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Odd Differences

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

Triangular Triples

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

Iff

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

Robert's Spreadsheet

Age 14 to 16 Challenge Level:

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:
 
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?