Can you describe this route to infinity? Where will the arrows take you next?

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?

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.

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

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

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

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

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. . . .

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 connects consecutive triangular numbers?

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. . . .

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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

On the 3D grid a strange (and deadly) animal is lurking. Using the tracking system can you locate this creature as quickly as possible?

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

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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. . . .

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?

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?

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

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

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?

Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?

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

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

Square It game for an adult and child. Can you come up with a way of always winning this game?

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 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. . . .

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 article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

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!

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. . . .

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.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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?

Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .

Can you mark 4 points on a flat surface so that there are only two different distances between them?

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

Here is a solitaire type environment for you to experiment with. Which targets can you reach?

Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

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