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Triangular Triples

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

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Iff

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

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Smith and Jones

Mr Smith and Mr Jones are two maths teachers. By asking questions, the answers to which may be right or wrong, Mr Jones is able to find the number of the house Mr Smith lives in... Or not!

Robert's Spreadsheet

Age 14 to 16 Challenge Level:

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?