An Algebra Curiosity
By Dr. Mike Ecker

If you take the product of any three consecutive integers and then add the middle number, the result is always a perfect cube (of the middle number).

For example,
3*4*5 + 4 = 64 = 4³
4*5*6 + 5 = 125 = 5³

Why does this work?

Use some algebra to prove it:
(n - 1) n (n + 1) + n =
n(n² - 1) + n =
n³ - n + n = n³

Send any comments or questions to: David Pleacher