IBM Ponder This #319 · November 2024

This riddle was suggested by Hugo Pfoertner - thanks Hugo!

A tetrahedron is a polyhedron composed of four triangular faces, six straight edges, and four vertices. We can write the lengths of the edges as a list, e.g., [2,2,2,2,2,2].

The volume $V$ of a tetrahedron with edge lengths [2,2,2,2,2,2] is an irrational number. However, setting the constant $c=144$, we find $c{V}^2=128$.

In the following, we keep c=144; do not pass another value of c as part of your solution.

Your goal: Find a tetrahedron with integer edge lengths different than [2,2,2,2,2,2], which also has a volume $V$ satisfying $c{V}^2=128$. Provide your solution as the list of edge lengths.

A bonus "*" will be given for finding a tetrahedron with integer edge lengths satisfying $c{V}^2=655$.

Solution

Best opened after a real attempt

To be added.