $\angle AEB=\angle BEC = \angle CED = 60^\circ$, and $AE=24$. [asy]
pair A, B, C, D, E;
A=(0,20.785);
B=(0,0);
C=(9,-5.196);
D=(13.5,-2.598);
E=(12,0);
draw(A--B--C--D--E--A);
draw(B--E);
draw(C--E);
label("A", A, N);
label("B", B, W);
label("C", C, SW);
label("D", D, dir(0));
label("E", E, NE);
[/asy] Find the length of $CE.$
We find $CE$ by first finding $BE$.

Since $AE = 24$ and $\angle AEB = 60^\circ$ and $AEB$ is a right triangle, then we can see that $AE$ is the hypotenuse and $BE$ is the shorter leg, so $BE = \dfrac{1}{2} \cdot 24 = 12.$

Likewise, since $BE = 12$ and $\angle BEC = 60^\circ$, then $CE = \dfrac{1}{2} \cdot 12 = \boxed{6}$.