Solution:

Laura should extend the line $A M$ beyond $M$. Measure $A M$ and find the point $P$ on the extension of $A M$ beyond $M$ such that $A M = M P$. Vertical angles $\angle C M P = \angle D M A$, $C M = M D$ and $A M = M P$ so $\triangle P M C$ is congruent to $\triangle A M D$ by $\mathrm{SAS}$.

Because of the triangle congruence, $\angle C P M = \angle D A M$. This means that the transversal $A P$ makes equal angles with $P C$ and $A D$ so $P C$ will be parallel to $A D$. The line $B C$ is another line through $C$ that is parallel to $A D$ so it is the same as line $P C$, so $P$ lies on the line containing $B, C$, and $N$.

Again, by the congruence of the triangles, $C P = A D$ and $A D = B C = B N$, so if we use the magic ruler to divide $P N$ into three equal parts, the division points must correspond to the missing points $B$ and $C$. By extending $C M$ and measuring off an additional length of $C M$ on the other side of $M$, Laura can construct the final missing point $D$.