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

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 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?

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

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

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

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.

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

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. . . .

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

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!

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

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 find a rule which relates triangular numbers to square numbers?

Can you find a rule which connects consecutive triangular numbers?

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

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.

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. . . .

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

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

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

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?

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface. . . .

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

A rectangular field has two posts with a ring on top of each post. There are two quarrelsome goats and plenty of ropes which you can tie to their collars. How can you secure them so they can't. . . .

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together?

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

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

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?

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

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. . . .

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. . . .

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.

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

A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?

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

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

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?

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

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?