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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Comments
Interesting and presented well. Thanks for sharing.
239 can also be expressed as 64+64+27+27+27+27+1+1+1, which uses the 64 that you warned against in the hint.
Right you are, Dewtell!
I could try to weasel out and claim that the clue just helps players find my given solution, but the truth is I was not aware of your solution. Well done!
Thanks for posting the alternate.
I could try to weasel out and claim that the clue just helps players find my given solution, but the truth is I was not aware of your solution. Well done!
Thanks for posting the alternate.
Thanks for sharing.
I managed to solve it with scip (a linear, mixed integer and nonlinear programming solver). The model file that I used is available at pastebin for anyone who might be interested to play with it.
https://pastebin.com/hMbRXEq2
I managed to solve it with scip (a linear, mixed integer and nonlinear programming solver). The model file that I used is available at pastebin for anyone who might be interested to play with it.
https://pastebin.com/hMbRXEq2
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