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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?

Generating Triples

Stage: 4 Challenge Level: Challenge Level:1
Here is an extract from Charlie's table:

Where is the $5^2$?
Where is the $12^2$?
Where is the $13^2$?

Where might we find a similar set of related square numbers?