The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.

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 ?

Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?

Have you got the Mach knack? Discover the mathematics behind exceeding the sound barrier.

A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.

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

How efficiently can various flat shapes be fitted together?

A game for 2 people. Take turns joining two dots, until your opponent is unable to move.

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.

In how many different ways can I colour the five edges of a pentagon red, blue and green so that no two adjacent edges are the same colour?

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

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. . . .

Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . .

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

This is a simple version of an ancient game played all over the world. It is also called Mancala. What tactics will increase your chances of winning?

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

Can you find a rule which connects consecutive triangular numbers?

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

Show that all pentagonal numbers are one third of a triangular number.

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . .

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?

A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.

On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!

The net of a cube is to be cut from a sheet of card 100 cm square. What is the maximum volume cube that can be made from a single piece of card?

Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .

I found these clocks in the Arts Centre at the University of Warwick intriguing - do they really need four clocks and what times would be ambiguous with only two or three of them?

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Can you make sense of the charts and diagrams that are created and used by sports competitors, trainers and statisticians?

Can you describe this route to infinity? Where will the arrows take you next?

Square It game for an adult and child. Can you come up with a way of always winning this game?

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

Use the diagram to investigate the classical Pythagorean means.

This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .

This article explores ths history of theories about the shape of our planet. It is the first in a series of articles looking at the significance of geometric shapes in the history of astronomy.

See if you can anticipate successive 'generations' of the two animals shown here.