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I was trying to solve a code challenge but could not wrap my head around the math.

Your task is to construct a building which will be a pile of n cubes. The cube at the bottom will have a volume of $n^3$, the cube above will have volume of $(n-1)^3$ and so on until the top which will have a volume of $1^3$.

You are given the total volume $m$ of the building. Being given $m$ can you find the number $n$ of cubes you will have to build?

So the equation for the volume of a stack is $n^3 + (n-1)^3 + ... + 1^3 = m$.

How do I find the value of $n$?

Community
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domi91c
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2 Answers2

4

As explained and proven here, the sum of the first $n$ cubes is $1^3+2^3+\dots n^3=\left( n(n+1)/2 \right)^2$.

Now given:$$m=\left( n(n+1)/2 \right)^2$$ you need to find $n$. So first take the square root to get: $$2m^{1/2}=n^2+n$$ rearrange to $$n^2+n-2m^{1/2}=0$$ Now solve this quadratic and you want the positive root

Conrad Turner
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if it is a programming problem, then what is asked is

function find_n_from(m) {
   var n = 0;
   var volume = 0;
   while(volume < m) {
      n++;
      volume += n*n*n;
   }
   if (volume != m) "problem..";
   return n;
}
reuns
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    Of course if it is a programming problem it is off topic (and trivial, as you demonstrate, and it is pretty elementary even as a maths problem). – Conrad Turner May 21 '16 at 08:23
  • @ConradTurner : what is not off topic is telling to the mathematicians that there are not "only the maths" :) – reuns May 21 '16 at 08:24
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    Would you like to try that last comment again in English? – Conrad Turner May 21 '16 at 08:26
  • @ConradTurner : I can't do much better sry – reuns May 21 '16 at 08:27
  • @ConradTurner : how would you say it ? there are many interesting things beyond the maths ? – reuns May 21 '16 at 08:29
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    Thanks, that makes it clear, but does that not support the suggestion that it is off topic? It is not as though maths:SE is the only forum here, or that many of the posters here do not hang out at some of those. Also, completely separate from the rest of this comment: It might be obvious to us what programming language or languages you code is in but should you not say what it is? – Conrad Turner May 21 '16 at 08:44