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

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

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

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?

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

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?

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

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?

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?

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?

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

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?

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

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

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

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

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

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

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.

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

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

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

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

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

An investigation that gives you the opportunity to make and justify predictions.

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?

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

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

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

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

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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?

Find the sum of all three-digit numbers each of whose digits is odd.

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

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?

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?

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

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

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?