### Brain Teasers

# Waring It Well

All natural numbers can be written as the sum of cubes of other natural numbers.

For example:

37 = 27 + 8 + 1 + 1 = 3^3 + 2^3 + 1^3 + 1^3

All but two natural numbers can be written as the sum of eight or fewer cubes.

Two numbers require the sum of at least nine cubes.

a) One such number is 239. Write 239 as the sum of nine cubes.

b) What is the other natural number that requires a sum of at least nine cubes?

For example:

37 = 27 + 8 + 1 + 1 = 3^3 + 2^3 + 1^3 + 1^3

All but two natural numbers can be written as the sum of eight or fewer cubes.

Two numbers require the sum of at least nine cubes.

a) One such number is 239. Write 239 as the sum of nine cubes.

b) What is the other natural number that requires a sum of at least nine cubes?

### Hint

a) Don't use 4^3 = 64.b) It is fairly small. Start at 1 and test each number.

### Answer

a) 239 = 125 + 27 + 27 + 27 + 8 + 8 + 8 + 8 + 1b) 23 = 8 + 8 + 1 + 1 + 1 + 1 + 1 + 1 + 1

In 1770, Edward Waring asked if for every number k, there is a number s such that every natural number can be represented as the sum of at most s powers of k.

It was not until 1909 that a proof of Waring's problem was developed, the Hilbert-Waring Theorem.

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