Ponder This Challenge - November 2024 - Tetrahedron volumes

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 $cV^2=128$.

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 $cV^2=128$. Provide your solution as the list of edge lengths.

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

Solution

• Click here to view the solution

  November 2024 Solution:

  The solutions are [177, 245, 137, 280, 217, 69]

  and [345, 252, 245, 94, 102, 9]

  A common way to solve the puzzle is through the expansion of the Cayley-Menger determinant

  $288 V^2 = \begin{vmatrix} 0 & 1 & 1 & 1 & 1\\ 1 & 0 & d_{12}^2 & d_{13}^2 & d_{14}^2\\ 1 & d_{21}^2 & 0 & d_{23}^2 & d_{24}^2\\ 1 & d_{31}^2 & d_{32}^2 & 0 & d_{34}^2\\ 1 & d_{41}^2 & d_{42}^2 & d_{43}^2 & 0\\ \end{vmatrix}$

  Many solvers had problems due to the limits of floating point precision computations. An interesting example arises for the "solution" [1,x,x,x-1,x-1,2] which gives the determinant - 32 for every value of x, but when using numpy's determinat method yields 255.99999991524982 for x = 524283. It was better to avoid numpy's implementation and use the [Bareiss algorithm] which allows one to use only integer operations, or even compute the determinant naively using recursion on minors.

Solvers

• *Lazar Ilic (30/10/2024 6:59 PM IDT)
• *King Pig (31/10/2024 5:45 AM IDT)
• *Bertram Felgenhauer (31/10/2024 1:11 PM IDT)
• *Jenna Talia (1/11/2024 3:36 AM IDT)
• *Serge Batalov (1/11/2024 4:10 AM IDT)
• *Arthur Vause (2/11/2024 12:14 AM IDT)
• *Nolan Jiang (2/11/2024 1:32 AM IDT)
• *Yi Jiang (4/11/2024 2:56 PM IDT)
• *Dieter Beckerle (5/11/2024 10:44 AM IDT)
• *Joaquim Carrapa (5/11/2024 1:00 PM IDT)
• *Yan-Wu He (5/11/2024 4:25 PM IDT)
• *Adrian Neacsu (6/11/2024 9:18 AM IDT)
• *Richard Gosiorovsky (6/11/2024 8:47 PM IDT)
• *Sri Mallikarjun J (8/11/2024 3:33 PM IDT)
• *Marco Bellocchi (10/11/2024 7:50 PM IDT)
• *Radu-Alexandru Todor (10/11/2024 9:38 PM IDT)
• Michael Liepelt (10/11/2024 11:38 PM IDT)
• *John Spurgeon & Kipp Johnson (11/11/2024 5:16 AM IDT)
• *Sanandan Swaminathan (13/11/2024 1:31 AM IDT)
• *Daniel Bitin (13/11/2024 6:12 PM IDT)
• *Andreas Stiller (13/11/2024 7:27 PM IDT)
• *John Tromp (13/11/2024 9:46 PM IDT)
• Gary M. Gerken (15/11/2024 1:35 PM IDT)
• *Alper Halbutogullari (16/11/2024 3:37 AM IDT)
• *Robin Guilliou (17/11/2024 4:25 PM IDT)
• Victor Chang (18/11/2024 4:35 AM IDT)
• *Dan Dima (20/11/2024 8:41 AM IDT)
• *Latchezar Christov (20/11/2024 3:17 PM IDT)
• *David F.H. Dunkley (21/11/2024 1:48 PM IDT)
• *Harald Bögeholz (21/11/2024 7:48 PM IDT)
• *David Greer (24/11/2024 11:45 AM IDT)
• *Michael Vahle (25/11/2024 1:52 PM IDT)
• *Tim Walters (26/11/2024 3:18 AM IDT)
• *Daniel Chong Jyh Tar (26/11/2024 4:20 PM IDT)
• Stéphane Higueret (27/11/2024 11:55 AM IDT)
• *Karl-Heinz Hofmann (28/11/2024 3:22 AM IDT)
• Lorenz Reichel (29/11/2024 8:29 AM IDT)
• *Kang Jin Cho (30/11/2024 7:55 AM IDT)
• Chris Shannon (30/11/2024 5:16 PM IDT)
• Paul Lupascu (1/12/2024 10:26 AM IDT)
• *Stephen Ebert & Matthew Lee (2/12/2024 7:20 AM IDT)
• *Evan Semet (2/12/2024 5:37 PM IDT)
• *Blaine Hill (2/12/2024 9:28 PM IDT)
• *Julian Ma (3/12/2024 11:05 AM IDT)
• *Li Li (4/12/2024 7:22 PM IDT)
• Matt Cristina (5/12/2024 6:59 AM IDT)
• *BC & E. Lam (6/12/2024 5:05 PM IDT)