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I was writing a blog post today, and I ended up asking the question of how many layers tall a human pyramid would be if it contained all of the people who use Facebook, approximately 750 million.

First I had to define how the pyramid would work. Basically, I ended up with $n$ being the number of layers from the top, and $n^2$ being the number of people in that layer.

So the top layer would be $n=1$ and would contain $n^2=1$ people. Next layer would be $n=2$ and would contain $n^2=4$ people. I ended up writing a simple python script to answer the question, but now I'm wondering about a more generalized answer.

Given $x$ people, how tall would the pyramid be? There's quite possibly a very simple answer to this, but I don't know what it would be.

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    Took it upon myself to retag as algebra-precalculus.2011-08-24

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As yunone has pointed out, there is a formula for the sum of squares, namely

$\sum_{k=1}^n k^2 = \frac{n(n + 1)(2n + 1)}{6} .$

So if the sum is about $x$ then $n$ is slightly less than $\sqrt[3]{3x}$, and for large $x$, $\sqrt[3]{3x}-\tfrac{1}{2}$ is a good estimate. With $x=750,000,000$, this suggests something about $1309.87$. Indeed the the sum of the first $1310$ squares is $750,221,935$.

You will also need to multiply by the average height of each layer (remembering that most human pyramids stand on shoulders rather than heads).

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    The $1/2$ comes from the fact that it is the only value $a$ such that the error between $n^3+\frac{3}{2}n^2+\frac{1}{2}n$ and $(n+a)^3$ is linear instead of quadratic in $n$.2011-08-23
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The sum of the squares of the first $n$ natural numbers is given by:

$x = \frac{n(n+1)(2n+1)}{6}$

Hence, you need to find $n$ in terms of $x$ using the following equation:

$ n(n+1)(2n+1) - 6x = 0$

Hence you need to find the real root of the cubic function

$2n^3 + 2n^2 + n^2 + n - 6x$

given by $n = $

enter image description here

Thus, for $x = 750,000,000$, we get $n = \lfloor 1309.9 \rfloor $ or $1309$ levels.

Using the average height of a male in the US (1.776m), that's a 2324.8m high pyramid made of $748,505,835$ people!

Here's the closed-form solution in pseudocode:

n = 1/(12*((3*x)/2 + ((9*x^2)/4 - 1/1728)^(1/2))^(1/3)) + ((3*x)/2 + ((9*x^2)/4 - 1/1728)^(1/2))^(1/3) - 1/2 
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    Haha true. That would be $\approx 35$ tons per person!2011-08-24