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

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.

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.

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?

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

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 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. . . .

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?

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 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.

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

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

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

Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?

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 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. . . .

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?

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

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

Can you find a rule which connects consecutive triangular numbers?

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?

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?

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. . . .

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?

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

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

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

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

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 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!

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

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

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?

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. . . .

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

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?

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?

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.

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

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

A triangle PQR, right angled at P, slides on a horizontal floor with Q and R in contact with perpendicular walls. What is the locus of P?

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. . . .

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

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

Use the diagram to investigate the classical Pythagorean means.