Explaining, convincing and proving

Sarah's average speed for a journey was 2 mph, and her return average speed was 4 mph. What is her average speed for the whole journey?

The 80 spokes of The London Eye are made from 4 miles of cable. What is the approximate circumference of the wheel?

Peter wrote a list of all the numbers that can be formed by changing one digit of the number 200. How many of Peter's numbers are prime?

Take four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Four jewellers share their stock. Can you work out the relative values of their gems?

What can you say about the lengths of the sides of a quadrilateral whose vertices are on a unit circle?

Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. Does this always work? Can you prove or disprove this conjecture?