MathDB

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?

Problem Statement