Prove that if a is a natural number and the square root of a is
rational, then it is a square number (an integer n^2 for some
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.
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Robert noticed some interesting patterns when he highlighted square
numbers in a spreadsheet. Can you prove that the patterns will
Which numbers can we write as a sum of square numbers?
Discover a way to sum square numbers by building cuboids from small
cubes. Can you picture how the sequence will grow?