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

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

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

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?

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.

A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .

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

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

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

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?

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.

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.

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

Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

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

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.

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

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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?

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?

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

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.

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

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

Mark a point P inside a closed curve. Is it always possible to find two points that lie on the curve, such that P is the mid point of the line joining these two points?

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?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

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

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

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 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?

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

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?

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

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.

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

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