Given only the cuboid's space diagonal $S$, it it possible to calculate the volume of the cuboid? This question could be reframed like this: do two cuboids exist such that they have the same volume, but different space diagonals (or vice versa)?
To prove that it is possible to do that we have to prove that the following system of equations has no real solutions for $a,b,c ≠ d,e,f$:
$$a^2 + b^2 + c^2 = d^2 + e^2 + f^2$$ $$abc = def$$
Any ideas about this?
$$ 27 = 3^2+3^2+3^2 = 1^2 + 1^2 + 5^2 $$ and $$ 3 \cdot 3 \cdot 3 = 27 \neq 5 = 1 \cdot 1 \cdot 5. $$ On the other hand, $$ 2 \cdot 3 \cdot 6 = 36 = 2 \cdot 2 \cdot 9 $$ and $$ 2^2+3^2+6^2 = 49 \neq 89 = 2^2+2^2+9^2. $$