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.

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

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?

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 spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

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. . . .

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

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?

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

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?

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

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

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

Use the diagram to investigate the classical Pythagorean means.

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?

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

This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .

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

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

How efficiently can various flat shapes be fitted together?

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?

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?

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!

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?

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.

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 cheap and simple toy with lots of mathematics. Can you interpret the images that are produced? Can you predict the pattern that will be produced using different wheels?

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.

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

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

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

Can you find a rule which connects consecutive triangular numbers?

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?

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