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?

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

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

Can you work out how to win this game of Nim? Does it matter if you go first or second?

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 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.

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.

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.

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.

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.

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

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.

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

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

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

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

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

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?

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

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 find a way of counting the spheres in these arrangements?

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?

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

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

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

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.

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

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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

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?

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

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

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.

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

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Surprise your friends with this magic square trick.

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

This task follows on from Build it Up and takes the ideas into three dimensions!

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

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

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

How many centimetres of rope will I need to make another mat just like the one I have here?