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.

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?

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

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.

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

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

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

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.

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

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.

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

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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?

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

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

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

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

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

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

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

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

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

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

Can you find sets of sloping lines that enclose a square?

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

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.

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

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

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

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

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

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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

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

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

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?

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

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

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.

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

Can all unit fractions be written as the sum of two unit fractions?

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

Can you explain the strategy for winning this game with any target?

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.