Use the diagram to investigate the classical Pythagorean means.

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

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 make a tetrahedron whose faces all have the same perimeter?

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

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. . . .

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

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.

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

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

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

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

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

Can you find a rule which connects consecutive triangular numbers?

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?

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?

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.

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.

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?

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

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.

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.

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?

Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .

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?

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

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. . . .

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?

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

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

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?

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. . . .

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?

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. . . .

Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

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

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 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 ?

Some diagrammatic 'proofs' of algebraic identities and inequalities.

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?

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?

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

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

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?