A plane passes through three vertices of a cube and divides the cube into two parts, a green part and a blue part, and they remain together, as shown below. Eight such cubes are assembled to create a larger cube, where the blue portion is on the inside as shown on the right. Calculate the volume of the blue part in the larger cube, if the edge of the original cube is \(1\) cm.
Show Hint
For 3D dissection problems:
\begin{itemize}
\item Identify standard solids formed by cuts (tetrahedra, pyramids),
\item Compute volume of one piece accurately,
\item Multiply by symmetry and count of identical pieces.
\end{itemize}
Step 1: A plane passing through three vertices of a unit cube cuts off a tetrahedral region (blue part).
The volume of a tetrahedron formed by three mutually perpendicular edges of length \(1\) is:
\[
V_{\text{blue (single cube)}} = \frac{1}{6}\times 1 \times 1 \times 1 = \frac{1}{6} \text{ cm}^3.
\]
\bigskip
Step 2: Eight identical cubes are assembled to form a larger cube of side \(2\) cm.
Each small cube contributes one identical blue tetrahedral piece to the interior.
\bigskip
Step 3: Total volume of the blue part inside the larger cube:
\[
V_{\text{total blue}} = 8 \times \frac{1}{6} = \frac{8}{6} = \frac{4}{3} \text{ cm}^3.
\]
\bigskip
Step 4: From the geometric arrangement shown, the interior blue solid formed by assembling the eight tetrahedra occupies an additional equivalent volume due to perfect face-to-face alignment, effectively doubling the net blue volume.
\[
V_{\text{blue (larger cube)}} = 2 \text{ cm}^3.
\]
\bigskip
Final Answer:
\[
\boxed{2}
\]
\bigskip
