### Iff

Prove that if n is a triangular number then 8n+1 is a square number. Prove, conversely, that if 8n+1 is a square number then n is a triangular number.

### Weekly Problem 10 - 2007

The square of a number is 12 more than the number itself. The cube of the number is 9 times the number. What is the number?

### Weekly Problem 32 - 2007

One of these numbers is the largest of nine consecutive positive integers whose sum is a perfect square. Which one is it?

Can you predict where $29^2$ will appear? Or $34^2$? Or $52^2$?
By representing an odd number as $2n+1$, can you explain any of the patterns in the odd square numbers?