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?

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?

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?

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?

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

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. . . .

A challenge that requires you to apply your knowledge of the properties of numbers. Can you fill all the squares on the board?