Explaining, convincing and proving

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Here is a Sudoku with a difference! Use information about lowest common multiples to help you solve it.

You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

What is the same and what is different about these circle questions? What connections can you make?

Explore the relationships between different paper sizes.

What do you get when you raise a quadratic to the power of a quadratic?

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?