Brain Teasers
Sum of Cubes
According to Waring’s theorem, any positive integer can be represented as the sum of nine
or fewer perfect cubes (not necessarily distinct).
For
instance, 89 can be represented as the sum of four perfect cubes: 27 + 27 + 27
+ 8 = 89.
Can
you express 239 as a sum of nine or fewer perfect cubes?
Solution: 64 +
64 + 27
+ 27 +
27 + 27
+ 1 +
1 + 1.
As you may have guessed, the number 239 wasn’t chosen at
random. It has the unusual property of being the largest number that cannot be
represented with fewer than nine
cubes (23 is the only other number requiring nine).
The only numbers that can be used to add to 239 are 1, 8,
27, 64, 125, and 216 — the cubes of the first six integers, respectively,
each of which is less than 239. But the confounding thing about 239 is
that the obvious candidates don’t work. Starting with 216 or 125 goes
nowhere — using these with eight more cubes will get you to 238 but not to
239.
The representation we’re looking for starts with 64 and goes
like this:
239 = 64 +
64 + 27
+ 27 +
27 + 27
+ 1 +
1 + 1
= 43 + 43
+ 33 + 33 + 33 + 33
+
13 + 13 + 13
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