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

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?

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?

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. . . .

A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?

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. . . .

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. . . .

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

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

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

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?

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?

A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?

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

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

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

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?

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. . . .

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

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

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.

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?

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

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

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

Use the diagram to investigate the classical Pythagorean means.

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

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.

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.

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?

Can you find a rule which connects consecutive triangular numbers?

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

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. . . .

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?

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

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

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?

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

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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.

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

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.

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. . . .