Nicomachus’ Theorem: How Squares and Cubes Meet
In 100 C.E., Nicomachus of Gerasa observed that
1³ + 2³ + 3³ + … + n³ = (1 + 2 + 3 + … + n)²
Or “the sum of the cubes of 1 to n is the same as the square of their sum.”
Counting the individual squares shows that
1 × 1² + 2 × 2² + 3 × 3² + 4 × 4² + 5 × 5² + 6 × 6² =
1³ + 2³ + 3³ + 4³ + 5³ + 6³ = (1 + 2 + 3 + 4 + 5 + 6)²
(The even-numbered squares had to be cut in half at the borders of the diagram.)
credits:
http://www.futilitycloset.com/2014/12/30/nicomachus-theorem/http://en.wikipedia.org/wiki/Nicomachus - Nicomachus (Greek c. 60 – c. 120 CE) was
an important mathematician in the ancient world and is best known for his works
Introduction to Arithmetic and Manual of Harmonics in Greek. He was born in Gerasa,
in the Roman province of Syria (now Jerash, Jordan), and was strongly influenced by Aristotle.
He was a Neopythagorean, who wrote about the mystical properties of numbers.