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

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?

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

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?

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

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

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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 find a way of counting the spheres in these arrangements?

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

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

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.

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

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

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.

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

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

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

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?

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

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

Find out what a "fault-free" rectangle is and try to make some of your own.

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?

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

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

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 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?

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

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 asks you to imagine a snake coiling on itself.

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

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

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

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?

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?

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?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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?

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

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

Here are two kinds of spirals for you to explore. What do you notice?

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.

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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

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

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