A puzzle by James M., an operations researcher at the National Security Agency:
Frosty the Snowman wants to create a small snowman friend for himself. The new snowman needs a base, torso, and a head, all three of which should be spheres. The torso should be no larger than the base and the head should be no larger than the torso.
For building material, Frosty has a spherical snowball with a 6inch radius. Since Frosty likes to keep things simple, he also wants the radius of each of the three pieces to be a positive integer. Can Frosty accomplish this?

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Yes! The formula for the volume of a sphere with radius r is . Let a, b and c be the radii of the base, torso, and head, respectively, which are integers greater than 0. The problem then amounts to Frosty solving the equation
By canceling out the factor of from both sides, we’re left with
As stated in the problem, a ≥ b ≥ c. The biggest that a can be is 5, so let’s try that first. Subtracting 5^{3} = 125 from both sides gives us
The biggest that b can be is 4, so let’s try that next. Subtracting 4^{3} = 64 from both sides gives us
Frosty is in luck, since 3^{3} = 27. Thus we have that 6^{3} = 5^{3} + 4^{3} + 3^{3}. Now Frosty can build his new snowman friend to his specifications. In fact, (5, 4, 3) is the only combination of numbers that will work.
From NSA’s Puzzle Periodical series.
