Scaling Spheres

Let's call $a,b,c$ and $d$ the radii of the spheres with $100>a>b>c>d$ .

The weight of a sphere with homogeneously distributed mass with radius $r$ will be proportional to $r^3$ and hence we will have $a^3+d^3 = b^3+c^3$ (we cannot neither have $a^3+b^3 = c^3+d^3$ since $a>c$ and $b>d$ nor $a^3+c^3 = b^3+d^3$ since $a>b$ and $c>d$ .)

Numbers that can be written as a sum of cubes in more than a way are called taxicab numbers . We can compute those (cf. python code at the end) for which all terms are $<100^3$ and in the output, we can see that only the integer $24$ can appear as the smallest term or the second smallest or the second largest or the largest term. So only if one of the radii is $r=24$ can we guessed it with absolute certainty.

$\color{#D61F06}24\color{#333333}^3+2^3=18^3+20^3 \\ 27^3+10^3=19^3+\color{#D61F06}24\color{#333333}^3 \\ 55^3+17^3={\color{#D61F06}24}^3+54^3 \\ 80^3+{\color{#D61F06}24}^3=62^3+66^3 \\ 98^3+{\color{#D61F06}24}^3=63^3+89^3$

We can also see that only when $24$ is the smallest term can we have two different solutions so the answer is that $r=24$ is the radius of $\boxed{\text{ the smallest sphere }}$

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from collections import defaultdict
from itertools import product
from pprint import pprint as pp

cube2n = {x**3:x for x in range(1, 99)}
sum2cubes = defaultdict(set)
for c1, c2 in product(cube2n, cube2n):
    if c1 >= c2: sum2cubes[c1 + c2].add((cube2n[c1], cube2n[c2]))

taxied = sorted((k, v) for k,v in sum2cubes.items() if len(v) >= 2)

for t in enumerate(taxied[:100], 1):
    pp(t)