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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

Robert made a spreadsheet with eight columns. He arranged the numbers from 1 to 1000 in his spreadsheet, and then coloured in all the square numbers.
Here is a picture showing the first few rows of Robert's spreadsheet:
Robert noticed some interesting patterns beginning to emerge.
Why not create your own copy of Robert's spreadsheet and see what patterns you notice?
Can you explain the patterns you find?
Will the patterns continue? How can you be sure?