TITLE: How many of these lines lie entirely in the interior of the original cube?
QUESTION [5 upvotes]: A portion of a wooden cube is sawed off at each vertex so that a small equilateral triangle is formed at each corner with vertices on the edges of the cube. The $24$ vertices of the new object are all connected to each other by straight lines. How many of these lines (with the exception, of course, of their end-points) lie entirely in the interior of the original cube?

REPLY [5 votes]: Alternatively, consider four-storey building with $8,4,4,8$ points labeled $1$ through $24$:
$\hspace{3cm}$
First floor: Number $1$ can connect with none on the first floor. $1$ can connect with $\{11,12\}$ on the second floor, with $\{15,16\}$ on the third floor and with $\{19,20,21,22,23,24\}$ on the fourth floor. So, $1$ can connect with $2+2+6=10$ points, so does each of $2-8$, hence: $8\cdot 10=\color{red}{80}$.
Second floor: Number $9$ can connect with none on the first floor. $9$ can connect with $\{11\}$ on the second floor, so does $10$ with $12$. Hence: $\color{red}{2}$ connections on the second floor only. $9$ can connect with $\{15\}$ on the third floor and with $\{19,20,21,22\}$ on the fourth floor. So, $9$ can connect with $1+4=5$ points on the third and fourth floors, so does each each of $\{10,11,12\}$, hence: $4\cdot 5=\color{red}{20}$. Overall, $\color{red}{22}$ connections.
Third floor: $13$ can connect with none on the first or second floor. $13$ can connect with $\{15\}$ on the third floor, so does $14$ with $16$, hence $\color{red}{2}$ connections on the third floor only. $13$ can connect with $\{19,20,21,22\}$ on the fourth floor. So, $13$ can connect with $4$ points on the fourth floor, so does each of $\{14,15,16\}$, hence: $4\cdot 4=\color{red}{16}$. Overall, $\color{red}{18}$ connections.
Fourth floor: $17$ can connect with none on the first, second, third or fourth floor. So does each other on the fourth floor.
Hence, the number of inner connections is: $\color{red}{80}+\color{red}{22}+\color{red}{18}=120$.