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?

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

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

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 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?

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

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

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.

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

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

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.

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

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?

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.

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

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.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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.

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

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.

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

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

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 see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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

These tasks give learners chance to generalise, which involves identifying an underlying structure.

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

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?

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

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.

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

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?

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

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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

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

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

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

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

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.

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.

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

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

This challenge asks you to imagine a snake coiling on itself.