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

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?

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.

How efficiently can various flat shapes be fitted together?

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. . . .

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.

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

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

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

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. . . .

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

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

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

Use the diagram to investigate the classical Pythagorean means.

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

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

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

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

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. . . .

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 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.

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

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

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

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.

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

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

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

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

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?

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!

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.

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 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?

Can you find a rule which connects consecutive triangular numbers?

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?

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?

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?

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

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

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

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

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

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. . . .