$ABCDEF$ is a regular hexagon. Let $R$ be the overlap between $\vartriangle ACE$ and $\vartriangle BDF$. What is the area of $R$ divided by the area of $ABCDEF$?
1. **Understanding the Problem:**
   We are given a regular hexagon $ABCDEF$ and need to find the ratio of the area of the overlap $R$ between $\triangle ACE$ and $\triangle BDF$ to the area of the hexagon $ABCDEF$.

2. **Properties of Regular Hexagon:**
   A regular hexagon can be divided into 6 equilateral triangles. If the side length of the hexagon is $x$, then each of these equilateral triangles has a side length $x$.

3. **Area of Regular Hexagon:**
   The area of a regular hexagon with side length $x$ can be calculated as:
   \[
   \text{Area of hexagon} = 6 \times \text{Area of one equilateral triangle}
   \]
   The area of one equilateral triangle with side length $x$ is:
   \[
   \text{Area of one equilateral triangle} = \frac{\sqrt{3}}{4} x^2
   \]
   Therefore, the area of the hexagon is:
   \[
   \text{Area of hexagon} = 6 \times \frac{\sqrt{3}}{4} x^2 = \frac{3\sqrt{3}}{2} x^2
   \]

4. **Overlap Area $R$:**
   The overlap $R$ between $\triangle ACE$ and $\triangle BDF$ forms another regular hexagon inside the original hexagon. The vertices of this inner hexagon are the midpoints of the sides of the original hexagon.

5. **Side Length of Inner Hexagon:**
   The side length of the inner hexagon is the distance between the midpoints of two adjacent sides of the original hexagon. This distance is:
   \[
   \text{Side length of inner hexagon} = \frac{\sqrt{3}}{3} x
   \]

6. **Area of Inner Hexagon:**
   Using the same formula for the area of a regular hexagon, the area of the inner hexagon with side length $\frac{\sqrt{3}}{3} x$ is:
   \[
   \text{Area of inner hexagon} = 6 \times \frac{\sqrt{3}}{4} \left(\frac{\sqrt{3}}{3} x\right)^2 = 6 \times \frac{\sqrt{3}}{4} \times \frac{1}{3} x^2 = 6 \times \frac{\sqrt{3}}{12} x^2 = \frac{\sqrt{3}}{2} x^2
   \]

7. **Ratio of Areas:**
   The ratio of the area of the inner hexagon to the area of the original hexagon is:
   \[
   \text{Ratio} = \frac{\text{Area of inner hexagon}}{\text{Area of original hexagon}} = \frac{\frac{\sqrt{3}}{2} x^2}{\frac{3\sqrt{3}}{2} x^2} = \frac{\frac{\sqrt{3}}{2}}{\frac{3\sqrt{3}}{2}} = \frac{1}{3}
   \]

Conclusion:
\[
\boxed{\frac{1}{3}}
\]