Visualising and representing

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns around and heads back to the starting point where he meets the runner who is just finishing his first circuit. Find the ratio of their speeds.

Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?

Find all the ways to cut out a 'net' of six squares that can be folded into a cube.

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

Take a line segment of length 1. Remove the middle third. Remove the middle thirds of what you have left. Repeat infinitely many times, and you have the Cantor Set. Can you picture it?

A solid 'star' shape is created. How many faces does it have?

What 3D shapes occur in nature. How efficiently can you pack these shapes together?

Draw three straight lines to separate these shapes into four groups - each group must contain one of each shape.