Problem

Source: Bundeswettbewerb Mathematik 2024, Round 1 - Problem 4

Tags: combinatorics, combinatorics proposed



For positive integers $p$, $q$ and $r$ we are given $p \cdot q \cdot r$ unit cubes. We drill a hole along the space diagonal of each of these cubes and then tie them to a very thin thread of length $p \cdot q \cdot r \cdot \sqrt{3}$ like a string of pearls. We now want to construct a cuboid of side lengths $p$, $q$ and $r$ out of the cubes, without tearing the thread. a) For which numbers $p$, $q$ and $r$ is this possible? b) For which numbers $p$, $q$ and $r$ is this possible in a way such that both ends of the thread coincide?