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.

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.

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

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

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

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 centimetres of rope will I need to make another mat just like the one I have here?

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?

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

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

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?

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

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

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

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

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

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

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?

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?

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

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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

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?

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'.

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?

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?

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

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?

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

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

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

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

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

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

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?

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

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

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

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

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

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.

Benâ€™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?

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?