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.

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

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?

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

A square patio was tiled with square tiles all the same size. Some of the tiles were removed from the middle of the patio in order to make a square flower bed, but the number of the remaining tiles. . . .

How many four digit square numbers are composed of even numerals? What four digit square numbers can be reversed and become the square of another number?

A woman was born in a year that was a square number, lived a square number of years and died in a year that was also a square number. When was she born?

Using your knowledge of the properties of numbers, can you fill all the squares on the board?

What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A