prove that $1^3+2^3+\ldots+n^3= \left(\frac{n(n+1)}{2}\right)^2$ using geometrically

Question

For all integers $n\geq1$ prove the following identity using geometrically.

$$1^3+2^3+\ldots+n^3= \left(\frac{n(n+1)}{2}\right)^2$$

I can prove the statement using mathematical induction. I want to find a geometrically proof

Thanks!

Answer 1

The usual approach here is to slice the cubes into layers of n(n-1), except for the last, which is n².

So you put the 1.1.1 cube, and around it two rectangles 2.1.1, and the square 2.2.1. This makes a square whose sides are (1+2).

The 3-cube gives 3 slices of 3.3.1. Since the side of the square is now 3, we can place these squares to double the existing square.

The 4.4.4 cube is cut into a 4.4.1 layer, and the remainder is cut 4.3.1 four times. We can put two of these 4.3.1 so that the 3 side is against the existing square, and the 4 side enongens the edge, a pair of these will fill the inner edge, and the 4.4.1 array goes in the corner far from 1.

Repeat steps 3, 4 for odd and even numbers, as far as necessary. The shape formed will be a square, of the sum of numbers.