Visualising and representing

Can you arrange the shapes in a chain so that each one shares a face (or faces) that are the same shape as the one that follows it?

An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

Can you find a rule which relates triangular numbers to square numbers?

Which hexagons tessellate?

Ever thought of playing three dimensional Noughts and Crosses? This problem might help you visualise what's involved.

In this problem we see how many pieces we can cut a cube of cheese into using a limited number of slices. How many pieces will you be able to make?

A cube is made from smaller cubes, 5 by 5 by 5, then some of those cubes are removed. Can you make the specified shapes, and what is the most and least number of cubes required ?

Mary is driving to Birmingham Airport. Using her average speed for the entire journey, find how long her journey took.

Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?