The second in a series of articles on visualising and modelling shapes in the history of astronomy.

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.

Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

How efficiently can various flat shapes be fitted together?

A visualisation problem in which you search for vectors which sum to zero from a jumble of arrows. Will your eyes be quicker than algebra?

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

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

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?

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

Which of the following cubes can be made from these nets?

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

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the. . . .

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

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

Some treasure has been hidden in a three-dimensional grid! Can you work out a strategy to find it as efficiently as possible?

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.

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?

A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.

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.

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.

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. . . .

Can you find a rule which connects consecutive triangular numbers?

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. . . .

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?

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. . . .

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?

A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?

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?

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. . . .

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

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

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

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?

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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

Use the diagram to investigate the classical Pythagorean means.

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

Can you find the link between these beautiful circle patterns and Farey Sequences?

What can you see? What do you notice? What questions can you ask?

What's the largest volume of box you can make from a square of paper?

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?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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