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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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.

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?

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

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?

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

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

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?

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

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?

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.

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.

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

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

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

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

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

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

This article for primary teachers discusses how we can help learners generalise and prove, using NRICH tasks as examples.

Are these statements relating to odd and even numbers always true, sometimes true or never true?

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?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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?