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 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. . . .

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?

In how many different ways can I colour the five edges of a pentagon red, blue and green so that no two adjacent edges are the same colour?

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.

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

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

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

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

The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.

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?

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. . . .

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

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.

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.

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?

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

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?

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

Use the diagram to investigate the classical Pythagorean means.

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

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

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

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

Three frogs hopped onto the table. A red frog on the left a green in the middle and a blue frog on the right. Then frogs started jumping randomly over any adjacent frog. Is it possible for them to. . . .

Two angles ABC and PQR are floating in a box so that AB//PQ and BC//QR. Prove that the two angles are equal.

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. . . .

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

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.

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?

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?

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

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.

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

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 a rule which connects consecutive triangular numbers?

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.

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

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?

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 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?

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