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

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?

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

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?

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

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?

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

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

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

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

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

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

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?

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

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.

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

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

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?

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

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

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

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

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.

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

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

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.

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

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.

A game for 2 players with similaritlies 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.

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?

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

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?

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

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

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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

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

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

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?

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

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

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

What happens when you round these numbers to the nearest whole number?

What happens when you round these three-digit numbers to the nearest 100?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

This activity involves rounding four-digit numbers to the nearest thousand.