Squares & Cubes

Marvelously, the square of the sum of natural numbers is the sum of their cubes! Equivalently, the sum of their cubes is the square of their sum. This mathematical gem is attributed to Nicomachus of Gerasa who lived almost 2000 years ago.

For example,

(1+2+3)^2 = 36 = 1^3 +2^3 + 3^3.

More generally,

(1 + 2 + \ldots + n)^2 = 1^3 + 2^3 + \cdots n^3


\left( \sum n \right)^2 = \sum n^3.

The accompanying animation illustrates the identity, where the cubes can be rearranged into either a square or a sequence of composite cubes of the same total volume.

Square of the sum is the sum of the cubes

Square of the sum is the sum of the cubes

About John F. Lindner

John F. Lindner was born in Sleepy Hollow, New York, and educated at the University of Vermont and Caltech. He is an emeritus professor of physics and astronomy at The College of Wooster and a visiting professor at North Carolina State University. He has enjoyed multiple yearlong sabbaticals at Georgia Tech, University of Portland, University of Hawai'i, and NCSU. His research interests include nonlinear dynamics, celestial mechanics, and neural networks.
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