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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.

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Relative Powers

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?

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Nine Sum to Square

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?

Robert's Spreadsheet

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Can you predict where $29^2$ will appear? Or $34^2$? Or $52^2$?
Where are the odd square numbers?
Are there diagrams that could help you to show what happens when you square an odd number (or an even number)?
By representing an odd number as $2n+1$, can you explain any of the patterns in the odd square numbers?