Can someone explain this “visual proof” of $(1 + 2 + 3 +…+ n)^2 = 1^3 + 2^3 + 3^3 +…+ n^3$?

Question

I see a lot of people saying this visual proof is beautiful and I really want to be able to understand it. If anyone could help me out, I'd really appreciate it! Thanks in advance!

Answer 1

The lower left figure (red and gray) has $(1+2+3+...+n)$ little cubes. (In the diagram, $n=6$.)

Going across the bottom row from left to right, we see that the figures are $1$ thick, $2$ thick, $...$, $n$ thick.

So this is $(1+2+3+...+n)$ little cubes, repeated $(1+2+3+...+n)$ times, which is $(1+2+3+...+n)^2$.

The top row is a bit easier to visualize: $1^3 + 2^3 + 3^3 + ... + n^3$.

There is the same number of each color of cube in the top and bottom row.

For $n=1$ just the gray cube is used: one gray cube on the bottom; one gray cube on the top.

For $n=2$ look at the top two layers of cubes in the two leftmost figures in the bottom row (gray and red cubes). Seven gray cubes and two red ones are reconfigured to make the $1\times 1 \times 1$ and $2\times 2 \times 2$ cubes in the top row.

For $n=3$ look at the top three layers of cubes in the three leftmost figures in the bottom row ($25$ gray, $5$ red, and $6$ yellow cubes).

And so forth.