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On the lateral edge of the pyramid, two points are taken, dividing the edge into three equal parts. Planes are drawn through these points, parallel to the base. Find the volume of the part of the pyramid enclosed between these planes, if the volume of the entire pyramid is 1.
Let points $A_1$ and $A_2$ lie on the lateral edge $AD$ of the triangular pyramid $ABCD$ with vertex $D$, such that $DA_1 = A_1A_2 = A_2A$. The plane passing through point $A_1$ and parallel to the plane $ABC$ intersects the lateral edges $BD$ and $BC$ at points $B_1$ and $C_1$, respectively, and the plane passing through point $A_2$ and parallel to the plane $ABC$ intersects the lateral edges $BD$ and $BC$ at points $B_2$ and $C_2$. Therefore, the pyramid $DA_1B_1C_1$ is similar to the given pyramid $DABC$ with a similarity ratio of $\frac{DA_1}{DA} = \frac{1}{3}$, so

$$
V_{DA_1B_1C_1} = \left(\frac{1}{3}\right)^3 \cdot V_{DABC} = \frac{1}{27}
$$

The pyramid $DA_2B_2C_2$ is similar to the given pyramid $DABC$ with a similarity ratio of $\frac{DA_2}{DA} = \frac{2}{3}$, so

$$
V_{DA_2B_2C_2} = \left(\frac{2}{3}\right)^3 \cdot V_{DABC} = \frac{8}{27}
$$

Therefore,

$$
V_{A_1B_1C_1A_2B_2C_2} = V_{DA_2B_2C_2} - V_{DA_1B_1C_1} = \frac{8}{27} - \frac{1}{27} = \frac{7}{27}
$$

Similarly for the case of an arbitrary $n$-sided pyramid.

## Answer

$\frac{7}{27}$.