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?

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.

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

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

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

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

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Delight your friends with this cunning trick! Can you explain how it works?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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.

It would be nice to have a strategy for disentangling any tangled ropes...

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Can you find a way of counting the spheres in these arrangements?

How many moves does it take to swap over some red and blue frogs? Do you have a method?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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

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

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

Can you find the values at the vertices when you know the values on the edges?

Got It game for an adult and child. How can you play so that you know you will always win?

Surprise your friends with this magic square trick.

Watch this animation. What do you see? Can you explain why this happens?

Are these statements always true, sometimes true or never true?

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

It starts quite simple but great opportunities for number discoveries and patterns!