Square numbers

problem
Robert's spreadsheet

Age

14 to 16

Challenge level

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?
problem
Filling the gaps

Age

14 to 16

Challenge level

Which numbers can we write as a sum of square numbers?
problem
Iff

Age

14 to 18

Challenge level

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

Age

16 to 18

Challenge level

Can you find some Pythagorean Triples where the two smaller numbers differ by 1?