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

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

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. . . .

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?

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. . . .

To avoid losing think of another very well known game where the patterns of play are similar.

Takes you through the systematic way in which you can begin to solve a mixed up Cubic Net. How close will you come to a solution?

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?

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?

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?

A bicycle passes along a path and leaves some tracks. Is it possible to say which track was made by the front wheel and which by the back wheel?

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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?

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.

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

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

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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!

Imagine a rectangular tray lying flat on a table. Suppose that a plate lies on the tray and rolls around, in contact with the sides as it rolls. What can we say about the motion?

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.

A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?

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

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?

Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?

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.

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. . . .

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 connects consecutive triangular numbers?

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

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?

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

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?

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?

How efficiently can various flat shapes be fitted together?