Consider those planes that are parallel to the face planes of the tetrahedron $\mathcal{T}$, and the small tetrahedrons cut off from $\mathcal{T}$ by these planes have a volume that is $1 / 3$ of the volume of $\mathcal{T}$. What fraction of the volume of $\mathcal{T}$ remains if we cut off its vertices with these planes?
A tetrahedron and the tetrahedron cut by a plane parallel to one of its faces are centrally similar, with the center of similarity being the common vertex of the two tetrahedra. The corresponding sides of the tetrahedra are in the same ratio as the cube root of the ratio of their volumes, meaning the edges of the cut tetrahedron are $\sqrt[3]{\frac{1}{3}} \approx 0.69$ times the corresponding edges of the original tetrahedron. This is why the tetrahedra cut from $\mathcal{T}$ "overlap," intersecting each other.

Since the centroid $S$ of $\mathcal{T}$ divides the medians in the ratio $3:1$, and $\frac{3}{4} > \sqrt[3]{\frac{1}{3}}$, $S$ is not inside any of the cut tetrahedra. After the cuts, the remaining solid is entirely within the interior of $\mathcal{T}$, because $\sqrt[3]{\frac{1}{3}} > \frac{2}{3}$, meaning we have removed every point on the faces of $\mathcal{T}$ (Figure 1). Therefore, the remaining solid is bounded by 4 planes, making it a tetrahedron. From the similarity of the cut small tetrahedra and $\mathcal{T}$, it follows that the distance between the corresponding faces of the remaining tetrahedron and $\mathcal{T}$ is divided by $S$ in the ratio $\left(\frac{3}{4} - \sqrt[3]{\frac{1}{3}}\right) : \frac{1}{4}$ for all four pairs of faces (Figure 2).

Thus, the central similarity with center $S$ and ratio $\lambda = -\left(3 - 4 \sqrt[3]{\frac{1}{3}}\right)$ maps $\mathcal{T}$ to the remaining tetrahedron. The ratio of the volumes of similar solids is equal to the cube of the absolute value of the similarity ratio, so the volume of the remaining solid is $\left(3 - 4 \sqrt[3]{\frac{1}{3}}\right)^3 \approx 0.0116$ of the volume of $\mathcal{T}$.

Based on the work of Márton Hablicsek (Fazekas M. Főv. Gyak. Gimn., 9th grade)