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

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?

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 ...?

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.

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?

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

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

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

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

This challenge is about finding the difference between numbers which have the same tens digit.

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.

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

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?

Stop the Clock game for an adult and child. How can you make sure you always win 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 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?

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

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.

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

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 is a game for two players. Can you find out how to be the first to get to 12 o'clock?

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 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.

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

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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

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

Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?

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

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

This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

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

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

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?

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?

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

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

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

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?