How efficiently can various flat shapes be fitted together?

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

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

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

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?

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 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.

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.

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

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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 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?

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

Small circles nestle under touching parent circles when they sit on the axis at neighbouring points in a Farey sequence.

Can you make a tetrahedron whose faces all have the same perimeter?

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?

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

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

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?

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

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?

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

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

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

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 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 ?

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?

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?

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?

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?

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?

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

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

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

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.

Consider a watch face which has identical hands and identical marks for the hours. It is opposite to a mirror. When is the time as read direct and in the mirror exactly the same between 6 and 7?

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.

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

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?

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!

Can you find a rule which connects consecutive triangular numbers?

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

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

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

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