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 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?

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

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

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

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

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

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

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

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

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?

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

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

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

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?

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?

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

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

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

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

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

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

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.

Delight your friends with this cunning trick! Can you explain how it works?

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.

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

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

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

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

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

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

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

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?

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?

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

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?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

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.

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

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?