A Visual Proof of Nicomachus's Theorem

step

2

mesh

Nicomachus's theorem states that

3

1

+

3

2

+…+

3

n

=

2

(1+2+…+n)

, where

n

is a positive integer. In words, the sum of the cubes from 1 to

n

is equal to the square of the sum from 1 to

n

.

For a visual proof, calculate the total area in the figure in two different ways: First, count the unit squares from the center to an edge to get

1+2+3+...+n

, so that the total area is

4

2

(1+2+...+n)

. Second, consider that each square ring consists of

4k

squares of side

k

, with area

4

3

k

.